Research ArticleThe Origin of Chern-Simons Modified Gravity froman 11 + 3-Dimensional Manifold
J A Helayeumll-Neto1 Alireza Sepehri2 and Tooraj Ghaffary3
1Centro Brasileiro de Pesquisas Fisicas Rua Dr Xavier Sigaud 150 Urca RJ 22290-180 Brazil2Research Institute for Astronomy and Astrophysics of Maragha (RIAAM) PO Box 55134-441 Maragha Iran3Department of Science Shiraz Branch Islamic Azad University Shiraz Iran
Correspondence should be addressed to J A Helayel-Neto helayelcbpfbr
Received 14 April 2017 Accepted 25 September 2017 Published 20 December 2017
Academic Editor Piero Nicolini
Copyright copy 2017 J A Helayel-Neto et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited The publication of this article was funded by SCOAP3
It is our aim to show that the Chern-Simons terms of modified gravity can be understood as generated by the addition of a 3-dimensional algebraic manifold to an initial 11-dimensional space-time manifold this builds up an 11 + 3-dimensional space-timeIn this system firstly some fields living in the bulk join the fields that live on the 11-dimensional manifold so that the rank ofthe gauge fields exceeds the dimension of the algebra consequently there emerges an anomaly To solve this problem another11-dimensional manifold is included in the 11 + 3-dimensional space-time and it interacts with the initial manifold by exchangingChern-Simon fields This mechanism is able to remove the anomaly Chern-Simons terms actually produce an extra manifold inthe pair of 11-dimensional manifolds of the 11 + 3-space-time Summing up the topology of both the 11-dimensional manifolds andthe topology of the exchanged Chern-Simons manifold in the bulk we conclude that the total topology shrinks to one which is inagreement with the main idea of the Big Bang theory
1 Introduction
Some authors have recently extended general relativity andproposed a Chern-Simons modified gravity in which theEinstein-Hilbert action is supplemented by a parity-violatingChern-Simons term which couples to gravity via a scalarfield The parity-violating Chern-Simons term is defined asa contraction of the Riemann curvature tensor with its dualand the Chern-Simons scalar field [1] Ever since a great dealof contributions and discussions on this particular model hasappeared in the literature For example the authors of [2] havestudied the combined effects of the Lorentz-symmetry violat-ing Chern-Simons and Ricci-Cotton actions for the Einstein-Hilbert model in the second-order formalism extended bythe inclusion of higher-derivative terms and considered theirconsequences on the spectrum In another investigation theauthors have argued about rotating black hole solutions inthe (3 + 1)-dimensional Chern-Simons modified gravity bytaking account of perturbations around the Schwarzschildsolution [3]They have obtained the zenith-angle dependence
of a metric function that corresponds to the frame-draggingeffect by using a constraint equation without choosingthe embedding coordinate system Also a conserved andsymmetric energy-momentum (pseudo)tensor for Chern-Simons modified gravity has been built up and it has beenshown that the model is Lorentz invariant [4] In anotherarticle the authors have considered the effect of Chern-Simons modified gravity on the quantum phase shift of deBroglie waves in neutron interferometry by applying a unifiedapproach of optical-mechanical analogy in a semiclassicalmodel [5] In a different scenario the authors have assertedthe consistency of the Godel-type solutions within the four-dimensional Chern-Simons modified gravity with the non-dynamical Chern-Simons coefficient for various shapes ofscalar matter and electromagnetic fields [6] Finally in oneof the latest versions of the Chern-Simons gravity the Chern-Simons scalar fields are treated as dynamical fields possessingtheir own stress energy tensor and an evolution equationThisversion has been named Dynamical Chern-Simons ModifiedGravity (DCSMG) [7ndash9] Now a question arises on what this
HindawiAdvances in High Energy PhysicsVolume 2017 Article ID 6021419 13 pageshttpsdoiorg10115520176021419
2 Advances in High Energy Physics
tensor is andwhat would be the origin of these Chern-Simonsterms We shall here show that our universe is a part of an11-dimensional manifold which is connected with another 11-dimensional manifold by an extra 3-dimensional space The11-dimensional manifolds interact with one another via theexchange of Chern-Simons fields which move along the 3-dimensional manifold
Our model is in fact a generalization of Kaluza-Kleintheory to 11 + 3-dimensional space-time Until now manydiscussions have been done on this subject For examplein one paper some very interesting features of the large 119863expansion of a Kaluza-Klein theory in 4 + 119863 dimensionshave been considered This model exhibits a nontrivial large119863 scaling in particular it has been found that the four-dimensional effective cosmologicalΛ constant is of order 1119863[10] In other researches the properties of different types ofblack holes in119863-dimensional Kaluza-Klein theory have beenconsidered It is observed that by reducing dimensions tofour these black holes achieve the same properties of normalblack holes in 4-dimensional gravity [11ndash13]
The main reason for considering higher dimensionalworld is responding to some main questions and removingsome puzzles in field theory and cosmology For examplewhat is the reason for the emergence of the difference betweenfermions and bosons What is the origin of the emergenceof extra terms in generalized uncertainty principles in 4-dimensional field theoryWhat is the origin of the emergenceof very big energy at the Big Bang In 10-dimensional stringtheory some of these questions have had a response howeverthis theory had some anomalies In 1995 by generalizationsof the number of dimensions to 11 the anomalies in 10-dimensional string theory have been removed [14] Howeverthis theory contains two stable objects like 3-dimensional1198722-brane and 6-dimensional 1198725-brane and designing a4-dimensional universe is very hard Also this 119872-theoryincludes some other anomalies which can be removed intheories with more than 11 dimensions like 119866-theory in 14dimensions [15]
Recently a new theory has been proposed in 14-dimensional space-time that responds to many questionsremoves the anomalies in 11-dimensional 119872-theory andconsiders the evolution of universe from nothing to presentstage In this theory at the beginning two types of 1198660-branes one with positive energy and one with negativeenergy are created from nothing in fourteen dimensionsThen these branes are compacted on three circles via twodifferent ways (symmetrically and antisymmetrically) andtwo bosonic and fermionic parts of action for 1198720-branesare created By joining 1198720-branes supersymmetric 119872119901-branes are produced which include the equal number ofdegrees of freedom for fermions and bosons Our universeis built on one of 119872119901-branes and other 119872119901-brane and extraenergy play the role of bulk By dissolving extra energy whichis created by compacting actions of 119866119901-branes into ouruniverse the number of degrees of freedom on it and also itsscale factor increase and universe expands We test 119866-theorywith observations andfind that themagnitude of the slow-rollparameters and the tensor-to-scalar ratio in this model are alot smaller than one which are in agreement with predictions
of experimental data Finally we consider the origin of theextended theories of gravity in 119866-theory and show that thesetheories could be anomaly-free And finally one of the mainresults of an extension to 14 dimensions yields the predictedterms in the generalized uncertainty principle [16] Thistheory gives the exact form of GUP and explains the reasonfor the birth of extra terms and their growing in this principleIn this paper we will show that the physics of 11-dimensionalspacingmanifold + 3-dimensional algebraicmanifold is equalto the physics of 14-dimensional manifold This helps us tounderstand why 119872-theory with Lie-three algebra is a truetheory and solvemany problems in physics In fact119872-theorywith Lie-three algebra lives on 14 dimensional manifoldwhere a 3-dimensional part of it corresponds to Lie-3-algebraand the other 11-dimensional part is related to space-timeAlso we show that 119872-theory on 11-dimensional manifoldcould be the anomaly-free if a three-dimensional algebraicmanifold is added to it or its spacial time is increased to 14dimensions This 14-dimensional manifold can be broken totwo parallel 11-dimensional manifolds which are connectedby a three-dimensional Chern-Simons manifold
In our model there is a 14-dimensional manifold whichcan be divided into smaller parts Each of these parts can forma new smaller manifold In Horava-Witten mechanism mostof anomalies are removed on an 11-dimensional manifoldWe will show that there are more anomalies that may beremoved in a new systemwhich is constructed of two parallel11-dimensional manifolds which are connected by a three-dimensional manifold This system can be created in a 14-dimensional space-time In fact this system is similar to Bionin string theory Bion is a system which has been constructedof two branes which are connected by a wormhole Now ournew Bion has been constructed of two 11-dimensional man-ifolds which have been connected by a three-dimensionalmanifold This new Bion can be a part of a 14-dimensionalmanifold
In this model we will use a generalization of the conceptof Lie algebra to an 119899-array bracket This algebra gives usthis opportunity to produce all types of gauge fields by usingthe relation between brackets and derivatives with respect tostrings In 119872-theory Lie-three algebra has been used whichinclude 3-dimensional brackets (brackets with 3 arrays) Wewill show that there is a direct relation between dimensionsof brackets and dimensions of manifold and by increasingthe number of dimensions of manifold the number ofarrays (number of dimensions) of algebra is increased Toremove anomalies we have to increase number of dimensionsof manifolds Consequently the number of dimensions ofbrackets (number of arrays) should be increased
Maybe this question arises can all anomalies be removedin 10-dimensional superstring theory In this theory anoma-lies depend on the difference between numbers of degrees offreedoms of fermions and bosons If the number of degreesof freedoms of fermions is equal to the number of degrees offreedoms of bosons all anomalies can be removed HoweverHorava and Witten show that some anomalies appeared dueto axial fields and also interactions between fermions that canbe removed by extending dimensions to eleven [14 17] In thispaper it is shown that 11-dimensional theory also has some
Advances in High Energy Physics 3
anomalies that can be removed by extending dimensions to14 Also we will show that there is a direct relation betweennumber of dimensions and algebra and also by choosing asuitable algebra anomalies can be removed completely
Our paper is organized according to the following outlinein Section 2 we devote efforts to show that by adding up a3-dimensional manifold to eleven-dimensional gravity thereemerges a Chern-Simonsmodified gravity Next in Section 3we shall show that if the fields obey a special algebraChern-Simons modified gravity is shown to be anomaly-free However by increasing the rank of the fields otheranomalies show up In Section 4 we focus on the removalof the anomaly of this type of gravity in a system composedof two 11-dimensional spaces and a Chern-Simons manifoldthat connects them In the last section we cast a summaryand our final considerations
where 119877 is the curvature and 120601 is the Chern-Simons scalarfield
Now we are going to show that Chern-Simons modifiedgravity can be obtained from a supergravity which lives onan 11 + 3-dimensional manifold Actually we assume thatour four-dimensional universe is a part of an 11-dimensionalmanifold that interacts with the bulk in an 11+3-dimensionalspace-time by exchanging Chern-Simons fields For thisour departure point is the purely bosonic sector of eleven-dimensional supergravity and we show that by adding upa three-dimensional manifold Chern-Simons terms willappear
The bosonic piece of the action for a gravity which liveson an eleven-dimensional manifold is given by [14 17]
119878Bosonic-SUGRA= 11205812 int11988911119909radic119892(minus12119877 minus 148119866119868119869119870119871119866119868119869119870119871) + 119878119862119866119866
119878119862119866119866 = minus radic234561205812 int11987211 1198891111990912057611986811198682 sdotsdotsdot119868111198621198681119868211986831198661198684 sdotsdotsdot11986871198661198688 sdotsdotsdot11986811 (2)
where the curvature (119877) and 119866119868119869119870119871 and 119862119868111986821198683 given in termsof the gauge field119860 and its field-strength 119865 are cast in whatfollows [17]
119866119868119869119870119871 = minus 3radic2 12058121205822 120576 (11990911) (119865119868119869119865119870119871 minus 119877119868119869119877119870119871) + sdot sdot sdot
120575119862119860119861119862 = minus 12058126radic21205822 120575 (11990911) tr (120598119862119865119860119861 minus 120598119862119877119860119861)11986611119860119861119862 = (12059711119862119860119861119862 plusmn 23 permutations)
+ cyclic permutations of 119860 119861 119862119865119868119869 = 120597119868119860119869 minus 120597119869119860119868119877119868119869 = 120597119868Γ120573119869120573 minus 120597119869Γ120573119868120573 + Γ120572119869120573Γ120573119868120572 minus Γ120572119868120573Γ120573119869120572Γ119868119869119870 = 120597119868119892119869119870 + 120597119870119892119868119869 minus 120597119869119892119868119870119866119868119869 = 119877119868119869 minus 12119877119892119868119869
(3)
Here 120576(11990911) is 1 for 11990911 gt 0 and 1 for 11990911 lt 0 and120575(11990911) = 12059712057612059711990911 Both capitalized Latin (eg 119868 119869) and Greek(eg 120573) indices act on the same manifold and we have onlyexhibited the free indices 119868 119869 119870 and the dummy ones (120572 120573)120598119862 is used as a vector in direction of119862This helps the equationthat becomes balanced from the indices point of view Thegauge variation of the 119862119866119866-action gives the following result[17]
120575119878119862119866119866100381610038161003816100381611= minus radic234561205812 int11987211 1198891111990912057611986811198682 sdotsdotsdot119868111205751198621198681119868211986831198661198684 sdotsdotsdot11986871198661198688 sdotsdotsdot11986811asymp minus 12058141281205826 int11987210
5sum119899=1
(tr119865119899 minus tr119877119899 + tr (1198651198991198775minus119899)) (4)
where tr119865119899 = tr(119865[11986811198682 sdot sdot sdot 1198651198682119899minus11198682119899]) =12059811986811198682 sdotsdotsdot1198682119899minus11198682119899 tr(11986511986811198682 sdot sdot sdot 1198651198682119899minus11198682119899) and tr119877119899 =tr(119877[11986811198682 sdot sdot sdot 1198771198682119899minus11198682119899]) = 12059811986811198682sdotsdotsdot1198682119899minus11198682119899 tr(11987711986811198682 sdot sdot sdot 1198771198682119899minus11198682119899) Theseterms above cancel the anomaly of (119878Bosonic-SUGRA) ineleven-dimensional manifold [17]
Thus 119878119862119866119866 is necessary for the anomaly cancelation solet us now go on and try to find a good rationale for it Alsowe shall answer the question related to the origin of 119862119866119866terms in 11-dimensional supergravity We actually propose ascenario in which the 119862119866119866 terms appear in the supergravityaction in a way that we do not add them up by hand Tothis end we choose a unified shape for all fields by using theNambu-Poisson brackets and the properties of string fields(119883) We define [15 18 19]
minus 12120598119868119894119868119895119868119896119868119898120601119877119868119894119868119895119868119896119868119898 + sdot sdot sdot (6)
where 120601 is the Chern-Simons scalar field 119860119868 is the gaugefield Γ is related to the curvature (119877) and 119868 is a unit vectorin the direction of the coordinate which can be expanded interms of derivatives of metric In fact the origin of all matterfields and strings is the same and they are equal to the unit
vectors (119868119869 = 119868120598119869) in addition to some fields (120601 119860119868) whichappear as fluctuations of space The latter may emerge by theinteraction of stringswhich breaks the initial symmetric stateWithout string interactions we have a symmetry that couldbe explained by a unit vector or a matrix We can first saythat in the static state all strings are equal to a unit vector ora matrix and then these strings interact with one anotherso that the symmetry is broken and fields emerge Also120598119868119894119868119895119868119896119868119898 is an antisymmetric tensor that has been attached toantisymmetric curvature and makes a symmetric part Thistensor causes that different states of curvature be regardedMaybe this question arises is 119883 used only for strings in 26-dimensional string theory In fact this could be a sign forbosonic strings in any dimension and is not related to 26 or 10dimensions Using four-dimensional brackets instead of two-dimensional ones we obtain the shape of the 119866119866-terms insupergravity as functions of strings (119883)
The equation above helps us to extract the 119862119866119866 termsfrom the 119866119866-terms in supergravity To this end we must adda three-dimensional manifold (related to a Lie-three-algebra)to eleven-dimensional supergravity by using the properties ofstrings (119883) in Nambu-Poisson brackets [15]
119883119868119894 = 119910119868119894 + 119860119868119894 minus 120598119868119894119869Γ120572120572119869 minus 120598119868119894119869 infinsum119899=1
where the integration has been carried out over a three-dimensional manifold with coordinates (1199101198685 1199101198686 1199101198687) andconsequently the integration can be done by using thatint1199101198685+1199101198686+1199101198687
= int1198891199101198685 int1198891199101198686 int1198891199101198687) The result above showsthat by ignoring fluctuations of space which yield productionof fields the area of each three-dimensional manifold canshrink to one and the result of the integration over thatmanifold goes to oneWhenwe addonemanifold to the otherthe integration will be the product of an integration over eachmanifold for the coordinates of the addedmanifolds increase
the elements of integration By adding the three-dimensionalmanifold of (8) to the eleven-dimensional manifold of (7) weget
This equation presents three results we should commenton (1) 119862119866119866 terms may appear in the action of supergravityby adding a three-dimensional manifold related to theLie-three-algebra added to eleven-dimensional supergrav-ity (2) 11-dimensional manifold + three-Lie algebra = 14-dimensional supergravity (3)Theshape of the119862-terms is nowclear in terms of the string fields (119883119894)
radic119892(12120598119868119894119868119895119868119896119868119898120601119877119868119896119868119898119868119897119868119899119877119868119897119868119899119868119894119868119895 minus 120597119868119894120601120597119868119895120601)minus 12 int11987211+119872119873=3
In the equation above the first integration is in agreementwith previous predictions of Chern-Simons gravity in [7ndash9] and can be reduced to the four-dimensional Chern-Simonsmodified gravity of (1) Also the second integration isrelated to the interaction of gauge fields with Chern-Simonsfields Thus this model not only produces the Chern-Simonsmodified gravity but also exhibits some modifications to itStill these results show that our universe is a part of one-eleven-dimensional manifold which interacts with a bulk ina 14-dimensional space-time by exchanging Chern-Simonsscalars
3 Anomalies in Chern-SimonsModified Gravity
In this section we shall consider various anomalies whichmay be induced in Chern-Simonsmodified gravity Althoughwe expect that terms in the gauge variation of the Chern-Simons action remove the anomaly in eleven-dimensionalsupergravity we will observe that some extra anomalies areproduced by the Chern-Simons field It is our goal to showthat these anomalies depend on the algebra and thus bychoosing a suitable algebra in this model all anomalies canbe removed To obtain the anomalies of the Chern-Simonstheory we should reobtain the gauge variation of the 119862119866119866-action in (4) in terms of field-strengths and curvatures To
this end by using (8) and (9) we can work out the gaugevariation of 119862 [15]
119883119868119894 = 119910119868119894 + 119860119868119894 minus 120598119868119894119869Γ120572120572119869 minus 120598119868119894119869 infinsum119899=1
minus 119877119868119894119868119895 + 120597119868119894120601120597119868119895120601 minus 12120598119868119894119868119895119868119896119868119898120601119877119868119894119868119895119868119896119868119898 + sdot sdot sdot)
(11)
Using the equation above and (7) we get the gaugevariation of the 119862119866119866 action given in (9)
The first line of this equation removes the anomaly on the 11-dimensional manifold of (4) however the second and thirdlines show that extra anomalies can emerge due to the Chern-Simons fields We can show that if we choose a suitablealgebra for the 11-dimensional manifold all anomalies can beswept out We can extend our discussion to a119863-dimensionalmanifold with a Lie-119873-algebra In fact we wish to obtain amethod that makes all supergravities with arbitrary dimen-sion anomaly-free To this end wemake use of the propertiesof Nambu-Poisson brackets and strings (119883) in (6) to obtaina unified definition for different terms in supergravity andrewrite action (4) as follows
In the equation above we only used the Lie-two-algebra with two-dimensional bracket however it is notclear whether this algebra is true In fact for 119872-theory Lie-three-algebra with three-dimensional bracket [18 19] is moresuitable To obtain the exact form of the Lie algebra whichis suitable for119863-dimensional space-time we shall generalizethe dimension of space-time from eleven to119863 and the algebrafrom two to119873 anduse the followingNambu-Poisson brackets[19]
In this equation we have added a new manifold relatedto the algebra to the world manifold In fact we have to
6 Advances in High Energy Physics
regard both algebraic (119872119873) and space-time (119872119863) manifoldsto achieve the exact results For the 119873-dimensional algebrawe introduce the following fields
120575[11986911198692sdotsdotsdot119869119873] = 12057511986911198692sdotsdotsdot119869119873 minus 12057511986921198691sdotsdotsdot119869119873 + sdot sdot sdot (16)
Here 120575 is the generalized Kronecker delta With defi-nitions in (15) we can obtain the explicit form of the 119873-dimensional Nambu-Poisson brackets in terms of fields
(11986511986911 sdotsdotsdot1198691119873 minus 12059711986915 sdot sdot sdot 120597119869111987311987711986911 sdotsdotsdot11986914)times (11986511986921 sdotsdotsdot1198692119873 minus 12059711986925 sdot sdot sdot 120597119869211987311987711986921 sdotsdotsdot11986924) sdot sdot sdot (11986511986911198632sdotsdotsdot1198691198632119873 minus 12059711986911986325sdot sdot sdot 1205971198691198632119873 11987711986911986321 sdotsdotsdot11986911986324 )
(18)
where 119885 is a constant related to the algebra This equationshows that the gauge variation of the action depends onthe rank-119873 field-strength The action above is not actuallydirectly zero and there emerges an anomaly Now we useproperties of 120598 and rewrite (18) as below
119882(119863119873) = ([(119863 + 2) (119863 minus 2)8 minus 119873(1198632 minus 1)]sdot [ (119863 + 2) (119863 minus 2)8 minus 1 minus 119873(1198632 minus 1)] sdot sdot sdot 1)sdot ([119873(1198632 minus 1)] [119873(1198632 minus 1) minus 1] sdot sdot sdot 1)119880 (120575)
(20)
where119880 is a function of the generalized Kronecker delta Onthe other hand 120575119878119862119866119866|119863+1 has been added to the main actionof supergravity to remove its anomaly Thus we can write
119882(119863119873) = ([(119863 + 2) (119863 minus 2)8 minus 119873(1198632 minus 1)]sdot [ (119863 + 2) (119863 minus 2)8 minus 1 minus 119873(1198632 minus 1)] sdot sdot sdot 1)times ([119873(1198632 minus 1)] [119873(1198632 minus 1) minus 1] sdot sdot sdot 1)119880 = 0 997904rArr
119873 le (119863 + 2) (119863 minus 2)8 (1198632 minus 1)
(21)
This equation indicates that for a (119863+1)-dimensional space-time the dimension of the Lie algebra should be equal to orless than a critical value Under these conditions the Chern-Simons gravity is free from anomalies and we do not needan extra manifold On the other hand as we show in (15)the dimension of the algebra determines the dimension ofthe field-strengthThis means that for a Lie-119873-algebra field-strengths should have at most 119873 indices For example for a
Advances in High Energy Physics 7
manifoldwith 11 dimensions the algebra can be of order threeas predicted in recent papers [18 19] and field-strengths mayhave three indices In fact in above equation we have shownthat the physics of an 11-dimensional spacing manifold plusa 3-dimensional algebraic manifold is equal to the physicsof 14-dimensional manifold This helps us to understandwhy 119872-theory with Lie-three algebra is a true theory andsolves many problems in physics In fact119872-theory with Lie-three algebra lives on 14-dimensional manifold where a 3-dimensional part of it corresponds to Lie-3-algebra and theother 11-dimensional part is related to space-time Also weshow that 119872-theory on 11-dimensional manifold could bethe anomaly-free if a three-dimensional algebraic manifoldis added to it or its spacial time is increased to 14 dimensionsThis 14-dimensional manifold can be broken to two parallel11-dimensional manifold which are connected by a three-dimensional Chern-Simons manifold
4 A Chern-Simons Manifold between Two11-Dimensional Manifolds in an 11 + 3Dimensional Space-Time
In the previous section we have found that for an eleven-dimensional manifold the suitable algebra which removesthe anomaly in Chern-Simons gravity is a three-dimensionalLie algebra This means that the rank of the fields can beof order two or three However (2) shows that the rank ofthe fields may be higher than three in eleven-dimensionalsupergravity Thus in Chern-Simons gravity theory whichlives on an eleven-dimensional manifold some extra anoma-lies are expected to show up To remove them we assumethat there is another 11-dimensional manifold in the 14-dimensional space-time which interacts with the first oneby exchanging Chern-Simons fields These fields producea Chern-Simons manifold that connects these two eleven-dimensional manifolds (see Figure 1)Thus in this model wehave two 119866119866 terms which live on 11-dimensional manifolds(see (2)) and two 119862119866119866 terms in the bulk so that each of theminteracts with one of the 11-dimensional manifolds
We can write the supergravity in 14-dimensional space-time as follows
119878SUGRA-14 = int119872119873=3
(int11987211
119866119866 + int11987211
119862119866119866+ int11987211
119862119866119866 + int11987211
119866119866) = (int11987214
119862119866119866+ int11987214
119862119866119866 + int11987214
119862119866119866 + int11987214
119862119866119866) (22)
In this equation119862119866119866 and119862119866119866 are related to the Chern-Simons gravities which live on the two eleven-dimensionalmanifolds and are extracted from 119866119866 and 119866119866 terms Also119862119866119866 and119862119866119866 correspond to the Chern-Simons fields whichare exchanged between the two manifolds in 14-dimensionalspace-time By generalizing the results of (3) (6) and (11) weget
119866119868119869119870119871 sim minus (119865119868119869119865119870119871 minus 119877119868119869119877119870119871) + 120597119868120601120597119869120601119877119870119871 sdot sdot sdot
119866119868119869119870119871 sim minus (119865119868119869119865119870119871 minus 119877119868119869119877119870119871) + 120597119868120601120597119869120601119877119870119871 sdot sdot sdot120575119862119868119894119868119895119868119896 sim minus120598119868119896 tr(119865119868119894119868119895 minus 119877119868119894119868119895 + 120597119868119894120601120597119868119895120601
minus 12120598119868119894119868119895119868119896119868119898119877119868119894119868119895119868119896119868119898 + sdot sdot sdot) (23)
Here the 119865rsquos 119877rsquos and 120601rsquos live on one of the supergravitymanifolds as depicted in Figure 1 whereas the 119865rsquos 119877rsquos and 120601rsquosare fields of the other supergravity manifold To obtain theirrelations we should make use of (12) and the gauge variationof the actions (22) in doing so we obtain
These results show that to remove the anomaly in 14-dimensional space-time coordinates and fields in one of theeleven-dimensionalmanifolds should be equal to coordinatesand fields in the othermanifold in addition to one extra 119894Thisimplies that time- or space-like coordinates and fields in onemanifold transmute into space- or time-like coordinates andfields of anothermanifold For example the zeroth coordinatewhich is known as time on one manifold will transmute intoa space coordinate of the other manifold If our universe withone time and three space coordinates is located on one of themanifolds an antiuniverse with one space and three times islocated in the other manifold
Now we shall show that by substituting the results of (24)into the action of (22) the topology of the 14-dimensionalmanifold tends to one This means that the world with allits matter began from a point and it may be thought of asa signature of Big Bang in our proposal To this end using(6) (7) (9) and (12) we rewrite 119862119866119866 terms in terms ofderivatives of scalar strings
On the other hand results in (24) show that (119883119868119894 119883119868119895 =minus119883119868119894 119883119868119895) Thus we can conclude that the action givenabove tends to an action on the three-dimensional manifold
119883119868119894 119883119868119895 = minus 119883119868119894 119883119868119895119878SUGRA-14 = 119896int
119872119873=3int11987211
(12057511 (119910)) = 119896int119872119873=3
997888rarr119896412058711987733
(28)
Advances in High Energy Physics 9
14-dimensionalspace-time
Chern-Simons manifold
11-dimensionalmanifold11-dimensional
manifold
Point
Figure 2 14-dimensional manifold shrinks to one point
where119877 is the radius ofmanifold By redefining scalar stringswe are able to show that the action of supergravity shrinks toone
This equation yields results that deserve our commentsIn fact two eleven-dimensional manifolds and the bulkinteract with each other via different types of 119862 119866 andChern-Simons-fields When we sum up supergravities thatlive on these manifolds and consider fields in the spacebetween them we get supergravity in 14-dimensional spaceBy canceling the anomaly in this new supergravity we canobtain the relations between fields By substituting theserelations into the action of the 14-dimensional supergravitywe simply obtain one This means that the 14-dimensionalmanifold with all its matter content can be topologicallyshrunk to one point (see Figure 2) In fact the system ofthe world began from this point and then expanded and
constructed a 14-dimensional world similar to what happensin a Big Bang theory
Another interesting result that comes out of our theoryconcerns the reduction of the action of the world to oneThismeans that the origin of allmatter or fields is the same In factat the first stage of the world we have only a constant energywhich lives on 14-dimensional manifold Then by internalinteractions two 11-dimensional manifolds are createdThesemanifolds interact with each other via an extraChern-Simonsmanifold By exchanging the energy between 11-dimensionalmanifolds their energies are changed and two differentactions are produced Also as due to interaction betweenstrings their shapes become different and different types offields like spinors and bosons are emerged
5 Summary and Final Considerations
In this paper we have shown that the Chern-Simons termsof modified gravity may be understood as due to the inter-action between two 11-dimensional manifolds in an 11 + 3-dimensional space-time where 3 is the dimension of a Lie-type-algebra We also argue that there is a direct relationbetween the dimension of the algebra and the dimensionof the manifold For example for 11-dimensional world thedimension of the Lie-algebra is three If the rank of the fieldswhich live in one manifold becomes larger than the rankof the algebra there emerges an anomaly This anomaly isproduced as an effect of connecting fields in the bulk tofields which live in the manifold To cancel this anomaly we
10 Advances in High Energy Physics
need to introduce another 11-dimensional manifold in the11+3-dimensional space-timewhich interacts with the initialmanifold by exchanging Chern-Simons terms These Chern-Simons terms produce an extra manifold If we sum up thetopology of the 11-dimensional manifolds and the topologyof the Chern-Simons manifold we can show that the totaltopology shrinks to one which is consistent with predictionsof the Big Bang theory
Our proposal is actually another version of the 119866-theory in [15 16] This approach removes anomalies in 11-dimensional 119872-theory by a generalization of dimension to14 In this research contribution we have shown that thephysics of an 11-dimensional spacing manifold plus a 3-dimensional algebraic manifold is the same as the physics of14-dimensional manifold In fact 11-dimensional 119872-theoryis a free-anomaly theory whenever a three-dimensionalalgebraic manifold is added to it This 14-dimensional man-ifold can be broken to two 11-dimensional manifolds whichare connected by a 3-dimensional Chern-Simons theoryAnother interesting result that comes out is the reduction ofthe action of the world to one This means that the originof all matter or fields is the same In fact at the first stageof the world we have only a constant energy which lives on14-dimensional manifold Then by internal interactions two11-dimensional manifolds come outThesemanifolds interactwith each other via one extra Chern-Simons manifold Byexchanging the energy between 11-dimensional manifoldstheir energies are changed and two different actions areproduced Also as due to interaction between strings theirshapes differ and diverse types of fields like spinors andbosons emerge
To conclude we would like to point out that that all ourtreatment has been restricted to the purely bosonic sectorof 11-dimensional supergravity whose on-shell multipletbesides the metric tensor and the 3-form gauge potentialincludes the gravitino fieldThe latter has not been consideredhere we have restricted ourselves to the bosonic sectorHowever it would be a further task to inspect how theinclusion of the gravitino would affect our developmentspossibly changing the dimension of the Lie algebra to beadded to the 11-dimensional manifold By including thefermion it is no longer ensured that the local supersymmetryof the 11 + 3-dimensional supergravity action remains validThe change in the dimension of the Lie algebra could in turngive rise to new terms so that the Chern-Simons modifiedgravity would be further extended as a result of including thegravitino We intend to pursue an investigation on this issueand to report on it elsewhere
In our model the three-Lie algebra that we add to passfrom 11 to 14119863 is geometrical It is shown that all anomalieswhich are produced by axial bosonic fields in 10- and 11-dimensional theories can be removed on 14-dimensionalmanifold Maybe a question arises ldquowhich is the effect offermionic anomalies on the number of dimensionsrdquo It mustbe clear that this paper is the first step for solving anomaliesin 11-dimensional theories In the next step by applyingthe model of this paper for fermions the exact number ofdimensions of the world can be estimated
Appendix
Details of Calculations
In this section we will bring details of calculations First letus disclose more aspects of (6)
1205981198681198941198681015840119894 (1205971198681015840119894119860119868119895 minus 1205971198681015840119894 (120598119868119895119868119896Γ120572120572119868119896) + 120597119868119894120601120597119868119895120601 + sdot sdot sdot)= 119865119868119894119868119895 minus 119877119868119894119868119895 + 120597119868119894120601120597119868119895120601 minus 12120598119868119894119868119895119868119896119868119898120601119877119868119894119868119895119868119896119868119898
+ sdot sdot sdot
(A1)
where 120601 is the Chern-Simons scalar field119860119868 is the gauge fieldΓ is related to the curvature (119877) and 119868 is a unit vector in thedirection of the coordinate which can be expanded in termsof derivatives of metric
Also it is needed to reobtain the results of (10) with doingsome mathematical calculations
In the equation above the first integration is in agreementwith previous predictions of Chern-Simons gravity in [7ndash9] and can be reduced to the four-dimensional Chern-Simonsmodified gravity of (1) Also the second integration isrelated to the interaction of gauge fields with Chern-Simonsfields Thus this model not only produces the Chern-Simonsmodified gravity but also exhibits some modifications to it
Now we can explain the mathematical model for obtain-ing results of (12)
The first line of this equation removes the anomaly on the 11-dimensional manifold of (4) however the second and thirdlines show that extra anomalies can emerge due to the Chern-Simons fields
At this stage we can consider the process of obtaining(24)
119878SUGRA-14 = 120575 (int11987214
119862119866119866 + int11987214
119862119866119866 + int11987214
119862119866119866+ int11987214
119862119866119866) = int11987214
5sum119899=1
(tr119865119899 minus tr119877119899 + tr (1198651198991198775minus119899))+ int11987214
5sum119899=1
119899sum119869=0
119899minus119869sum119898=0
(tr119865119869119865119898 minus tr119877119869119877119898 + tr (1198651198691198775minus1198691198651198981198775minus119898))+ int11987214
5sum119899=1
119899sum119869=0
119899minus119869sum119898=0
(tr119865119898119865119869 minus tr119877119898119877119869 + tr (1198651198981198775minus1198981198651198691198775minus119869))+ int11987214
5sum119899=1
(tr119865119899 minus tr119877119899 + tr (1198651198991198775minus119899))+ int11987210+119872119873=3
The authors declare that they have no conflicts of interest
Acknowledgments
The work of Alireza Sepehri has been supported financiallyby the Research Institute for Astronomy and Astrophysics ofMaragha (RIAAM) Iran under Research Project no 15237-75
References
[1] R Jackiw and S-Y Pi ldquoChern-Simons modification of generalrelativityrdquo Physical Review D Particles Fields Gravitation andCosmology vol 68 Article ID 104012 2003
[2] B Pereira-Dias C A Hernaski and J A Helayel-Neto ldquoProb-ing the effects of lorentz-symmetry violating chern-simons andricci-cotton terms in higher derivative gravityrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 83 no 82011
[3] K Konno T Matsuyama and S Tanda ldquoDoes a black holerotate in Chern-Simons modified gravityrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 76 no 2Article ID 024009 2007
[4] D Guarrera and A J Hariton ldquoPapapetrou energy-momentumtensor for Chern-Simons modified gravityrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 76 no 4 2007
Advances in High Energy Physics 13
[5] K K Nandi I R Kizirgulov O V Mikolaychuk N P Miko-laychuk and A A Potapov ldquoQuantum phase shift in Chern-Simons modified gravityrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 79 no 8 Article ID 0830062009
[6] P J Porfırio J B Fonseca-Neto J R Nascimento A Petrov JRicardo and A Santos ldquoChern-Simons modified gravity andclosed timelike curvesrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 94 no 4 2016
[7] S Alexander and N Yunes ldquoChernndashSimons modified generalrelativityrdquo Physics Reports vol 480 no 1-2 pp 1ndash55 2009
[8] S Chen and J Jing ldquoGeodetic precession and strong gravita-tional lensing in dynamical Chern-Simons-modified gravityrdquoClassical and QuantumGravity vol 27 no 22 225006 16 pages2010
[9] N Yunes and C F Sopuerta ldquoPerturbations of Schwarzschildblack holes in Chern-Simons modified gravityrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 77 no 62008
[10] F Canfora A Giacomini and A R Zerwekh ldquoKaluza-Kleintheory in the limit of large number of extra dimensionsrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 80 no 8 Article ID 084039 2009
[11] S Wu ldquoGeneral rotating charged Kaluza-Klein AdS black holesin higher dimensionsrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 83 no 12 2011
[12] T Houri and K Yamamoto ldquoKilling-Yano symmetry of Kaluza-KLEin black holes in five dimensionsrdquo Classical and QuantumGravity vol 30 no 7 075013 21 pages 2013
[13] T Tatsuoka H Ishihara M Kimura and K MatsunoldquoExtremal charged black holes with a twisted extra dimensionrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 85 no 4 2012
[14] P Horava and E Witten ldquoHeterotic and Type I string dynamicsfrom eleven dimensionsrdquo Nuclear Physics B vol 460 no 3Article ID 9510209 pp 506ndash524 1996
[15] A Sepehri and R Pincak ldquoThe birth of the universe in a newG-theory approachrdquo Modern Physics Letters A vol 32 no 5Article ID 1750033 1750033 32 pages 2017
[16] A Sepehri A Pradhan R Pincak F Rahaman A Beeshamand T Ghaffary ldquoBirth of the GUP and its effect on theentropy of the universe in Lie-N-algebrardquo International Journalof Geometric Methods in Modern Physics vol 14 no 9 ArticleID 1750130 2017
[17] P Horava and E Witten ldquoEleven-dimensional supergravity ona manifold with boundaryrdquo Nuclear Physics B vol 475 pp 94ndash114 1996
[18] J Bagger and N Lambert ldquoGauge symmetry and supersymme-try of multiple M2-branesrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 77 no 6 065008 6 pages 2008
[19] P Ho and Y Matsuo ldquoM5 from M2rdquo Journal of High EnergyPhysics vol 2008 no 06 p 105 2008
tensor is andwhat would be the origin of these Chern-Simonsterms We shall here show that our universe is a part of an11-dimensional manifold which is connected with another 11-dimensional manifold by an extra 3-dimensional space The11-dimensional manifolds interact with one another via theexchange of Chern-Simons fields which move along the 3-dimensional manifold
Our model is in fact a generalization of Kaluza-Kleintheory to 11 + 3-dimensional space-time Until now manydiscussions have been done on this subject For examplein one paper some very interesting features of the large 119863expansion of a Kaluza-Klein theory in 4 + 119863 dimensionshave been considered This model exhibits a nontrivial large119863 scaling in particular it has been found that the four-dimensional effective cosmologicalΛ constant is of order 1119863[10] In other researches the properties of different types ofblack holes in119863-dimensional Kaluza-Klein theory have beenconsidered It is observed that by reducing dimensions tofour these black holes achieve the same properties of normalblack holes in 4-dimensional gravity [11ndash13]
The main reason for considering higher dimensionalworld is responding to some main questions and removingsome puzzles in field theory and cosmology For examplewhat is the reason for the emergence of the difference betweenfermions and bosons What is the origin of the emergenceof extra terms in generalized uncertainty principles in 4-dimensional field theoryWhat is the origin of the emergenceof very big energy at the Big Bang In 10-dimensional stringtheory some of these questions have had a response howeverthis theory had some anomalies In 1995 by generalizationsof the number of dimensions to 11 the anomalies in 10-dimensional string theory have been removed [14] Howeverthis theory contains two stable objects like 3-dimensional1198722-brane and 6-dimensional 1198725-brane and designing a4-dimensional universe is very hard Also this 119872-theoryincludes some other anomalies which can be removed intheories with more than 11 dimensions like 119866-theory in 14dimensions [15]
Recently a new theory has been proposed in 14-dimensional space-time that responds to many questionsremoves the anomalies in 11-dimensional 119872-theory andconsiders the evolution of universe from nothing to presentstage In this theory at the beginning two types of 1198660-branes one with positive energy and one with negativeenergy are created from nothing in fourteen dimensionsThen these branes are compacted on three circles via twodifferent ways (symmetrically and antisymmetrically) andtwo bosonic and fermionic parts of action for 1198720-branesare created By joining 1198720-branes supersymmetric 119872119901-branes are produced which include the equal number ofdegrees of freedom for fermions and bosons Our universeis built on one of 119872119901-branes and other 119872119901-brane and extraenergy play the role of bulk By dissolving extra energy whichis created by compacting actions of 119866119901-branes into ouruniverse the number of degrees of freedom on it and also itsscale factor increase and universe expands We test 119866-theorywith observations andfind that themagnitude of the slow-rollparameters and the tensor-to-scalar ratio in this model are alot smaller than one which are in agreement with predictions
of experimental data Finally we consider the origin of theextended theories of gravity in 119866-theory and show that thesetheories could be anomaly-free And finally one of the mainresults of an extension to 14 dimensions yields the predictedterms in the generalized uncertainty principle [16] Thistheory gives the exact form of GUP and explains the reasonfor the birth of extra terms and their growing in this principleIn this paper we will show that the physics of 11-dimensionalspacingmanifold + 3-dimensional algebraicmanifold is equalto the physics of 14-dimensional manifold This helps us tounderstand why 119872-theory with Lie-three algebra is a truetheory and solvemany problems in physics In fact119872-theorywith Lie-three algebra lives on 14 dimensional manifoldwhere a 3-dimensional part of it corresponds to Lie-3-algebraand the other 11-dimensional part is related to space-timeAlso we show that 119872-theory on 11-dimensional manifoldcould be the anomaly-free if a three-dimensional algebraicmanifold is added to it or its spacial time is increased to 14dimensions This 14-dimensional manifold can be broken totwo parallel 11-dimensional manifolds which are connectedby a three-dimensional Chern-Simons manifold
In our model there is a 14-dimensional manifold whichcan be divided into smaller parts Each of these parts can forma new smaller manifold In Horava-Witten mechanism mostof anomalies are removed on an 11-dimensional manifoldWe will show that there are more anomalies that may beremoved in a new systemwhich is constructed of two parallel11-dimensional manifolds which are connected by a three-dimensional manifold This system can be created in a 14-dimensional space-time In fact this system is similar to Bionin string theory Bion is a system which has been constructedof two branes which are connected by a wormhole Now ournew Bion has been constructed of two 11-dimensional man-ifolds which have been connected by a three-dimensionalmanifold This new Bion can be a part of a 14-dimensionalmanifold
In this model we will use a generalization of the conceptof Lie algebra to an 119899-array bracket This algebra gives usthis opportunity to produce all types of gauge fields by usingthe relation between brackets and derivatives with respect tostrings In 119872-theory Lie-three algebra has been used whichinclude 3-dimensional brackets (brackets with 3 arrays) Wewill show that there is a direct relation between dimensionsof brackets and dimensions of manifold and by increasingthe number of dimensions of manifold the number ofarrays (number of dimensions) of algebra is increased Toremove anomalies we have to increase number of dimensionsof manifolds Consequently the number of dimensions ofbrackets (number of arrays) should be increased
Maybe this question arises can all anomalies be removedin 10-dimensional superstring theory In this theory anoma-lies depend on the difference between numbers of degrees offreedoms of fermions and bosons If the number of degreesof freedoms of fermions is equal to the number of degrees offreedoms of bosons all anomalies can be removed HoweverHorava and Witten show that some anomalies appeared dueto axial fields and also interactions between fermions that canbe removed by extending dimensions to eleven [14 17] In thispaper it is shown that 11-dimensional theory also has some
Advances in High Energy Physics 3
anomalies that can be removed by extending dimensions to14 Also we will show that there is a direct relation betweennumber of dimensions and algebra and also by choosing asuitable algebra anomalies can be removed completely
Our paper is organized according to the following outlinein Section 2 we devote efforts to show that by adding up a3-dimensional manifold to eleven-dimensional gravity thereemerges a Chern-Simonsmodified gravity Next in Section 3we shall show that if the fields obey a special algebraChern-Simons modified gravity is shown to be anomaly-free However by increasing the rank of the fields otheranomalies show up In Section 4 we focus on the removalof the anomaly of this type of gravity in a system composedof two 11-dimensional spaces and a Chern-Simons manifoldthat connects them In the last section we cast a summaryand our final considerations
where 119877 is the curvature and 120601 is the Chern-Simons scalarfield
Now we are going to show that Chern-Simons modifiedgravity can be obtained from a supergravity which lives onan 11 + 3-dimensional manifold Actually we assume thatour four-dimensional universe is a part of an 11-dimensionalmanifold that interacts with the bulk in an 11+3-dimensionalspace-time by exchanging Chern-Simons fields For thisour departure point is the purely bosonic sector of eleven-dimensional supergravity and we show that by adding upa three-dimensional manifold Chern-Simons terms willappear
The bosonic piece of the action for a gravity which liveson an eleven-dimensional manifold is given by [14 17]
119878Bosonic-SUGRA= 11205812 int11988911119909radic119892(minus12119877 minus 148119866119868119869119870119871119866119868119869119870119871) + 119878119862119866119866
119878119862119866119866 = minus radic234561205812 int11987211 1198891111990912057611986811198682 sdotsdotsdot119868111198621198681119868211986831198661198684 sdotsdotsdot11986871198661198688 sdotsdotsdot11986811 (2)
where the curvature (119877) and 119866119868119869119870119871 and 119862119868111986821198683 given in termsof the gauge field119860 and its field-strength 119865 are cast in whatfollows [17]
119866119868119869119870119871 = minus 3radic2 12058121205822 120576 (11990911) (119865119868119869119865119870119871 minus 119877119868119869119877119870119871) + sdot sdot sdot
120575119862119860119861119862 = minus 12058126radic21205822 120575 (11990911) tr (120598119862119865119860119861 minus 120598119862119877119860119861)11986611119860119861119862 = (12059711119862119860119861119862 plusmn 23 permutations)
+ cyclic permutations of 119860 119861 119862119865119868119869 = 120597119868119860119869 minus 120597119869119860119868119877119868119869 = 120597119868Γ120573119869120573 minus 120597119869Γ120573119868120573 + Γ120572119869120573Γ120573119868120572 minus Γ120572119868120573Γ120573119869120572Γ119868119869119870 = 120597119868119892119869119870 + 120597119870119892119868119869 minus 120597119869119892119868119870119866119868119869 = 119877119868119869 minus 12119877119892119868119869
(3)
Here 120576(11990911) is 1 for 11990911 gt 0 and 1 for 11990911 lt 0 and120575(11990911) = 12059712057612059711990911 Both capitalized Latin (eg 119868 119869) and Greek(eg 120573) indices act on the same manifold and we have onlyexhibited the free indices 119868 119869 119870 and the dummy ones (120572 120573)120598119862 is used as a vector in direction of119862This helps the equationthat becomes balanced from the indices point of view Thegauge variation of the 119862119866119866-action gives the following result[17]
120575119878119862119866119866100381610038161003816100381611= minus radic234561205812 int11987211 1198891111990912057611986811198682 sdotsdotsdot119868111205751198621198681119868211986831198661198684 sdotsdotsdot11986871198661198688 sdotsdotsdot11986811asymp minus 12058141281205826 int11987210
5sum119899=1
(tr119865119899 minus tr119877119899 + tr (1198651198991198775minus119899)) (4)
where tr119865119899 = tr(119865[11986811198682 sdot sdot sdot 1198651198682119899minus11198682119899]) =12059811986811198682 sdotsdotsdot1198682119899minus11198682119899 tr(11986511986811198682 sdot sdot sdot 1198651198682119899minus11198682119899) and tr119877119899 =tr(119877[11986811198682 sdot sdot sdot 1198771198682119899minus11198682119899]) = 12059811986811198682sdotsdotsdot1198682119899minus11198682119899 tr(11987711986811198682 sdot sdot sdot 1198771198682119899minus11198682119899) Theseterms above cancel the anomaly of (119878Bosonic-SUGRA) ineleven-dimensional manifold [17]
Thus 119878119862119866119866 is necessary for the anomaly cancelation solet us now go on and try to find a good rationale for it Alsowe shall answer the question related to the origin of 119862119866119866terms in 11-dimensional supergravity We actually propose ascenario in which the 119862119866119866 terms appear in the supergravityaction in a way that we do not add them up by hand Tothis end we choose a unified shape for all fields by using theNambu-Poisson brackets and the properties of string fields(119883) We define [15 18 19]
minus 12120598119868119894119868119895119868119896119868119898120601119877119868119894119868119895119868119896119868119898 + sdot sdot sdot (6)
where 120601 is the Chern-Simons scalar field 119860119868 is the gaugefield Γ is related to the curvature (119877) and 119868 is a unit vectorin the direction of the coordinate which can be expanded interms of derivatives of metric In fact the origin of all matterfields and strings is the same and they are equal to the unit
vectors (119868119869 = 119868120598119869) in addition to some fields (120601 119860119868) whichappear as fluctuations of space The latter may emerge by theinteraction of stringswhich breaks the initial symmetric stateWithout string interactions we have a symmetry that couldbe explained by a unit vector or a matrix We can first saythat in the static state all strings are equal to a unit vector ora matrix and then these strings interact with one anotherso that the symmetry is broken and fields emerge Also120598119868119894119868119895119868119896119868119898 is an antisymmetric tensor that has been attached toantisymmetric curvature and makes a symmetric part Thistensor causes that different states of curvature be regardedMaybe this question arises is 119883 used only for strings in 26-dimensional string theory In fact this could be a sign forbosonic strings in any dimension and is not related to 26 or 10dimensions Using four-dimensional brackets instead of two-dimensional ones we obtain the shape of the 119866119866-terms insupergravity as functions of strings (119883)
The equation above helps us to extract the 119862119866119866 termsfrom the 119866119866-terms in supergravity To this end we must adda three-dimensional manifold (related to a Lie-three-algebra)to eleven-dimensional supergravity by using the properties ofstrings (119883) in Nambu-Poisson brackets [15]
119883119868119894 = 119910119868119894 + 119860119868119894 minus 120598119868119894119869Γ120572120572119869 minus 120598119868119894119869 infinsum119899=1
where the integration has been carried out over a three-dimensional manifold with coordinates (1199101198685 1199101198686 1199101198687) andconsequently the integration can be done by using thatint1199101198685+1199101198686+1199101198687
= int1198891199101198685 int1198891199101198686 int1198891199101198687) The result above showsthat by ignoring fluctuations of space which yield productionof fields the area of each three-dimensional manifold canshrink to one and the result of the integration over thatmanifold goes to oneWhenwe addonemanifold to the otherthe integration will be the product of an integration over eachmanifold for the coordinates of the addedmanifolds increase
the elements of integration By adding the three-dimensionalmanifold of (8) to the eleven-dimensional manifold of (7) weget
This equation presents three results we should commenton (1) 119862119866119866 terms may appear in the action of supergravityby adding a three-dimensional manifold related to theLie-three-algebra added to eleven-dimensional supergrav-ity (2) 11-dimensional manifold + three-Lie algebra = 14-dimensional supergravity (3)Theshape of the119862-terms is nowclear in terms of the string fields (119883119894)
radic119892(12120598119868119894119868119895119868119896119868119898120601119877119868119896119868119898119868119897119868119899119877119868119897119868119899119868119894119868119895 minus 120597119868119894120601120597119868119895120601)minus 12 int11987211+119872119873=3
In the equation above the first integration is in agreementwith previous predictions of Chern-Simons gravity in [7ndash9] and can be reduced to the four-dimensional Chern-Simonsmodified gravity of (1) Also the second integration isrelated to the interaction of gauge fields with Chern-Simonsfields Thus this model not only produces the Chern-Simonsmodified gravity but also exhibits some modifications to itStill these results show that our universe is a part of one-eleven-dimensional manifold which interacts with a bulk ina 14-dimensional space-time by exchanging Chern-Simonsscalars
3 Anomalies in Chern-SimonsModified Gravity
In this section we shall consider various anomalies whichmay be induced in Chern-Simonsmodified gravity Althoughwe expect that terms in the gauge variation of the Chern-Simons action remove the anomaly in eleven-dimensionalsupergravity we will observe that some extra anomalies areproduced by the Chern-Simons field It is our goal to showthat these anomalies depend on the algebra and thus bychoosing a suitable algebra in this model all anomalies canbe removed To obtain the anomalies of the Chern-Simonstheory we should reobtain the gauge variation of the 119862119866119866-action in (4) in terms of field-strengths and curvatures To
this end by using (8) and (9) we can work out the gaugevariation of 119862 [15]
119883119868119894 = 119910119868119894 + 119860119868119894 minus 120598119868119894119869Γ120572120572119869 minus 120598119868119894119869 infinsum119899=1
minus 119877119868119894119868119895 + 120597119868119894120601120597119868119895120601 minus 12120598119868119894119868119895119868119896119868119898120601119877119868119894119868119895119868119896119868119898 + sdot sdot sdot)
(11)
Using the equation above and (7) we get the gaugevariation of the 119862119866119866 action given in (9)
The first line of this equation removes the anomaly on the 11-dimensional manifold of (4) however the second and thirdlines show that extra anomalies can emerge due to the Chern-Simons fields We can show that if we choose a suitablealgebra for the 11-dimensional manifold all anomalies can beswept out We can extend our discussion to a119863-dimensionalmanifold with a Lie-119873-algebra In fact we wish to obtain amethod that makes all supergravities with arbitrary dimen-sion anomaly-free To this end wemake use of the propertiesof Nambu-Poisson brackets and strings (119883) in (6) to obtaina unified definition for different terms in supergravity andrewrite action (4) as follows
In the equation above we only used the Lie-two-algebra with two-dimensional bracket however it is notclear whether this algebra is true In fact for 119872-theory Lie-three-algebra with three-dimensional bracket [18 19] is moresuitable To obtain the exact form of the Lie algebra whichis suitable for119863-dimensional space-time we shall generalizethe dimension of space-time from eleven to119863 and the algebrafrom two to119873 anduse the followingNambu-Poisson brackets[19]
In this equation we have added a new manifold relatedto the algebra to the world manifold In fact we have to
6 Advances in High Energy Physics
regard both algebraic (119872119873) and space-time (119872119863) manifoldsto achieve the exact results For the 119873-dimensional algebrawe introduce the following fields
120575[11986911198692sdotsdotsdot119869119873] = 12057511986911198692sdotsdotsdot119869119873 minus 12057511986921198691sdotsdotsdot119869119873 + sdot sdot sdot (16)
Here 120575 is the generalized Kronecker delta With defi-nitions in (15) we can obtain the explicit form of the 119873-dimensional Nambu-Poisson brackets in terms of fields
(11986511986911 sdotsdotsdot1198691119873 minus 12059711986915 sdot sdot sdot 120597119869111987311987711986911 sdotsdotsdot11986914)times (11986511986921 sdotsdotsdot1198692119873 minus 12059711986925 sdot sdot sdot 120597119869211987311987711986921 sdotsdotsdot11986924) sdot sdot sdot (11986511986911198632sdotsdotsdot1198691198632119873 minus 12059711986911986325sdot sdot sdot 1205971198691198632119873 11987711986911986321 sdotsdotsdot11986911986324 )
(18)
where 119885 is a constant related to the algebra This equationshows that the gauge variation of the action depends onthe rank-119873 field-strength The action above is not actuallydirectly zero and there emerges an anomaly Now we useproperties of 120598 and rewrite (18) as below
119882(119863119873) = ([(119863 + 2) (119863 minus 2)8 minus 119873(1198632 minus 1)]sdot [ (119863 + 2) (119863 minus 2)8 minus 1 minus 119873(1198632 minus 1)] sdot sdot sdot 1)sdot ([119873(1198632 minus 1)] [119873(1198632 minus 1) minus 1] sdot sdot sdot 1)119880 (120575)
(20)
where119880 is a function of the generalized Kronecker delta Onthe other hand 120575119878119862119866119866|119863+1 has been added to the main actionof supergravity to remove its anomaly Thus we can write
119882(119863119873) = ([(119863 + 2) (119863 minus 2)8 minus 119873(1198632 minus 1)]sdot [ (119863 + 2) (119863 minus 2)8 minus 1 minus 119873(1198632 minus 1)] sdot sdot sdot 1)times ([119873(1198632 minus 1)] [119873(1198632 minus 1) minus 1] sdot sdot sdot 1)119880 = 0 997904rArr
119873 le (119863 + 2) (119863 minus 2)8 (1198632 minus 1)
(21)
This equation indicates that for a (119863+1)-dimensional space-time the dimension of the Lie algebra should be equal to orless than a critical value Under these conditions the Chern-Simons gravity is free from anomalies and we do not needan extra manifold On the other hand as we show in (15)the dimension of the algebra determines the dimension ofthe field-strengthThis means that for a Lie-119873-algebra field-strengths should have at most 119873 indices For example for a
Advances in High Energy Physics 7
manifoldwith 11 dimensions the algebra can be of order threeas predicted in recent papers [18 19] and field-strengths mayhave three indices In fact in above equation we have shownthat the physics of an 11-dimensional spacing manifold plusa 3-dimensional algebraic manifold is equal to the physicsof 14-dimensional manifold This helps us to understandwhy 119872-theory with Lie-three algebra is a true theory andsolves many problems in physics In fact119872-theory with Lie-three algebra lives on 14-dimensional manifold where a 3-dimensional part of it corresponds to Lie-3-algebra and theother 11-dimensional part is related to space-time Also weshow that 119872-theory on 11-dimensional manifold could bethe anomaly-free if a three-dimensional algebraic manifoldis added to it or its spacial time is increased to 14 dimensionsThis 14-dimensional manifold can be broken to two parallel11-dimensional manifold which are connected by a three-dimensional Chern-Simons manifold
4 A Chern-Simons Manifold between Two11-Dimensional Manifolds in an 11 + 3Dimensional Space-Time
In the previous section we have found that for an eleven-dimensional manifold the suitable algebra which removesthe anomaly in Chern-Simons gravity is a three-dimensionalLie algebra This means that the rank of the fields can beof order two or three However (2) shows that the rank ofthe fields may be higher than three in eleven-dimensionalsupergravity Thus in Chern-Simons gravity theory whichlives on an eleven-dimensional manifold some extra anoma-lies are expected to show up To remove them we assumethat there is another 11-dimensional manifold in the 14-dimensional space-time which interacts with the first oneby exchanging Chern-Simons fields These fields producea Chern-Simons manifold that connects these two eleven-dimensional manifolds (see Figure 1)Thus in this model wehave two 119866119866 terms which live on 11-dimensional manifolds(see (2)) and two 119862119866119866 terms in the bulk so that each of theminteracts with one of the 11-dimensional manifolds
We can write the supergravity in 14-dimensional space-time as follows
119878SUGRA-14 = int119872119873=3
(int11987211
119866119866 + int11987211
119862119866119866+ int11987211
119862119866119866 + int11987211
119866119866) = (int11987214
119862119866119866+ int11987214
119862119866119866 + int11987214
119862119866119866 + int11987214
119862119866119866) (22)
In this equation119862119866119866 and119862119866119866 are related to the Chern-Simons gravities which live on the two eleven-dimensionalmanifolds and are extracted from 119866119866 and 119866119866 terms Also119862119866119866 and119862119866119866 correspond to the Chern-Simons fields whichare exchanged between the two manifolds in 14-dimensionalspace-time By generalizing the results of (3) (6) and (11) weget
119866119868119869119870119871 sim minus (119865119868119869119865119870119871 minus 119877119868119869119877119870119871) + 120597119868120601120597119869120601119877119870119871 sdot sdot sdot
119866119868119869119870119871 sim minus (119865119868119869119865119870119871 minus 119877119868119869119877119870119871) + 120597119868120601120597119869120601119877119870119871 sdot sdot sdot120575119862119868119894119868119895119868119896 sim minus120598119868119896 tr(119865119868119894119868119895 minus 119877119868119894119868119895 + 120597119868119894120601120597119868119895120601
minus 12120598119868119894119868119895119868119896119868119898119877119868119894119868119895119868119896119868119898 + sdot sdot sdot) (23)
Here the 119865rsquos 119877rsquos and 120601rsquos live on one of the supergravitymanifolds as depicted in Figure 1 whereas the 119865rsquos 119877rsquos and 120601rsquosare fields of the other supergravity manifold To obtain theirrelations we should make use of (12) and the gauge variationof the actions (22) in doing so we obtain
These results show that to remove the anomaly in 14-dimensional space-time coordinates and fields in one of theeleven-dimensionalmanifolds should be equal to coordinatesand fields in the othermanifold in addition to one extra 119894Thisimplies that time- or space-like coordinates and fields in onemanifold transmute into space- or time-like coordinates andfields of anothermanifold For example the zeroth coordinatewhich is known as time on one manifold will transmute intoa space coordinate of the other manifold If our universe withone time and three space coordinates is located on one of themanifolds an antiuniverse with one space and three times islocated in the other manifold
Now we shall show that by substituting the results of (24)into the action of (22) the topology of the 14-dimensionalmanifold tends to one This means that the world with allits matter began from a point and it may be thought of asa signature of Big Bang in our proposal To this end using(6) (7) (9) and (12) we rewrite 119862119866119866 terms in terms ofderivatives of scalar strings
On the other hand results in (24) show that (119883119868119894 119883119868119895 =minus119883119868119894 119883119868119895) Thus we can conclude that the action givenabove tends to an action on the three-dimensional manifold
119883119868119894 119883119868119895 = minus 119883119868119894 119883119868119895119878SUGRA-14 = 119896int
119872119873=3int11987211
(12057511 (119910)) = 119896int119872119873=3
997888rarr119896412058711987733
(28)
Advances in High Energy Physics 9
14-dimensionalspace-time
Chern-Simons manifold
11-dimensionalmanifold11-dimensional
manifold
Point
Figure 2 14-dimensional manifold shrinks to one point
where119877 is the radius ofmanifold By redefining scalar stringswe are able to show that the action of supergravity shrinks toone
This equation yields results that deserve our commentsIn fact two eleven-dimensional manifolds and the bulkinteract with each other via different types of 119862 119866 andChern-Simons-fields When we sum up supergravities thatlive on these manifolds and consider fields in the spacebetween them we get supergravity in 14-dimensional spaceBy canceling the anomaly in this new supergravity we canobtain the relations between fields By substituting theserelations into the action of the 14-dimensional supergravitywe simply obtain one This means that the 14-dimensionalmanifold with all its matter content can be topologicallyshrunk to one point (see Figure 2) In fact the system ofthe world began from this point and then expanded and
constructed a 14-dimensional world similar to what happensin a Big Bang theory
Another interesting result that comes out of our theoryconcerns the reduction of the action of the world to oneThismeans that the origin of allmatter or fields is the same In factat the first stage of the world we have only a constant energywhich lives on 14-dimensional manifold Then by internalinteractions two 11-dimensional manifolds are createdThesemanifolds interact with each other via an extraChern-Simonsmanifold By exchanging the energy between 11-dimensionalmanifolds their energies are changed and two differentactions are produced Also as due to interaction betweenstrings their shapes become different and different types offields like spinors and bosons are emerged
5 Summary and Final Considerations
In this paper we have shown that the Chern-Simons termsof modified gravity may be understood as due to the inter-action between two 11-dimensional manifolds in an 11 + 3-dimensional space-time where 3 is the dimension of a Lie-type-algebra We also argue that there is a direct relationbetween the dimension of the algebra and the dimensionof the manifold For example for 11-dimensional world thedimension of the Lie-algebra is three If the rank of the fieldswhich live in one manifold becomes larger than the rankof the algebra there emerges an anomaly This anomaly isproduced as an effect of connecting fields in the bulk tofields which live in the manifold To cancel this anomaly we
10 Advances in High Energy Physics
need to introduce another 11-dimensional manifold in the11+3-dimensional space-timewhich interacts with the initialmanifold by exchanging Chern-Simons terms These Chern-Simons terms produce an extra manifold If we sum up thetopology of the 11-dimensional manifolds and the topologyof the Chern-Simons manifold we can show that the totaltopology shrinks to one which is consistent with predictionsof the Big Bang theory
Our proposal is actually another version of the 119866-theory in [15 16] This approach removes anomalies in 11-dimensional 119872-theory by a generalization of dimension to14 In this research contribution we have shown that thephysics of an 11-dimensional spacing manifold plus a 3-dimensional algebraic manifold is the same as the physics of14-dimensional manifold In fact 11-dimensional 119872-theoryis a free-anomaly theory whenever a three-dimensionalalgebraic manifold is added to it This 14-dimensional man-ifold can be broken to two 11-dimensional manifolds whichare connected by a 3-dimensional Chern-Simons theoryAnother interesting result that comes out is the reduction ofthe action of the world to one This means that the originof all matter or fields is the same In fact at the first stageof the world we have only a constant energy which lives on14-dimensional manifold Then by internal interactions two11-dimensional manifolds come outThesemanifolds interactwith each other via one extra Chern-Simons manifold Byexchanging the energy between 11-dimensional manifoldstheir energies are changed and two different actions areproduced Also as due to interaction between strings theirshapes differ and diverse types of fields like spinors andbosons emerge
To conclude we would like to point out that that all ourtreatment has been restricted to the purely bosonic sectorof 11-dimensional supergravity whose on-shell multipletbesides the metric tensor and the 3-form gauge potentialincludes the gravitino fieldThe latter has not been consideredhere we have restricted ourselves to the bosonic sectorHowever it would be a further task to inspect how theinclusion of the gravitino would affect our developmentspossibly changing the dimension of the Lie algebra to beadded to the 11-dimensional manifold By including thefermion it is no longer ensured that the local supersymmetryof the 11 + 3-dimensional supergravity action remains validThe change in the dimension of the Lie algebra could in turngive rise to new terms so that the Chern-Simons modifiedgravity would be further extended as a result of including thegravitino We intend to pursue an investigation on this issueand to report on it elsewhere
In our model the three-Lie algebra that we add to passfrom 11 to 14119863 is geometrical It is shown that all anomalieswhich are produced by axial bosonic fields in 10- and 11-dimensional theories can be removed on 14-dimensionalmanifold Maybe a question arises ldquowhich is the effect offermionic anomalies on the number of dimensionsrdquo It mustbe clear that this paper is the first step for solving anomaliesin 11-dimensional theories In the next step by applyingthe model of this paper for fermions the exact number ofdimensions of the world can be estimated
Appendix
Details of Calculations
In this section we will bring details of calculations First letus disclose more aspects of (6)
1205981198681198941198681015840119894 (1205971198681015840119894119860119868119895 minus 1205971198681015840119894 (120598119868119895119868119896Γ120572120572119868119896) + 120597119868119894120601120597119868119895120601 + sdot sdot sdot)= 119865119868119894119868119895 minus 119877119868119894119868119895 + 120597119868119894120601120597119868119895120601 minus 12120598119868119894119868119895119868119896119868119898120601119877119868119894119868119895119868119896119868119898
+ sdot sdot sdot
(A1)
where 120601 is the Chern-Simons scalar field119860119868 is the gauge fieldΓ is related to the curvature (119877) and 119868 is a unit vector in thedirection of the coordinate which can be expanded in termsof derivatives of metric
Also it is needed to reobtain the results of (10) with doingsome mathematical calculations
In the equation above the first integration is in agreementwith previous predictions of Chern-Simons gravity in [7ndash9] and can be reduced to the four-dimensional Chern-Simonsmodified gravity of (1) Also the second integration isrelated to the interaction of gauge fields with Chern-Simonsfields Thus this model not only produces the Chern-Simonsmodified gravity but also exhibits some modifications to it
Now we can explain the mathematical model for obtain-ing results of (12)
The first line of this equation removes the anomaly on the 11-dimensional manifold of (4) however the second and thirdlines show that extra anomalies can emerge due to the Chern-Simons fields
At this stage we can consider the process of obtaining(24)
119878SUGRA-14 = 120575 (int11987214
119862119866119866 + int11987214
119862119866119866 + int11987214
119862119866119866+ int11987214
119862119866119866) = int11987214
5sum119899=1
(tr119865119899 minus tr119877119899 + tr (1198651198991198775minus119899))+ int11987214
5sum119899=1
119899sum119869=0
119899minus119869sum119898=0
(tr119865119869119865119898 minus tr119877119869119877119898 + tr (1198651198691198775minus1198691198651198981198775minus119898))+ int11987214
5sum119899=1
119899sum119869=0
119899minus119869sum119898=0
(tr119865119898119865119869 minus tr119877119898119877119869 + tr (1198651198981198775minus1198981198651198691198775minus119869))+ int11987214
5sum119899=1
(tr119865119899 minus tr119877119899 + tr (1198651198991198775minus119899))+ int11987210+119872119873=3
The authors declare that they have no conflicts of interest
Acknowledgments
The work of Alireza Sepehri has been supported financiallyby the Research Institute for Astronomy and Astrophysics ofMaragha (RIAAM) Iran under Research Project no 15237-75
References
[1] R Jackiw and S-Y Pi ldquoChern-Simons modification of generalrelativityrdquo Physical Review D Particles Fields Gravitation andCosmology vol 68 Article ID 104012 2003
[2] B Pereira-Dias C A Hernaski and J A Helayel-Neto ldquoProb-ing the effects of lorentz-symmetry violating chern-simons andricci-cotton terms in higher derivative gravityrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 83 no 82011
[3] K Konno T Matsuyama and S Tanda ldquoDoes a black holerotate in Chern-Simons modified gravityrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 76 no 2Article ID 024009 2007
[4] D Guarrera and A J Hariton ldquoPapapetrou energy-momentumtensor for Chern-Simons modified gravityrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 76 no 4 2007
Advances in High Energy Physics 13
[5] K K Nandi I R Kizirgulov O V Mikolaychuk N P Miko-laychuk and A A Potapov ldquoQuantum phase shift in Chern-Simons modified gravityrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 79 no 8 Article ID 0830062009
[6] P J Porfırio J B Fonseca-Neto J R Nascimento A Petrov JRicardo and A Santos ldquoChern-Simons modified gravity andclosed timelike curvesrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 94 no 4 2016
[7] S Alexander and N Yunes ldquoChernndashSimons modified generalrelativityrdquo Physics Reports vol 480 no 1-2 pp 1ndash55 2009
[8] S Chen and J Jing ldquoGeodetic precession and strong gravita-tional lensing in dynamical Chern-Simons-modified gravityrdquoClassical and QuantumGravity vol 27 no 22 225006 16 pages2010
[9] N Yunes and C F Sopuerta ldquoPerturbations of Schwarzschildblack holes in Chern-Simons modified gravityrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 77 no 62008
[10] F Canfora A Giacomini and A R Zerwekh ldquoKaluza-Kleintheory in the limit of large number of extra dimensionsrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 80 no 8 Article ID 084039 2009
[11] S Wu ldquoGeneral rotating charged Kaluza-Klein AdS black holesin higher dimensionsrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 83 no 12 2011
[12] T Houri and K Yamamoto ldquoKilling-Yano symmetry of Kaluza-KLEin black holes in five dimensionsrdquo Classical and QuantumGravity vol 30 no 7 075013 21 pages 2013
[13] T Tatsuoka H Ishihara M Kimura and K MatsunoldquoExtremal charged black holes with a twisted extra dimensionrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 85 no 4 2012
[14] P Horava and E Witten ldquoHeterotic and Type I string dynamicsfrom eleven dimensionsrdquo Nuclear Physics B vol 460 no 3Article ID 9510209 pp 506ndash524 1996
[15] A Sepehri and R Pincak ldquoThe birth of the universe in a newG-theory approachrdquo Modern Physics Letters A vol 32 no 5Article ID 1750033 1750033 32 pages 2017
[16] A Sepehri A Pradhan R Pincak F Rahaman A Beeshamand T Ghaffary ldquoBirth of the GUP and its effect on theentropy of the universe in Lie-N-algebrardquo International Journalof Geometric Methods in Modern Physics vol 14 no 9 ArticleID 1750130 2017
[17] P Horava and E Witten ldquoEleven-dimensional supergravity ona manifold with boundaryrdquo Nuclear Physics B vol 475 pp 94ndash114 1996
[18] J Bagger and N Lambert ldquoGauge symmetry and supersymme-try of multiple M2-branesrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 77 no 6 065008 6 pages 2008
[19] P Ho and Y Matsuo ldquoM5 from M2rdquo Journal of High EnergyPhysics vol 2008 no 06 p 105 2008
anomalies that can be removed by extending dimensions to14 Also we will show that there is a direct relation betweennumber of dimensions and algebra and also by choosing asuitable algebra anomalies can be removed completely
Our paper is organized according to the following outlinein Section 2 we devote efforts to show that by adding up a3-dimensional manifold to eleven-dimensional gravity thereemerges a Chern-Simonsmodified gravity Next in Section 3we shall show that if the fields obey a special algebraChern-Simons modified gravity is shown to be anomaly-free However by increasing the rank of the fields otheranomalies show up In Section 4 we focus on the removalof the anomaly of this type of gravity in a system composedof two 11-dimensional spaces and a Chern-Simons manifoldthat connects them In the last section we cast a summaryand our final considerations
where 119877 is the curvature and 120601 is the Chern-Simons scalarfield
Now we are going to show that Chern-Simons modifiedgravity can be obtained from a supergravity which lives onan 11 + 3-dimensional manifold Actually we assume thatour four-dimensional universe is a part of an 11-dimensionalmanifold that interacts with the bulk in an 11+3-dimensionalspace-time by exchanging Chern-Simons fields For thisour departure point is the purely bosonic sector of eleven-dimensional supergravity and we show that by adding upa three-dimensional manifold Chern-Simons terms willappear
The bosonic piece of the action for a gravity which liveson an eleven-dimensional manifold is given by [14 17]
119878Bosonic-SUGRA= 11205812 int11988911119909radic119892(minus12119877 minus 148119866119868119869119870119871119866119868119869119870119871) + 119878119862119866119866
119878119862119866119866 = minus radic234561205812 int11987211 1198891111990912057611986811198682 sdotsdotsdot119868111198621198681119868211986831198661198684 sdotsdotsdot11986871198661198688 sdotsdotsdot11986811 (2)
where the curvature (119877) and 119866119868119869119870119871 and 119862119868111986821198683 given in termsof the gauge field119860 and its field-strength 119865 are cast in whatfollows [17]
119866119868119869119870119871 = minus 3radic2 12058121205822 120576 (11990911) (119865119868119869119865119870119871 minus 119877119868119869119877119870119871) + sdot sdot sdot
120575119862119860119861119862 = minus 12058126radic21205822 120575 (11990911) tr (120598119862119865119860119861 minus 120598119862119877119860119861)11986611119860119861119862 = (12059711119862119860119861119862 plusmn 23 permutations)
+ cyclic permutations of 119860 119861 119862119865119868119869 = 120597119868119860119869 minus 120597119869119860119868119877119868119869 = 120597119868Γ120573119869120573 minus 120597119869Γ120573119868120573 + Γ120572119869120573Γ120573119868120572 minus Γ120572119868120573Γ120573119869120572Γ119868119869119870 = 120597119868119892119869119870 + 120597119870119892119868119869 minus 120597119869119892119868119870119866119868119869 = 119877119868119869 minus 12119877119892119868119869
(3)
Here 120576(11990911) is 1 for 11990911 gt 0 and 1 for 11990911 lt 0 and120575(11990911) = 12059712057612059711990911 Both capitalized Latin (eg 119868 119869) and Greek(eg 120573) indices act on the same manifold and we have onlyexhibited the free indices 119868 119869 119870 and the dummy ones (120572 120573)120598119862 is used as a vector in direction of119862This helps the equationthat becomes balanced from the indices point of view Thegauge variation of the 119862119866119866-action gives the following result[17]
120575119878119862119866119866100381610038161003816100381611= minus radic234561205812 int11987211 1198891111990912057611986811198682 sdotsdotsdot119868111205751198621198681119868211986831198661198684 sdotsdotsdot11986871198661198688 sdotsdotsdot11986811asymp minus 12058141281205826 int11987210
5sum119899=1
(tr119865119899 minus tr119877119899 + tr (1198651198991198775minus119899)) (4)
where tr119865119899 = tr(119865[11986811198682 sdot sdot sdot 1198651198682119899minus11198682119899]) =12059811986811198682 sdotsdotsdot1198682119899minus11198682119899 tr(11986511986811198682 sdot sdot sdot 1198651198682119899minus11198682119899) and tr119877119899 =tr(119877[11986811198682 sdot sdot sdot 1198771198682119899minus11198682119899]) = 12059811986811198682sdotsdotsdot1198682119899minus11198682119899 tr(11987711986811198682 sdot sdot sdot 1198771198682119899minus11198682119899) Theseterms above cancel the anomaly of (119878Bosonic-SUGRA) ineleven-dimensional manifold [17]
Thus 119878119862119866119866 is necessary for the anomaly cancelation solet us now go on and try to find a good rationale for it Alsowe shall answer the question related to the origin of 119862119866119866terms in 11-dimensional supergravity We actually propose ascenario in which the 119862119866119866 terms appear in the supergravityaction in a way that we do not add them up by hand Tothis end we choose a unified shape for all fields by using theNambu-Poisson brackets and the properties of string fields(119883) We define [15 18 19]
minus 12120598119868119894119868119895119868119896119868119898120601119877119868119894119868119895119868119896119868119898 + sdot sdot sdot (6)
where 120601 is the Chern-Simons scalar field 119860119868 is the gaugefield Γ is related to the curvature (119877) and 119868 is a unit vectorin the direction of the coordinate which can be expanded interms of derivatives of metric In fact the origin of all matterfields and strings is the same and they are equal to the unit
vectors (119868119869 = 119868120598119869) in addition to some fields (120601 119860119868) whichappear as fluctuations of space The latter may emerge by theinteraction of stringswhich breaks the initial symmetric stateWithout string interactions we have a symmetry that couldbe explained by a unit vector or a matrix We can first saythat in the static state all strings are equal to a unit vector ora matrix and then these strings interact with one anotherso that the symmetry is broken and fields emerge Also120598119868119894119868119895119868119896119868119898 is an antisymmetric tensor that has been attached toantisymmetric curvature and makes a symmetric part Thistensor causes that different states of curvature be regardedMaybe this question arises is 119883 used only for strings in 26-dimensional string theory In fact this could be a sign forbosonic strings in any dimension and is not related to 26 or 10dimensions Using four-dimensional brackets instead of two-dimensional ones we obtain the shape of the 119866119866-terms insupergravity as functions of strings (119883)
The equation above helps us to extract the 119862119866119866 termsfrom the 119866119866-terms in supergravity To this end we must adda three-dimensional manifold (related to a Lie-three-algebra)to eleven-dimensional supergravity by using the properties ofstrings (119883) in Nambu-Poisson brackets [15]
119883119868119894 = 119910119868119894 + 119860119868119894 minus 120598119868119894119869Γ120572120572119869 minus 120598119868119894119869 infinsum119899=1
where the integration has been carried out over a three-dimensional manifold with coordinates (1199101198685 1199101198686 1199101198687) andconsequently the integration can be done by using thatint1199101198685+1199101198686+1199101198687
= int1198891199101198685 int1198891199101198686 int1198891199101198687) The result above showsthat by ignoring fluctuations of space which yield productionof fields the area of each three-dimensional manifold canshrink to one and the result of the integration over thatmanifold goes to oneWhenwe addonemanifold to the otherthe integration will be the product of an integration over eachmanifold for the coordinates of the addedmanifolds increase
the elements of integration By adding the three-dimensionalmanifold of (8) to the eleven-dimensional manifold of (7) weget
This equation presents three results we should commenton (1) 119862119866119866 terms may appear in the action of supergravityby adding a three-dimensional manifold related to theLie-three-algebra added to eleven-dimensional supergrav-ity (2) 11-dimensional manifold + three-Lie algebra = 14-dimensional supergravity (3)Theshape of the119862-terms is nowclear in terms of the string fields (119883119894)
radic119892(12120598119868119894119868119895119868119896119868119898120601119877119868119896119868119898119868119897119868119899119877119868119897119868119899119868119894119868119895 minus 120597119868119894120601120597119868119895120601)minus 12 int11987211+119872119873=3
In the equation above the first integration is in agreementwith previous predictions of Chern-Simons gravity in [7ndash9] and can be reduced to the four-dimensional Chern-Simonsmodified gravity of (1) Also the second integration isrelated to the interaction of gauge fields with Chern-Simonsfields Thus this model not only produces the Chern-Simonsmodified gravity but also exhibits some modifications to itStill these results show that our universe is a part of one-eleven-dimensional manifold which interacts with a bulk ina 14-dimensional space-time by exchanging Chern-Simonsscalars
3 Anomalies in Chern-SimonsModified Gravity
In this section we shall consider various anomalies whichmay be induced in Chern-Simonsmodified gravity Althoughwe expect that terms in the gauge variation of the Chern-Simons action remove the anomaly in eleven-dimensionalsupergravity we will observe that some extra anomalies areproduced by the Chern-Simons field It is our goal to showthat these anomalies depend on the algebra and thus bychoosing a suitable algebra in this model all anomalies canbe removed To obtain the anomalies of the Chern-Simonstheory we should reobtain the gauge variation of the 119862119866119866-action in (4) in terms of field-strengths and curvatures To
this end by using (8) and (9) we can work out the gaugevariation of 119862 [15]
119883119868119894 = 119910119868119894 + 119860119868119894 minus 120598119868119894119869Γ120572120572119869 minus 120598119868119894119869 infinsum119899=1
minus 119877119868119894119868119895 + 120597119868119894120601120597119868119895120601 minus 12120598119868119894119868119895119868119896119868119898120601119877119868119894119868119895119868119896119868119898 + sdot sdot sdot)
(11)
Using the equation above and (7) we get the gaugevariation of the 119862119866119866 action given in (9)
The first line of this equation removes the anomaly on the 11-dimensional manifold of (4) however the second and thirdlines show that extra anomalies can emerge due to the Chern-Simons fields We can show that if we choose a suitablealgebra for the 11-dimensional manifold all anomalies can beswept out We can extend our discussion to a119863-dimensionalmanifold with a Lie-119873-algebra In fact we wish to obtain amethod that makes all supergravities with arbitrary dimen-sion anomaly-free To this end wemake use of the propertiesof Nambu-Poisson brackets and strings (119883) in (6) to obtaina unified definition for different terms in supergravity andrewrite action (4) as follows
In the equation above we only used the Lie-two-algebra with two-dimensional bracket however it is notclear whether this algebra is true In fact for 119872-theory Lie-three-algebra with three-dimensional bracket [18 19] is moresuitable To obtain the exact form of the Lie algebra whichis suitable for119863-dimensional space-time we shall generalizethe dimension of space-time from eleven to119863 and the algebrafrom two to119873 anduse the followingNambu-Poisson brackets[19]
In this equation we have added a new manifold relatedto the algebra to the world manifold In fact we have to
6 Advances in High Energy Physics
regard both algebraic (119872119873) and space-time (119872119863) manifoldsto achieve the exact results For the 119873-dimensional algebrawe introduce the following fields
120575[11986911198692sdotsdotsdot119869119873] = 12057511986911198692sdotsdotsdot119869119873 minus 12057511986921198691sdotsdotsdot119869119873 + sdot sdot sdot (16)
Here 120575 is the generalized Kronecker delta With defi-nitions in (15) we can obtain the explicit form of the 119873-dimensional Nambu-Poisson brackets in terms of fields
(11986511986911 sdotsdotsdot1198691119873 minus 12059711986915 sdot sdot sdot 120597119869111987311987711986911 sdotsdotsdot11986914)times (11986511986921 sdotsdotsdot1198692119873 minus 12059711986925 sdot sdot sdot 120597119869211987311987711986921 sdotsdotsdot11986924) sdot sdot sdot (11986511986911198632sdotsdotsdot1198691198632119873 minus 12059711986911986325sdot sdot sdot 1205971198691198632119873 11987711986911986321 sdotsdotsdot11986911986324 )
(18)
where 119885 is a constant related to the algebra This equationshows that the gauge variation of the action depends onthe rank-119873 field-strength The action above is not actuallydirectly zero and there emerges an anomaly Now we useproperties of 120598 and rewrite (18) as below
119882(119863119873) = ([(119863 + 2) (119863 minus 2)8 minus 119873(1198632 minus 1)]sdot [ (119863 + 2) (119863 minus 2)8 minus 1 minus 119873(1198632 minus 1)] sdot sdot sdot 1)sdot ([119873(1198632 minus 1)] [119873(1198632 minus 1) minus 1] sdot sdot sdot 1)119880 (120575)
(20)
where119880 is a function of the generalized Kronecker delta Onthe other hand 120575119878119862119866119866|119863+1 has been added to the main actionof supergravity to remove its anomaly Thus we can write
119882(119863119873) = ([(119863 + 2) (119863 minus 2)8 minus 119873(1198632 minus 1)]sdot [ (119863 + 2) (119863 minus 2)8 minus 1 minus 119873(1198632 minus 1)] sdot sdot sdot 1)times ([119873(1198632 minus 1)] [119873(1198632 minus 1) minus 1] sdot sdot sdot 1)119880 = 0 997904rArr
119873 le (119863 + 2) (119863 minus 2)8 (1198632 minus 1)
(21)
This equation indicates that for a (119863+1)-dimensional space-time the dimension of the Lie algebra should be equal to orless than a critical value Under these conditions the Chern-Simons gravity is free from anomalies and we do not needan extra manifold On the other hand as we show in (15)the dimension of the algebra determines the dimension ofthe field-strengthThis means that for a Lie-119873-algebra field-strengths should have at most 119873 indices For example for a
Advances in High Energy Physics 7
manifoldwith 11 dimensions the algebra can be of order threeas predicted in recent papers [18 19] and field-strengths mayhave three indices In fact in above equation we have shownthat the physics of an 11-dimensional spacing manifold plusa 3-dimensional algebraic manifold is equal to the physicsof 14-dimensional manifold This helps us to understandwhy 119872-theory with Lie-three algebra is a true theory andsolves many problems in physics In fact119872-theory with Lie-three algebra lives on 14-dimensional manifold where a 3-dimensional part of it corresponds to Lie-3-algebra and theother 11-dimensional part is related to space-time Also weshow that 119872-theory on 11-dimensional manifold could bethe anomaly-free if a three-dimensional algebraic manifoldis added to it or its spacial time is increased to 14 dimensionsThis 14-dimensional manifold can be broken to two parallel11-dimensional manifold which are connected by a three-dimensional Chern-Simons manifold
4 A Chern-Simons Manifold between Two11-Dimensional Manifolds in an 11 + 3Dimensional Space-Time
In the previous section we have found that for an eleven-dimensional manifold the suitable algebra which removesthe anomaly in Chern-Simons gravity is a three-dimensionalLie algebra This means that the rank of the fields can beof order two or three However (2) shows that the rank ofthe fields may be higher than three in eleven-dimensionalsupergravity Thus in Chern-Simons gravity theory whichlives on an eleven-dimensional manifold some extra anoma-lies are expected to show up To remove them we assumethat there is another 11-dimensional manifold in the 14-dimensional space-time which interacts with the first oneby exchanging Chern-Simons fields These fields producea Chern-Simons manifold that connects these two eleven-dimensional manifolds (see Figure 1)Thus in this model wehave two 119866119866 terms which live on 11-dimensional manifolds(see (2)) and two 119862119866119866 terms in the bulk so that each of theminteracts with one of the 11-dimensional manifolds
We can write the supergravity in 14-dimensional space-time as follows
119878SUGRA-14 = int119872119873=3
(int11987211
119866119866 + int11987211
119862119866119866+ int11987211
119862119866119866 + int11987211
119866119866) = (int11987214
119862119866119866+ int11987214
119862119866119866 + int11987214
119862119866119866 + int11987214
119862119866119866) (22)
In this equation119862119866119866 and119862119866119866 are related to the Chern-Simons gravities which live on the two eleven-dimensionalmanifolds and are extracted from 119866119866 and 119866119866 terms Also119862119866119866 and119862119866119866 correspond to the Chern-Simons fields whichare exchanged between the two manifolds in 14-dimensionalspace-time By generalizing the results of (3) (6) and (11) weget
119866119868119869119870119871 sim minus (119865119868119869119865119870119871 minus 119877119868119869119877119870119871) + 120597119868120601120597119869120601119877119870119871 sdot sdot sdot
119866119868119869119870119871 sim minus (119865119868119869119865119870119871 minus 119877119868119869119877119870119871) + 120597119868120601120597119869120601119877119870119871 sdot sdot sdot120575119862119868119894119868119895119868119896 sim minus120598119868119896 tr(119865119868119894119868119895 minus 119877119868119894119868119895 + 120597119868119894120601120597119868119895120601
minus 12120598119868119894119868119895119868119896119868119898119877119868119894119868119895119868119896119868119898 + sdot sdot sdot) (23)
Here the 119865rsquos 119877rsquos and 120601rsquos live on one of the supergravitymanifolds as depicted in Figure 1 whereas the 119865rsquos 119877rsquos and 120601rsquosare fields of the other supergravity manifold To obtain theirrelations we should make use of (12) and the gauge variationof the actions (22) in doing so we obtain
These results show that to remove the anomaly in 14-dimensional space-time coordinates and fields in one of theeleven-dimensionalmanifolds should be equal to coordinatesand fields in the othermanifold in addition to one extra 119894Thisimplies that time- or space-like coordinates and fields in onemanifold transmute into space- or time-like coordinates andfields of anothermanifold For example the zeroth coordinatewhich is known as time on one manifold will transmute intoa space coordinate of the other manifold If our universe withone time and three space coordinates is located on one of themanifolds an antiuniverse with one space and three times islocated in the other manifold
Now we shall show that by substituting the results of (24)into the action of (22) the topology of the 14-dimensionalmanifold tends to one This means that the world with allits matter began from a point and it may be thought of asa signature of Big Bang in our proposal To this end using(6) (7) (9) and (12) we rewrite 119862119866119866 terms in terms ofderivatives of scalar strings
On the other hand results in (24) show that (119883119868119894 119883119868119895 =minus119883119868119894 119883119868119895) Thus we can conclude that the action givenabove tends to an action on the three-dimensional manifold
119883119868119894 119883119868119895 = minus 119883119868119894 119883119868119895119878SUGRA-14 = 119896int
119872119873=3int11987211
(12057511 (119910)) = 119896int119872119873=3
997888rarr119896412058711987733
(28)
Advances in High Energy Physics 9
14-dimensionalspace-time
Chern-Simons manifold
11-dimensionalmanifold11-dimensional
manifold
Point
Figure 2 14-dimensional manifold shrinks to one point
where119877 is the radius ofmanifold By redefining scalar stringswe are able to show that the action of supergravity shrinks toone
This equation yields results that deserve our commentsIn fact two eleven-dimensional manifolds and the bulkinteract with each other via different types of 119862 119866 andChern-Simons-fields When we sum up supergravities thatlive on these manifolds and consider fields in the spacebetween them we get supergravity in 14-dimensional spaceBy canceling the anomaly in this new supergravity we canobtain the relations between fields By substituting theserelations into the action of the 14-dimensional supergravitywe simply obtain one This means that the 14-dimensionalmanifold with all its matter content can be topologicallyshrunk to one point (see Figure 2) In fact the system ofthe world began from this point and then expanded and
constructed a 14-dimensional world similar to what happensin a Big Bang theory
Another interesting result that comes out of our theoryconcerns the reduction of the action of the world to oneThismeans that the origin of allmatter or fields is the same In factat the first stage of the world we have only a constant energywhich lives on 14-dimensional manifold Then by internalinteractions two 11-dimensional manifolds are createdThesemanifolds interact with each other via an extraChern-Simonsmanifold By exchanging the energy between 11-dimensionalmanifolds their energies are changed and two differentactions are produced Also as due to interaction betweenstrings their shapes become different and different types offields like spinors and bosons are emerged
5 Summary and Final Considerations
In this paper we have shown that the Chern-Simons termsof modified gravity may be understood as due to the inter-action between two 11-dimensional manifolds in an 11 + 3-dimensional space-time where 3 is the dimension of a Lie-type-algebra We also argue that there is a direct relationbetween the dimension of the algebra and the dimensionof the manifold For example for 11-dimensional world thedimension of the Lie-algebra is three If the rank of the fieldswhich live in one manifold becomes larger than the rankof the algebra there emerges an anomaly This anomaly isproduced as an effect of connecting fields in the bulk tofields which live in the manifold To cancel this anomaly we
10 Advances in High Energy Physics
need to introduce another 11-dimensional manifold in the11+3-dimensional space-timewhich interacts with the initialmanifold by exchanging Chern-Simons terms These Chern-Simons terms produce an extra manifold If we sum up thetopology of the 11-dimensional manifolds and the topologyof the Chern-Simons manifold we can show that the totaltopology shrinks to one which is consistent with predictionsof the Big Bang theory
Our proposal is actually another version of the 119866-theory in [15 16] This approach removes anomalies in 11-dimensional 119872-theory by a generalization of dimension to14 In this research contribution we have shown that thephysics of an 11-dimensional spacing manifold plus a 3-dimensional algebraic manifold is the same as the physics of14-dimensional manifold In fact 11-dimensional 119872-theoryis a free-anomaly theory whenever a three-dimensionalalgebraic manifold is added to it This 14-dimensional man-ifold can be broken to two 11-dimensional manifolds whichare connected by a 3-dimensional Chern-Simons theoryAnother interesting result that comes out is the reduction ofthe action of the world to one This means that the originof all matter or fields is the same In fact at the first stageof the world we have only a constant energy which lives on14-dimensional manifold Then by internal interactions two11-dimensional manifolds come outThesemanifolds interactwith each other via one extra Chern-Simons manifold Byexchanging the energy between 11-dimensional manifoldstheir energies are changed and two different actions areproduced Also as due to interaction between strings theirshapes differ and diverse types of fields like spinors andbosons emerge
To conclude we would like to point out that that all ourtreatment has been restricted to the purely bosonic sectorof 11-dimensional supergravity whose on-shell multipletbesides the metric tensor and the 3-form gauge potentialincludes the gravitino fieldThe latter has not been consideredhere we have restricted ourselves to the bosonic sectorHowever it would be a further task to inspect how theinclusion of the gravitino would affect our developmentspossibly changing the dimension of the Lie algebra to beadded to the 11-dimensional manifold By including thefermion it is no longer ensured that the local supersymmetryof the 11 + 3-dimensional supergravity action remains validThe change in the dimension of the Lie algebra could in turngive rise to new terms so that the Chern-Simons modifiedgravity would be further extended as a result of including thegravitino We intend to pursue an investigation on this issueand to report on it elsewhere
In our model the three-Lie algebra that we add to passfrom 11 to 14119863 is geometrical It is shown that all anomalieswhich are produced by axial bosonic fields in 10- and 11-dimensional theories can be removed on 14-dimensionalmanifold Maybe a question arises ldquowhich is the effect offermionic anomalies on the number of dimensionsrdquo It mustbe clear that this paper is the first step for solving anomaliesin 11-dimensional theories In the next step by applyingthe model of this paper for fermions the exact number ofdimensions of the world can be estimated
Appendix
Details of Calculations
In this section we will bring details of calculations First letus disclose more aspects of (6)
1205981198681198941198681015840119894 (1205971198681015840119894119860119868119895 minus 1205971198681015840119894 (120598119868119895119868119896Γ120572120572119868119896) + 120597119868119894120601120597119868119895120601 + sdot sdot sdot)= 119865119868119894119868119895 minus 119877119868119894119868119895 + 120597119868119894120601120597119868119895120601 minus 12120598119868119894119868119895119868119896119868119898120601119877119868119894119868119895119868119896119868119898
+ sdot sdot sdot
(A1)
where 120601 is the Chern-Simons scalar field119860119868 is the gauge fieldΓ is related to the curvature (119877) and 119868 is a unit vector in thedirection of the coordinate which can be expanded in termsof derivatives of metric
Also it is needed to reobtain the results of (10) with doingsome mathematical calculations
In the equation above the first integration is in agreementwith previous predictions of Chern-Simons gravity in [7ndash9] and can be reduced to the four-dimensional Chern-Simonsmodified gravity of (1) Also the second integration isrelated to the interaction of gauge fields with Chern-Simonsfields Thus this model not only produces the Chern-Simonsmodified gravity but also exhibits some modifications to it
Now we can explain the mathematical model for obtain-ing results of (12)
The first line of this equation removes the anomaly on the 11-dimensional manifold of (4) however the second and thirdlines show that extra anomalies can emerge due to the Chern-Simons fields
At this stage we can consider the process of obtaining(24)
119878SUGRA-14 = 120575 (int11987214
119862119866119866 + int11987214
119862119866119866 + int11987214
119862119866119866+ int11987214
119862119866119866) = int11987214
5sum119899=1
(tr119865119899 minus tr119877119899 + tr (1198651198991198775minus119899))+ int11987214
5sum119899=1
119899sum119869=0
119899minus119869sum119898=0
(tr119865119869119865119898 minus tr119877119869119877119898 + tr (1198651198691198775minus1198691198651198981198775minus119898))+ int11987214
5sum119899=1
119899sum119869=0
119899minus119869sum119898=0
(tr119865119898119865119869 minus tr119877119898119877119869 + tr (1198651198981198775minus1198981198651198691198775minus119869))+ int11987214
5sum119899=1
(tr119865119899 minus tr119877119899 + tr (1198651198991198775minus119899))+ int11987210+119872119873=3
The authors declare that they have no conflicts of interest
Acknowledgments
The work of Alireza Sepehri has been supported financiallyby the Research Institute for Astronomy and Astrophysics ofMaragha (RIAAM) Iran under Research Project no 15237-75
References
[1] R Jackiw and S-Y Pi ldquoChern-Simons modification of generalrelativityrdquo Physical Review D Particles Fields Gravitation andCosmology vol 68 Article ID 104012 2003
[2] B Pereira-Dias C A Hernaski and J A Helayel-Neto ldquoProb-ing the effects of lorentz-symmetry violating chern-simons andricci-cotton terms in higher derivative gravityrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 83 no 82011
[3] K Konno T Matsuyama and S Tanda ldquoDoes a black holerotate in Chern-Simons modified gravityrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 76 no 2Article ID 024009 2007
[4] D Guarrera and A J Hariton ldquoPapapetrou energy-momentumtensor for Chern-Simons modified gravityrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 76 no 4 2007
Advances in High Energy Physics 13
[5] K K Nandi I R Kizirgulov O V Mikolaychuk N P Miko-laychuk and A A Potapov ldquoQuantum phase shift in Chern-Simons modified gravityrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 79 no 8 Article ID 0830062009
[6] P J Porfırio J B Fonseca-Neto J R Nascimento A Petrov JRicardo and A Santos ldquoChern-Simons modified gravity andclosed timelike curvesrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 94 no 4 2016
[7] S Alexander and N Yunes ldquoChernndashSimons modified generalrelativityrdquo Physics Reports vol 480 no 1-2 pp 1ndash55 2009
[8] S Chen and J Jing ldquoGeodetic precession and strong gravita-tional lensing in dynamical Chern-Simons-modified gravityrdquoClassical and QuantumGravity vol 27 no 22 225006 16 pages2010
[9] N Yunes and C F Sopuerta ldquoPerturbations of Schwarzschildblack holes in Chern-Simons modified gravityrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 77 no 62008
[10] F Canfora A Giacomini and A R Zerwekh ldquoKaluza-Kleintheory in the limit of large number of extra dimensionsrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 80 no 8 Article ID 084039 2009
[11] S Wu ldquoGeneral rotating charged Kaluza-Klein AdS black holesin higher dimensionsrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 83 no 12 2011
[12] T Houri and K Yamamoto ldquoKilling-Yano symmetry of Kaluza-KLEin black holes in five dimensionsrdquo Classical and QuantumGravity vol 30 no 7 075013 21 pages 2013
[13] T Tatsuoka H Ishihara M Kimura and K MatsunoldquoExtremal charged black holes with a twisted extra dimensionrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 85 no 4 2012
[14] P Horava and E Witten ldquoHeterotic and Type I string dynamicsfrom eleven dimensionsrdquo Nuclear Physics B vol 460 no 3Article ID 9510209 pp 506ndash524 1996
[15] A Sepehri and R Pincak ldquoThe birth of the universe in a newG-theory approachrdquo Modern Physics Letters A vol 32 no 5Article ID 1750033 1750033 32 pages 2017
[16] A Sepehri A Pradhan R Pincak F Rahaman A Beeshamand T Ghaffary ldquoBirth of the GUP and its effect on theentropy of the universe in Lie-N-algebrardquo International Journalof Geometric Methods in Modern Physics vol 14 no 9 ArticleID 1750130 2017
[17] P Horava and E Witten ldquoEleven-dimensional supergravity ona manifold with boundaryrdquo Nuclear Physics B vol 475 pp 94ndash114 1996
[18] J Bagger and N Lambert ldquoGauge symmetry and supersymme-try of multiple M2-branesrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 77 no 6 065008 6 pages 2008
[19] P Ho and Y Matsuo ldquoM5 from M2rdquo Journal of High EnergyPhysics vol 2008 no 06 p 105 2008
minus 12120598119868119894119868119895119868119896119868119898120601119877119868119894119868119895119868119896119868119898 + sdot sdot sdot (6)
where 120601 is the Chern-Simons scalar field 119860119868 is the gaugefield Γ is related to the curvature (119877) and 119868 is a unit vectorin the direction of the coordinate which can be expanded interms of derivatives of metric In fact the origin of all matterfields and strings is the same and they are equal to the unit
vectors (119868119869 = 119868120598119869) in addition to some fields (120601 119860119868) whichappear as fluctuations of space The latter may emerge by theinteraction of stringswhich breaks the initial symmetric stateWithout string interactions we have a symmetry that couldbe explained by a unit vector or a matrix We can first saythat in the static state all strings are equal to a unit vector ora matrix and then these strings interact with one anotherso that the symmetry is broken and fields emerge Also120598119868119894119868119895119868119896119868119898 is an antisymmetric tensor that has been attached toantisymmetric curvature and makes a symmetric part Thistensor causes that different states of curvature be regardedMaybe this question arises is 119883 used only for strings in 26-dimensional string theory In fact this could be a sign forbosonic strings in any dimension and is not related to 26 or 10dimensions Using four-dimensional brackets instead of two-dimensional ones we obtain the shape of the 119866119866-terms insupergravity as functions of strings (119883)
The equation above helps us to extract the 119862119866119866 termsfrom the 119866119866-terms in supergravity To this end we must adda three-dimensional manifold (related to a Lie-three-algebra)to eleven-dimensional supergravity by using the properties ofstrings (119883) in Nambu-Poisson brackets [15]
119883119868119894 = 119910119868119894 + 119860119868119894 minus 120598119868119894119869Γ120572120572119869 minus 120598119868119894119869 infinsum119899=1
where the integration has been carried out over a three-dimensional manifold with coordinates (1199101198685 1199101198686 1199101198687) andconsequently the integration can be done by using thatint1199101198685+1199101198686+1199101198687
= int1198891199101198685 int1198891199101198686 int1198891199101198687) The result above showsthat by ignoring fluctuations of space which yield productionof fields the area of each three-dimensional manifold canshrink to one and the result of the integration over thatmanifold goes to oneWhenwe addonemanifold to the otherthe integration will be the product of an integration over eachmanifold for the coordinates of the addedmanifolds increase
the elements of integration By adding the three-dimensionalmanifold of (8) to the eleven-dimensional manifold of (7) weget
This equation presents three results we should commenton (1) 119862119866119866 terms may appear in the action of supergravityby adding a three-dimensional manifold related to theLie-three-algebra added to eleven-dimensional supergrav-ity (2) 11-dimensional manifold + three-Lie algebra = 14-dimensional supergravity (3)Theshape of the119862-terms is nowclear in terms of the string fields (119883119894)
radic119892(12120598119868119894119868119895119868119896119868119898120601119877119868119896119868119898119868119897119868119899119877119868119897119868119899119868119894119868119895 minus 120597119868119894120601120597119868119895120601)minus 12 int11987211+119872119873=3
In the equation above the first integration is in agreementwith previous predictions of Chern-Simons gravity in [7ndash9] and can be reduced to the four-dimensional Chern-Simonsmodified gravity of (1) Also the second integration isrelated to the interaction of gauge fields with Chern-Simonsfields Thus this model not only produces the Chern-Simonsmodified gravity but also exhibits some modifications to itStill these results show that our universe is a part of one-eleven-dimensional manifold which interacts with a bulk ina 14-dimensional space-time by exchanging Chern-Simonsscalars
3 Anomalies in Chern-SimonsModified Gravity
In this section we shall consider various anomalies whichmay be induced in Chern-Simonsmodified gravity Althoughwe expect that terms in the gauge variation of the Chern-Simons action remove the anomaly in eleven-dimensionalsupergravity we will observe that some extra anomalies areproduced by the Chern-Simons field It is our goal to showthat these anomalies depend on the algebra and thus bychoosing a suitable algebra in this model all anomalies canbe removed To obtain the anomalies of the Chern-Simonstheory we should reobtain the gauge variation of the 119862119866119866-action in (4) in terms of field-strengths and curvatures To
this end by using (8) and (9) we can work out the gaugevariation of 119862 [15]
119883119868119894 = 119910119868119894 + 119860119868119894 minus 120598119868119894119869Γ120572120572119869 minus 120598119868119894119869 infinsum119899=1
minus 119877119868119894119868119895 + 120597119868119894120601120597119868119895120601 minus 12120598119868119894119868119895119868119896119868119898120601119877119868119894119868119895119868119896119868119898 + sdot sdot sdot)
(11)
Using the equation above and (7) we get the gaugevariation of the 119862119866119866 action given in (9)
The first line of this equation removes the anomaly on the 11-dimensional manifold of (4) however the second and thirdlines show that extra anomalies can emerge due to the Chern-Simons fields We can show that if we choose a suitablealgebra for the 11-dimensional manifold all anomalies can beswept out We can extend our discussion to a119863-dimensionalmanifold with a Lie-119873-algebra In fact we wish to obtain amethod that makes all supergravities with arbitrary dimen-sion anomaly-free To this end wemake use of the propertiesof Nambu-Poisson brackets and strings (119883) in (6) to obtaina unified definition for different terms in supergravity andrewrite action (4) as follows
In the equation above we only used the Lie-two-algebra with two-dimensional bracket however it is notclear whether this algebra is true In fact for 119872-theory Lie-three-algebra with three-dimensional bracket [18 19] is moresuitable To obtain the exact form of the Lie algebra whichis suitable for119863-dimensional space-time we shall generalizethe dimension of space-time from eleven to119863 and the algebrafrom two to119873 anduse the followingNambu-Poisson brackets[19]
In this equation we have added a new manifold relatedto the algebra to the world manifold In fact we have to
6 Advances in High Energy Physics
regard both algebraic (119872119873) and space-time (119872119863) manifoldsto achieve the exact results For the 119873-dimensional algebrawe introduce the following fields
120575[11986911198692sdotsdotsdot119869119873] = 12057511986911198692sdotsdotsdot119869119873 minus 12057511986921198691sdotsdotsdot119869119873 + sdot sdot sdot (16)
Here 120575 is the generalized Kronecker delta With defi-nitions in (15) we can obtain the explicit form of the 119873-dimensional Nambu-Poisson brackets in terms of fields
(11986511986911 sdotsdotsdot1198691119873 minus 12059711986915 sdot sdot sdot 120597119869111987311987711986911 sdotsdotsdot11986914)times (11986511986921 sdotsdotsdot1198692119873 minus 12059711986925 sdot sdot sdot 120597119869211987311987711986921 sdotsdotsdot11986924) sdot sdot sdot (11986511986911198632sdotsdotsdot1198691198632119873 minus 12059711986911986325sdot sdot sdot 1205971198691198632119873 11987711986911986321 sdotsdotsdot11986911986324 )
(18)
where 119885 is a constant related to the algebra This equationshows that the gauge variation of the action depends onthe rank-119873 field-strength The action above is not actuallydirectly zero and there emerges an anomaly Now we useproperties of 120598 and rewrite (18) as below
119882(119863119873) = ([(119863 + 2) (119863 minus 2)8 minus 119873(1198632 minus 1)]sdot [ (119863 + 2) (119863 minus 2)8 minus 1 minus 119873(1198632 minus 1)] sdot sdot sdot 1)sdot ([119873(1198632 minus 1)] [119873(1198632 minus 1) minus 1] sdot sdot sdot 1)119880 (120575)
(20)
where119880 is a function of the generalized Kronecker delta Onthe other hand 120575119878119862119866119866|119863+1 has been added to the main actionof supergravity to remove its anomaly Thus we can write
119882(119863119873) = ([(119863 + 2) (119863 minus 2)8 minus 119873(1198632 minus 1)]sdot [ (119863 + 2) (119863 minus 2)8 minus 1 minus 119873(1198632 minus 1)] sdot sdot sdot 1)times ([119873(1198632 minus 1)] [119873(1198632 minus 1) minus 1] sdot sdot sdot 1)119880 = 0 997904rArr
119873 le (119863 + 2) (119863 minus 2)8 (1198632 minus 1)
(21)
This equation indicates that for a (119863+1)-dimensional space-time the dimension of the Lie algebra should be equal to orless than a critical value Under these conditions the Chern-Simons gravity is free from anomalies and we do not needan extra manifold On the other hand as we show in (15)the dimension of the algebra determines the dimension ofthe field-strengthThis means that for a Lie-119873-algebra field-strengths should have at most 119873 indices For example for a
Advances in High Energy Physics 7
manifoldwith 11 dimensions the algebra can be of order threeas predicted in recent papers [18 19] and field-strengths mayhave three indices In fact in above equation we have shownthat the physics of an 11-dimensional spacing manifold plusa 3-dimensional algebraic manifold is equal to the physicsof 14-dimensional manifold This helps us to understandwhy 119872-theory with Lie-three algebra is a true theory andsolves many problems in physics In fact119872-theory with Lie-three algebra lives on 14-dimensional manifold where a 3-dimensional part of it corresponds to Lie-3-algebra and theother 11-dimensional part is related to space-time Also weshow that 119872-theory on 11-dimensional manifold could bethe anomaly-free if a three-dimensional algebraic manifoldis added to it or its spacial time is increased to 14 dimensionsThis 14-dimensional manifold can be broken to two parallel11-dimensional manifold which are connected by a three-dimensional Chern-Simons manifold
4 A Chern-Simons Manifold between Two11-Dimensional Manifolds in an 11 + 3Dimensional Space-Time
In the previous section we have found that for an eleven-dimensional manifold the suitable algebra which removesthe anomaly in Chern-Simons gravity is a three-dimensionalLie algebra This means that the rank of the fields can beof order two or three However (2) shows that the rank ofthe fields may be higher than three in eleven-dimensionalsupergravity Thus in Chern-Simons gravity theory whichlives on an eleven-dimensional manifold some extra anoma-lies are expected to show up To remove them we assumethat there is another 11-dimensional manifold in the 14-dimensional space-time which interacts with the first oneby exchanging Chern-Simons fields These fields producea Chern-Simons manifold that connects these two eleven-dimensional manifolds (see Figure 1)Thus in this model wehave two 119866119866 terms which live on 11-dimensional manifolds(see (2)) and two 119862119866119866 terms in the bulk so that each of theminteracts with one of the 11-dimensional manifolds
We can write the supergravity in 14-dimensional space-time as follows
119878SUGRA-14 = int119872119873=3
(int11987211
119866119866 + int11987211
119862119866119866+ int11987211
119862119866119866 + int11987211
119866119866) = (int11987214
119862119866119866+ int11987214
119862119866119866 + int11987214
119862119866119866 + int11987214
119862119866119866) (22)
In this equation119862119866119866 and119862119866119866 are related to the Chern-Simons gravities which live on the two eleven-dimensionalmanifolds and are extracted from 119866119866 and 119866119866 terms Also119862119866119866 and119862119866119866 correspond to the Chern-Simons fields whichare exchanged between the two manifolds in 14-dimensionalspace-time By generalizing the results of (3) (6) and (11) weget
119866119868119869119870119871 sim minus (119865119868119869119865119870119871 minus 119877119868119869119877119870119871) + 120597119868120601120597119869120601119877119870119871 sdot sdot sdot
119866119868119869119870119871 sim minus (119865119868119869119865119870119871 minus 119877119868119869119877119870119871) + 120597119868120601120597119869120601119877119870119871 sdot sdot sdot120575119862119868119894119868119895119868119896 sim minus120598119868119896 tr(119865119868119894119868119895 minus 119877119868119894119868119895 + 120597119868119894120601120597119868119895120601
minus 12120598119868119894119868119895119868119896119868119898119877119868119894119868119895119868119896119868119898 + sdot sdot sdot) (23)
Here the 119865rsquos 119877rsquos and 120601rsquos live on one of the supergravitymanifolds as depicted in Figure 1 whereas the 119865rsquos 119877rsquos and 120601rsquosare fields of the other supergravity manifold To obtain theirrelations we should make use of (12) and the gauge variationof the actions (22) in doing so we obtain
These results show that to remove the anomaly in 14-dimensional space-time coordinates and fields in one of theeleven-dimensionalmanifolds should be equal to coordinatesand fields in the othermanifold in addition to one extra 119894Thisimplies that time- or space-like coordinates and fields in onemanifold transmute into space- or time-like coordinates andfields of anothermanifold For example the zeroth coordinatewhich is known as time on one manifold will transmute intoa space coordinate of the other manifold If our universe withone time and three space coordinates is located on one of themanifolds an antiuniverse with one space and three times islocated in the other manifold
Now we shall show that by substituting the results of (24)into the action of (22) the topology of the 14-dimensionalmanifold tends to one This means that the world with allits matter began from a point and it may be thought of asa signature of Big Bang in our proposal To this end using(6) (7) (9) and (12) we rewrite 119862119866119866 terms in terms ofderivatives of scalar strings
On the other hand results in (24) show that (119883119868119894 119883119868119895 =minus119883119868119894 119883119868119895) Thus we can conclude that the action givenabove tends to an action on the three-dimensional manifold
119883119868119894 119883119868119895 = minus 119883119868119894 119883119868119895119878SUGRA-14 = 119896int
119872119873=3int11987211
(12057511 (119910)) = 119896int119872119873=3
997888rarr119896412058711987733
(28)
Advances in High Energy Physics 9
14-dimensionalspace-time
Chern-Simons manifold
11-dimensionalmanifold11-dimensional
manifold
Point
Figure 2 14-dimensional manifold shrinks to one point
where119877 is the radius ofmanifold By redefining scalar stringswe are able to show that the action of supergravity shrinks toone
This equation yields results that deserve our commentsIn fact two eleven-dimensional manifolds and the bulkinteract with each other via different types of 119862 119866 andChern-Simons-fields When we sum up supergravities thatlive on these manifolds and consider fields in the spacebetween them we get supergravity in 14-dimensional spaceBy canceling the anomaly in this new supergravity we canobtain the relations between fields By substituting theserelations into the action of the 14-dimensional supergravitywe simply obtain one This means that the 14-dimensionalmanifold with all its matter content can be topologicallyshrunk to one point (see Figure 2) In fact the system ofthe world began from this point and then expanded and
constructed a 14-dimensional world similar to what happensin a Big Bang theory
Another interesting result that comes out of our theoryconcerns the reduction of the action of the world to oneThismeans that the origin of allmatter or fields is the same In factat the first stage of the world we have only a constant energywhich lives on 14-dimensional manifold Then by internalinteractions two 11-dimensional manifolds are createdThesemanifolds interact with each other via an extraChern-Simonsmanifold By exchanging the energy between 11-dimensionalmanifolds their energies are changed and two differentactions are produced Also as due to interaction betweenstrings their shapes become different and different types offields like spinors and bosons are emerged
5 Summary and Final Considerations
In this paper we have shown that the Chern-Simons termsof modified gravity may be understood as due to the inter-action between two 11-dimensional manifolds in an 11 + 3-dimensional space-time where 3 is the dimension of a Lie-type-algebra We also argue that there is a direct relationbetween the dimension of the algebra and the dimensionof the manifold For example for 11-dimensional world thedimension of the Lie-algebra is three If the rank of the fieldswhich live in one manifold becomes larger than the rankof the algebra there emerges an anomaly This anomaly isproduced as an effect of connecting fields in the bulk tofields which live in the manifold To cancel this anomaly we
10 Advances in High Energy Physics
need to introduce another 11-dimensional manifold in the11+3-dimensional space-timewhich interacts with the initialmanifold by exchanging Chern-Simons terms These Chern-Simons terms produce an extra manifold If we sum up thetopology of the 11-dimensional manifolds and the topologyof the Chern-Simons manifold we can show that the totaltopology shrinks to one which is consistent with predictionsof the Big Bang theory
Our proposal is actually another version of the 119866-theory in [15 16] This approach removes anomalies in 11-dimensional 119872-theory by a generalization of dimension to14 In this research contribution we have shown that thephysics of an 11-dimensional spacing manifold plus a 3-dimensional algebraic manifold is the same as the physics of14-dimensional manifold In fact 11-dimensional 119872-theoryis a free-anomaly theory whenever a three-dimensionalalgebraic manifold is added to it This 14-dimensional man-ifold can be broken to two 11-dimensional manifolds whichare connected by a 3-dimensional Chern-Simons theoryAnother interesting result that comes out is the reduction ofthe action of the world to one This means that the originof all matter or fields is the same In fact at the first stageof the world we have only a constant energy which lives on14-dimensional manifold Then by internal interactions two11-dimensional manifolds come outThesemanifolds interactwith each other via one extra Chern-Simons manifold Byexchanging the energy between 11-dimensional manifoldstheir energies are changed and two different actions areproduced Also as due to interaction between strings theirshapes differ and diverse types of fields like spinors andbosons emerge
To conclude we would like to point out that that all ourtreatment has been restricted to the purely bosonic sectorof 11-dimensional supergravity whose on-shell multipletbesides the metric tensor and the 3-form gauge potentialincludes the gravitino fieldThe latter has not been consideredhere we have restricted ourselves to the bosonic sectorHowever it would be a further task to inspect how theinclusion of the gravitino would affect our developmentspossibly changing the dimension of the Lie algebra to beadded to the 11-dimensional manifold By including thefermion it is no longer ensured that the local supersymmetryof the 11 + 3-dimensional supergravity action remains validThe change in the dimension of the Lie algebra could in turngive rise to new terms so that the Chern-Simons modifiedgravity would be further extended as a result of including thegravitino We intend to pursue an investigation on this issueand to report on it elsewhere
In our model the three-Lie algebra that we add to passfrom 11 to 14119863 is geometrical It is shown that all anomalieswhich are produced by axial bosonic fields in 10- and 11-dimensional theories can be removed on 14-dimensionalmanifold Maybe a question arises ldquowhich is the effect offermionic anomalies on the number of dimensionsrdquo It mustbe clear that this paper is the first step for solving anomaliesin 11-dimensional theories In the next step by applyingthe model of this paper for fermions the exact number ofdimensions of the world can be estimated
Appendix
Details of Calculations
In this section we will bring details of calculations First letus disclose more aspects of (6)
1205981198681198941198681015840119894 (1205971198681015840119894119860119868119895 minus 1205971198681015840119894 (120598119868119895119868119896Γ120572120572119868119896) + 120597119868119894120601120597119868119895120601 + sdot sdot sdot)= 119865119868119894119868119895 minus 119877119868119894119868119895 + 120597119868119894120601120597119868119895120601 minus 12120598119868119894119868119895119868119896119868119898120601119877119868119894119868119895119868119896119868119898
+ sdot sdot sdot
(A1)
where 120601 is the Chern-Simons scalar field119860119868 is the gauge fieldΓ is related to the curvature (119877) and 119868 is a unit vector in thedirection of the coordinate which can be expanded in termsof derivatives of metric
Also it is needed to reobtain the results of (10) with doingsome mathematical calculations
In the equation above the first integration is in agreementwith previous predictions of Chern-Simons gravity in [7ndash9] and can be reduced to the four-dimensional Chern-Simonsmodified gravity of (1) Also the second integration isrelated to the interaction of gauge fields with Chern-Simonsfields Thus this model not only produces the Chern-Simonsmodified gravity but also exhibits some modifications to it
Now we can explain the mathematical model for obtain-ing results of (12)
The first line of this equation removes the anomaly on the 11-dimensional manifold of (4) however the second and thirdlines show that extra anomalies can emerge due to the Chern-Simons fields
At this stage we can consider the process of obtaining(24)
119878SUGRA-14 = 120575 (int11987214
119862119866119866 + int11987214
119862119866119866 + int11987214
119862119866119866+ int11987214
119862119866119866) = int11987214
5sum119899=1
(tr119865119899 minus tr119877119899 + tr (1198651198991198775minus119899))+ int11987214
5sum119899=1
119899sum119869=0
119899minus119869sum119898=0
(tr119865119869119865119898 minus tr119877119869119877119898 + tr (1198651198691198775minus1198691198651198981198775minus119898))+ int11987214
5sum119899=1
119899sum119869=0
119899minus119869sum119898=0
(tr119865119898119865119869 minus tr119877119898119877119869 + tr (1198651198981198775minus1198981198651198691198775minus119869))+ int11987214
5sum119899=1
(tr119865119899 minus tr119877119899 + tr (1198651198991198775minus119899))+ int11987210+119872119873=3
The authors declare that they have no conflicts of interest
Acknowledgments
The work of Alireza Sepehri has been supported financiallyby the Research Institute for Astronomy and Astrophysics ofMaragha (RIAAM) Iran under Research Project no 15237-75
References
[1] R Jackiw and S-Y Pi ldquoChern-Simons modification of generalrelativityrdquo Physical Review D Particles Fields Gravitation andCosmology vol 68 Article ID 104012 2003
[2] B Pereira-Dias C A Hernaski and J A Helayel-Neto ldquoProb-ing the effects of lorentz-symmetry violating chern-simons andricci-cotton terms in higher derivative gravityrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 83 no 82011
[3] K Konno T Matsuyama and S Tanda ldquoDoes a black holerotate in Chern-Simons modified gravityrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 76 no 2Article ID 024009 2007
[4] D Guarrera and A J Hariton ldquoPapapetrou energy-momentumtensor for Chern-Simons modified gravityrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 76 no 4 2007
Advances in High Energy Physics 13
[5] K K Nandi I R Kizirgulov O V Mikolaychuk N P Miko-laychuk and A A Potapov ldquoQuantum phase shift in Chern-Simons modified gravityrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 79 no 8 Article ID 0830062009
[6] P J Porfırio J B Fonseca-Neto J R Nascimento A Petrov JRicardo and A Santos ldquoChern-Simons modified gravity andclosed timelike curvesrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 94 no 4 2016
[7] S Alexander and N Yunes ldquoChernndashSimons modified generalrelativityrdquo Physics Reports vol 480 no 1-2 pp 1ndash55 2009
[8] S Chen and J Jing ldquoGeodetic precession and strong gravita-tional lensing in dynamical Chern-Simons-modified gravityrdquoClassical and QuantumGravity vol 27 no 22 225006 16 pages2010
[9] N Yunes and C F Sopuerta ldquoPerturbations of Schwarzschildblack holes in Chern-Simons modified gravityrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 77 no 62008
[10] F Canfora A Giacomini and A R Zerwekh ldquoKaluza-Kleintheory in the limit of large number of extra dimensionsrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 80 no 8 Article ID 084039 2009
[11] S Wu ldquoGeneral rotating charged Kaluza-Klein AdS black holesin higher dimensionsrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 83 no 12 2011
[12] T Houri and K Yamamoto ldquoKilling-Yano symmetry of Kaluza-KLEin black holes in five dimensionsrdquo Classical and QuantumGravity vol 30 no 7 075013 21 pages 2013
[13] T Tatsuoka H Ishihara M Kimura and K MatsunoldquoExtremal charged black holes with a twisted extra dimensionrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 85 no 4 2012
[14] P Horava and E Witten ldquoHeterotic and Type I string dynamicsfrom eleven dimensionsrdquo Nuclear Physics B vol 460 no 3Article ID 9510209 pp 506ndash524 1996
[15] A Sepehri and R Pincak ldquoThe birth of the universe in a newG-theory approachrdquo Modern Physics Letters A vol 32 no 5Article ID 1750033 1750033 32 pages 2017
[16] A Sepehri A Pradhan R Pincak F Rahaman A Beeshamand T Ghaffary ldquoBirth of the GUP and its effect on theentropy of the universe in Lie-N-algebrardquo International Journalof Geometric Methods in Modern Physics vol 14 no 9 ArticleID 1750130 2017
[17] P Horava and E Witten ldquoEleven-dimensional supergravity ona manifold with boundaryrdquo Nuclear Physics B vol 475 pp 94ndash114 1996
[18] J Bagger and N Lambert ldquoGauge symmetry and supersymme-try of multiple M2-branesrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 77 no 6 065008 6 pages 2008
[19] P Ho and Y Matsuo ldquoM5 from M2rdquo Journal of High EnergyPhysics vol 2008 no 06 p 105 2008
In the equation above the first integration is in agreementwith previous predictions of Chern-Simons gravity in [7ndash9] and can be reduced to the four-dimensional Chern-Simonsmodified gravity of (1) Also the second integration isrelated to the interaction of gauge fields with Chern-Simonsfields Thus this model not only produces the Chern-Simonsmodified gravity but also exhibits some modifications to itStill these results show that our universe is a part of one-eleven-dimensional manifold which interacts with a bulk ina 14-dimensional space-time by exchanging Chern-Simonsscalars
3 Anomalies in Chern-SimonsModified Gravity
In this section we shall consider various anomalies whichmay be induced in Chern-Simonsmodified gravity Althoughwe expect that terms in the gauge variation of the Chern-Simons action remove the anomaly in eleven-dimensionalsupergravity we will observe that some extra anomalies areproduced by the Chern-Simons field It is our goal to showthat these anomalies depend on the algebra and thus bychoosing a suitable algebra in this model all anomalies canbe removed To obtain the anomalies of the Chern-Simonstheory we should reobtain the gauge variation of the 119862119866119866-action in (4) in terms of field-strengths and curvatures To
this end by using (8) and (9) we can work out the gaugevariation of 119862 [15]
119883119868119894 = 119910119868119894 + 119860119868119894 minus 120598119868119894119869Γ120572120572119869 minus 120598119868119894119869 infinsum119899=1
minus 119877119868119894119868119895 + 120597119868119894120601120597119868119895120601 minus 12120598119868119894119868119895119868119896119868119898120601119877119868119894119868119895119868119896119868119898 + sdot sdot sdot)
(11)
Using the equation above and (7) we get the gaugevariation of the 119862119866119866 action given in (9)
The first line of this equation removes the anomaly on the 11-dimensional manifold of (4) however the second and thirdlines show that extra anomalies can emerge due to the Chern-Simons fields We can show that if we choose a suitablealgebra for the 11-dimensional manifold all anomalies can beswept out We can extend our discussion to a119863-dimensionalmanifold with a Lie-119873-algebra In fact we wish to obtain amethod that makes all supergravities with arbitrary dimen-sion anomaly-free To this end wemake use of the propertiesof Nambu-Poisson brackets and strings (119883) in (6) to obtaina unified definition for different terms in supergravity andrewrite action (4) as follows
In the equation above we only used the Lie-two-algebra with two-dimensional bracket however it is notclear whether this algebra is true In fact for 119872-theory Lie-three-algebra with three-dimensional bracket [18 19] is moresuitable To obtain the exact form of the Lie algebra whichis suitable for119863-dimensional space-time we shall generalizethe dimension of space-time from eleven to119863 and the algebrafrom two to119873 anduse the followingNambu-Poisson brackets[19]
In this equation we have added a new manifold relatedto the algebra to the world manifold In fact we have to
6 Advances in High Energy Physics
regard both algebraic (119872119873) and space-time (119872119863) manifoldsto achieve the exact results For the 119873-dimensional algebrawe introduce the following fields
120575[11986911198692sdotsdotsdot119869119873] = 12057511986911198692sdotsdotsdot119869119873 minus 12057511986921198691sdotsdotsdot119869119873 + sdot sdot sdot (16)
Here 120575 is the generalized Kronecker delta With defi-nitions in (15) we can obtain the explicit form of the 119873-dimensional Nambu-Poisson brackets in terms of fields
(11986511986911 sdotsdotsdot1198691119873 minus 12059711986915 sdot sdot sdot 120597119869111987311987711986911 sdotsdotsdot11986914)times (11986511986921 sdotsdotsdot1198692119873 minus 12059711986925 sdot sdot sdot 120597119869211987311987711986921 sdotsdotsdot11986924) sdot sdot sdot (11986511986911198632sdotsdotsdot1198691198632119873 minus 12059711986911986325sdot sdot sdot 1205971198691198632119873 11987711986911986321 sdotsdotsdot11986911986324 )
(18)
where 119885 is a constant related to the algebra This equationshows that the gauge variation of the action depends onthe rank-119873 field-strength The action above is not actuallydirectly zero and there emerges an anomaly Now we useproperties of 120598 and rewrite (18) as below
119882(119863119873) = ([(119863 + 2) (119863 minus 2)8 minus 119873(1198632 minus 1)]sdot [ (119863 + 2) (119863 minus 2)8 minus 1 minus 119873(1198632 minus 1)] sdot sdot sdot 1)sdot ([119873(1198632 minus 1)] [119873(1198632 minus 1) minus 1] sdot sdot sdot 1)119880 (120575)
(20)
where119880 is a function of the generalized Kronecker delta Onthe other hand 120575119878119862119866119866|119863+1 has been added to the main actionof supergravity to remove its anomaly Thus we can write
119882(119863119873) = ([(119863 + 2) (119863 minus 2)8 minus 119873(1198632 minus 1)]sdot [ (119863 + 2) (119863 minus 2)8 minus 1 minus 119873(1198632 minus 1)] sdot sdot sdot 1)times ([119873(1198632 minus 1)] [119873(1198632 minus 1) minus 1] sdot sdot sdot 1)119880 = 0 997904rArr
119873 le (119863 + 2) (119863 minus 2)8 (1198632 minus 1)
(21)
This equation indicates that for a (119863+1)-dimensional space-time the dimension of the Lie algebra should be equal to orless than a critical value Under these conditions the Chern-Simons gravity is free from anomalies and we do not needan extra manifold On the other hand as we show in (15)the dimension of the algebra determines the dimension ofthe field-strengthThis means that for a Lie-119873-algebra field-strengths should have at most 119873 indices For example for a
Advances in High Energy Physics 7
manifoldwith 11 dimensions the algebra can be of order threeas predicted in recent papers [18 19] and field-strengths mayhave three indices In fact in above equation we have shownthat the physics of an 11-dimensional spacing manifold plusa 3-dimensional algebraic manifold is equal to the physicsof 14-dimensional manifold This helps us to understandwhy 119872-theory with Lie-three algebra is a true theory andsolves many problems in physics In fact119872-theory with Lie-three algebra lives on 14-dimensional manifold where a 3-dimensional part of it corresponds to Lie-3-algebra and theother 11-dimensional part is related to space-time Also weshow that 119872-theory on 11-dimensional manifold could bethe anomaly-free if a three-dimensional algebraic manifoldis added to it or its spacial time is increased to 14 dimensionsThis 14-dimensional manifold can be broken to two parallel11-dimensional manifold which are connected by a three-dimensional Chern-Simons manifold
4 A Chern-Simons Manifold between Two11-Dimensional Manifolds in an 11 + 3Dimensional Space-Time
In the previous section we have found that for an eleven-dimensional manifold the suitable algebra which removesthe anomaly in Chern-Simons gravity is a three-dimensionalLie algebra This means that the rank of the fields can beof order two or three However (2) shows that the rank ofthe fields may be higher than three in eleven-dimensionalsupergravity Thus in Chern-Simons gravity theory whichlives on an eleven-dimensional manifold some extra anoma-lies are expected to show up To remove them we assumethat there is another 11-dimensional manifold in the 14-dimensional space-time which interacts with the first oneby exchanging Chern-Simons fields These fields producea Chern-Simons manifold that connects these two eleven-dimensional manifolds (see Figure 1)Thus in this model wehave two 119866119866 terms which live on 11-dimensional manifolds(see (2)) and two 119862119866119866 terms in the bulk so that each of theminteracts with one of the 11-dimensional manifolds
We can write the supergravity in 14-dimensional space-time as follows
119878SUGRA-14 = int119872119873=3
(int11987211
119866119866 + int11987211
119862119866119866+ int11987211
119862119866119866 + int11987211
119866119866) = (int11987214
119862119866119866+ int11987214
119862119866119866 + int11987214
119862119866119866 + int11987214
119862119866119866) (22)
In this equation119862119866119866 and119862119866119866 are related to the Chern-Simons gravities which live on the two eleven-dimensionalmanifolds and are extracted from 119866119866 and 119866119866 terms Also119862119866119866 and119862119866119866 correspond to the Chern-Simons fields whichare exchanged between the two manifolds in 14-dimensionalspace-time By generalizing the results of (3) (6) and (11) weget
119866119868119869119870119871 sim minus (119865119868119869119865119870119871 minus 119877119868119869119877119870119871) + 120597119868120601120597119869120601119877119870119871 sdot sdot sdot
119866119868119869119870119871 sim minus (119865119868119869119865119870119871 minus 119877119868119869119877119870119871) + 120597119868120601120597119869120601119877119870119871 sdot sdot sdot120575119862119868119894119868119895119868119896 sim minus120598119868119896 tr(119865119868119894119868119895 minus 119877119868119894119868119895 + 120597119868119894120601120597119868119895120601
minus 12120598119868119894119868119895119868119896119868119898119877119868119894119868119895119868119896119868119898 + sdot sdot sdot) (23)
Here the 119865rsquos 119877rsquos and 120601rsquos live on one of the supergravitymanifolds as depicted in Figure 1 whereas the 119865rsquos 119877rsquos and 120601rsquosare fields of the other supergravity manifold To obtain theirrelations we should make use of (12) and the gauge variationof the actions (22) in doing so we obtain
These results show that to remove the anomaly in 14-dimensional space-time coordinates and fields in one of theeleven-dimensionalmanifolds should be equal to coordinatesand fields in the othermanifold in addition to one extra 119894Thisimplies that time- or space-like coordinates and fields in onemanifold transmute into space- or time-like coordinates andfields of anothermanifold For example the zeroth coordinatewhich is known as time on one manifold will transmute intoa space coordinate of the other manifold If our universe withone time and three space coordinates is located on one of themanifolds an antiuniverse with one space and three times islocated in the other manifold
Now we shall show that by substituting the results of (24)into the action of (22) the topology of the 14-dimensionalmanifold tends to one This means that the world with allits matter began from a point and it may be thought of asa signature of Big Bang in our proposal To this end using(6) (7) (9) and (12) we rewrite 119862119866119866 terms in terms ofderivatives of scalar strings
On the other hand results in (24) show that (119883119868119894 119883119868119895 =minus119883119868119894 119883119868119895) Thus we can conclude that the action givenabove tends to an action on the three-dimensional manifold
119883119868119894 119883119868119895 = minus 119883119868119894 119883119868119895119878SUGRA-14 = 119896int
119872119873=3int11987211
(12057511 (119910)) = 119896int119872119873=3
997888rarr119896412058711987733
(28)
Advances in High Energy Physics 9
14-dimensionalspace-time
Chern-Simons manifold
11-dimensionalmanifold11-dimensional
manifold
Point
Figure 2 14-dimensional manifold shrinks to one point
where119877 is the radius ofmanifold By redefining scalar stringswe are able to show that the action of supergravity shrinks toone
This equation yields results that deserve our commentsIn fact two eleven-dimensional manifolds and the bulkinteract with each other via different types of 119862 119866 andChern-Simons-fields When we sum up supergravities thatlive on these manifolds and consider fields in the spacebetween them we get supergravity in 14-dimensional spaceBy canceling the anomaly in this new supergravity we canobtain the relations between fields By substituting theserelations into the action of the 14-dimensional supergravitywe simply obtain one This means that the 14-dimensionalmanifold with all its matter content can be topologicallyshrunk to one point (see Figure 2) In fact the system ofthe world began from this point and then expanded and
constructed a 14-dimensional world similar to what happensin a Big Bang theory
Another interesting result that comes out of our theoryconcerns the reduction of the action of the world to oneThismeans that the origin of allmatter or fields is the same In factat the first stage of the world we have only a constant energywhich lives on 14-dimensional manifold Then by internalinteractions two 11-dimensional manifolds are createdThesemanifolds interact with each other via an extraChern-Simonsmanifold By exchanging the energy between 11-dimensionalmanifolds their energies are changed and two differentactions are produced Also as due to interaction betweenstrings their shapes become different and different types offields like spinors and bosons are emerged
5 Summary and Final Considerations
In this paper we have shown that the Chern-Simons termsof modified gravity may be understood as due to the inter-action between two 11-dimensional manifolds in an 11 + 3-dimensional space-time where 3 is the dimension of a Lie-type-algebra We also argue that there is a direct relationbetween the dimension of the algebra and the dimensionof the manifold For example for 11-dimensional world thedimension of the Lie-algebra is three If the rank of the fieldswhich live in one manifold becomes larger than the rankof the algebra there emerges an anomaly This anomaly isproduced as an effect of connecting fields in the bulk tofields which live in the manifold To cancel this anomaly we
10 Advances in High Energy Physics
need to introduce another 11-dimensional manifold in the11+3-dimensional space-timewhich interacts with the initialmanifold by exchanging Chern-Simons terms These Chern-Simons terms produce an extra manifold If we sum up thetopology of the 11-dimensional manifolds and the topologyof the Chern-Simons manifold we can show that the totaltopology shrinks to one which is consistent with predictionsof the Big Bang theory
Our proposal is actually another version of the 119866-theory in [15 16] This approach removes anomalies in 11-dimensional 119872-theory by a generalization of dimension to14 In this research contribution we have shown that thephysics of an 11-dimensional spacing manifold plus a 3-dimensional algebraic manifold is the same as the physics of14-dimensional manifold In fact 11-dimensional 119872-theoryis a free-anomaly theory whenever a three-dimensionalalgebraic manifold is added to it This 14-dimensional man-ifold can be broken to two 11-dimensional manifolds whichare connected by a 3-dimensional Chern-Simons theoryAnother interesting result that comes out is the reduction ofthe action of the world to one This means that the originof all matter or fields is the same In fact at the first stageof the world we have only a constant energy which lives on14-dimensional manifold Then by internal interactions two11-dimensional manifolds come outThesemanifolds interactwith each other via one extra Chern-Simons manifold Byexchanging the energy between 11-dimensional manifoldstheir energies are changed and two different actions areproduced Also as due to interaction between strings theirshapes differ and diverse types of fields like spinors andbosons emerge
To conclude we would like to point out that that all ourtreatment has been restricted to the purely bosonic sectorof 11-dimensional supergravity whose on-shell multipletbesides the metric tensor and the 3-form gauge potentialincludes the gravitino fieldThe latter has not been consideredhere we have restricted ourselves to the bosonic sectorHowever it would be a further task to inspect how theinclusion of the gravitino would affect our developmentspossibly changing the dimension of the Lie algebra to beadded to the 11-dimensional manifold By including thefermion it is no longer ensured that the local supersymmetryof the 11 + 3-dimensional supergravity action remains validThe change in the dimension of the Lie algebra could in turngive rise to new terms so that the Chern-Simons modifiedgravity would be further extended as a result of including thegravitino We intend to pursue an investigation on this issueand to report on it elsewhere
In our model the three-Lie algebra that we add to passfrom 11 to 14119863 is geometrical It is shown that all anomalieswhich are produced by axial bosonic fields in 10- and 11-dimensional theories can be removed on 14-dimensionalmanifold Maybe a question arises ldquowhich is the effect offermionic anomalies on the number of dimensionsrdquo It mustbe clear that this paper is the first step for solving anomaliesin 11-dimensional theories In the next step by applyingthe model of this paper for fermions the exact number ofdimensions of the world can be estimated
Appendix
Details of Calculations
In this section we will bring details of calculations First letus disclose more aspects of (6)
1205981198681198941198681015840119894 (1205971198681015840119894119860119868119895 minus 1205971198681015840119894 (120598119868119895119868119896Γ120572120572119868119896) + 120597119868119894120601120597119868119895120601 + sdot sdot sdot)= 119865119868119894119868119895 minus 119877119868119894119868119895 + 120597119868119894120601120597119868119895120601 minus 12120598119868119894119868119895119868119896119868119898120601119877119868119894119868119895119868119896119868119898
+ sdot sdot sdot
(A1)
where 120601 is the Chern-Simons scalar field119860119868 is the gauge fieldΓ is related to the curvature (119877) and 119868 is a unit vector in thedirection of the coordinate which can be expanded in termsof derivatives of metric
Also it is needed to reobtain the results of (10) with doingsome mathematical calculations
In the equation above the first integration is in agreementwith previous predictions of Chern-Simons gravity in [7ndash9] and can be reduced to the four-dimensional Chern-Simonsmodified gravity of (1) Also the second integration isrelated to the interaction of gauge fields with Chern-Simonsfields Thus this model not only produces the Chern-Simonsmodified gravity but also exhibits some modifications to it
Now we can explain the mathematical model for obtain-ing results of (12)
The first line of this equation removes the anomaly on the 11-dimensional manifold of (4) however the second and thirdlines show that extra anomalies can emerge due to the Chern-Simons fields
At this stage we can consider the process of obtaining(24)
119878SUGRA-14 = 120575 (int11987214
119862119866119866 + int11987214
119862119866119866 + int11987214
119862119866119866+ int11987214
119862119866119866) = int11987214
5sum119899=1
(tr119865119899 minus tr119877119899 + tr (1198651198991198775minus119899))+ int11987214
5sum119899=1
119899sum119869=0
119899minus119869sum119898=0
(tr119865119869119865119898 minus tr119877119869119877119898 + tr (1198651198691198775minus1198691198651198981198775minus119898))+ int11987214
5sum119899=1
119899sum119869=0
119899minus119869sum119898=0
(tr119865119898119865119869 minus tr119877119898119877119869 + tr (1198651198981198775minus1198981198651198691198775minus119869))+ int11987214
5sum119899=1
(tr119865119899 minus tr119877119899 + tr (1198651198991198775minus119899))+ int11987210+119872119873=3
The authors declare that they have no conflicts of interest
Acknowledgments
The work of Alireza Sepehri has been supported financiallyby the Research Institute for Astronomy and Astrophysics ofMaragha (RIAAM) Iran under Research Project no 15237-75
References
[1] R Jackiw and S-Y Pi ldquoChern-Simons modification of generalrelativityrdquo Physical Review D Particles Fields Gravitation andCosmology vol 68 Article ID 104012 2003
[2] B Pereira-Dias C A Hernaski and J A Helayel-Neto ldquoProb-ing the effects of lorentz-symmetry violating chern-simons andricci-cotton terms in higher derivative gravityrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 83 no 82011
[3] K Konno T Matsuyama and S Tanda ldquoDoes a black holerotate in Chern-Simons modified gravityrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 76 no 2Article ID 024009 2007
[4] D Guarrera and A J Hariton ldquoPapapetrou energy-momentumtensor for Chern-Simons modified gravityrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 76 no 4 2007
Advances in High Energy Physics 13
[5] K K Nandi I R Kizirgulov O V Mikolaychuk N P Miko-laychuk and A A Potapov ldquoQuantum phase shift in Chern-Simons modified gravityrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 79 no 8 Article ID 0830062009
[6] P J Porfırio J B Fonseca-Neto J R Nascimento A Petrov JRicardo and A Santos ldquoChern-Simons modified gravity andclosed timelike curvesrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 94 no 4 2016
[7] S Alexander and N Yunes ldquoChernndashSimons modified generalrelativityrdquo Physics Reports vol 480 no 1-2 pp 1ndash55 2009
[8] S Chen and J Jing ldquoGeodetic precession and strong gravita-tional lensing in dynamical Chern-Simons-modified gravityrdquoClassical and QuantumGravity vol 27 no 22 225006 16 pages2010
[9] N Yunes and C F Sopuerta ldquoPerturbations of Schwarzschildblack holes in Chern-Simons modified gravityrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 77 no 62008
[10] F Canfora A Giacomini and A R Zerwekh ldquoKaluza-Kleintheory in the limit of large number of extra dimensionsrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 80 no 8 Article ID 084039 2009
[11] S Wu ldquoGeneral rotating charged Kaluza-Klein AdS black holesin higher dimensionsrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 83 no 12 2011
[12] T Houri and K Yamamoto ldquoKilling-Yano symmetry of Kaluza-KLEin black holes in five dimensionsrdquo Classical and QuantumGravity vol 30 no 7 075013 21 pages 2013
[13] T Tatsuoka H Ishihara M Kimura and K MatsunoldquoExtremal charged black holes with a twisted extra dimensionrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 85 no 4 2012
[14] P Horava and E Witten ldquoHeterotic and Type I string dynamicsfrom eleven dimensionsrdquo Nuclear Physics B vol 460 no 3Article ID 9510209 pp 506ndash524 1996
[15] A Sepehri and R Pincak ldquoThe birth of the universe in a newG-theory approachrdquo Modern Physics Letters A vol 32 no 5Article ID 1750033 1750033 32 pages 2017
[16] A Sepehri A Pradhan R Pincak F Rahaman A Beeshamand T Ghaffary ldquoBirth of the GUP and its effect on theentropy of the universe in Lie-N-algebrardquo International Journalof Geometric Methods in Modern Physics vol 14 no 9 ArticleID 1750130 2017
[17] P Horava and E Witten ldquoEleven-dimensional supergravity ona manifold with boundaryrdquo Nuclear Physics B vol 475 pp 94ndash114 1996
[18] J Bagger and N Lambert ldquoGauge symmetry and supersymme-try of multiple M2-branesrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 77 no 6 065008 6 pages 2008
[19] P Ho and Y Matsuo ldquoM5 from M2rdquo Journal of High EnergyPhysics vol 2008 no 06 p 105 2008
regard both algebraic (119872119873) and space-time (119872119863) manifoldsto achieve the exact results For the 119873-dimensional algebrawe introduce the following fields
120575[11986911198692sdotsdotsdot119869119873] = 12057511986911198692sdotsdotsdot119869119873 minus 12057511986921198691sdotsdotsdot119869119873 + sdot sdot sdot (16)
Here 120575 is the generalized Kronecker delta With defi-nitions in (15) we can obtain the explicit form of the 119873-dimensional Nambu-Poisson brackets in terms of fields
(11986511986911 sdotsdotsdot1198691119873 minus 12059711986915 sdot sdot sdot 120597119869111987311987711986911 sdotsdotsdot11986914)times (11986511986921 sdotsdotsdot1198692119873 minus 12059711986925 sdot sdot sdot 120597119869211987311987711986921 sdotsdotsdot11986924) sdot sdot sdot (11986511986911198632sdotsdotsdot1198691198632119873 minus 12059711986911986325sdot sdot sdot 1205971198691198632119873 11987711986911986321 sdotsdotsdot11986911986324 )
(18)
where 119885 is a constant related to the algebra This equationshows that the gauge variation of the action depends onthe rank-119873 field-strength The action above is not actuallydirectly zero and there emerges an anomaly Now we useproperties of 120598 and rewrite (18) as below
119882(119863119873) = ([(119863 + 2) (119863 minus 2)8 minus 119873(1198632 minus 1)]sdot [ (119863 + 2) (119863 minus 2)8 minus 1 minus 119873(1198632 minus 1)] sdot sdot sdot 1)sdot ([119873(1198632 minus 1)] [119873(1198632 minus 1) minus 1] sdot sdot sdot 1)119880 (120575)
(20)
where119880 is a function of the generalized Kronecker delta Onthe other hand 120575119878119862119866119866|119863+1 has been added to the main actionof supergravity to remove its anomaly Thus we can write
119882(119863119873) = ([(119863 + 2) (119863 minus 2)8 minus 119873(1198632 minus 1)]sdot [ (119863 + 2) (119863 minus 2)8 minus 1 minus 119873(1198632 minus 1)] sdot sdot sdot 1)times ([119873(1198632 minus 1)] [119873(1198632 minus 1) minus 1] sdot sdot sdot 1)119880 = 0 997904rArr
119873 le (119863 + 2) (119863 minus 2)8 (1198632 minus 1)
(21)
This equation indicates that for a (119863+1)-dimensional space-time the dimension of the Lie algebra should be equal to orless than a critical value Under these conditions the Chern-Simons gravity is free from anomalies and we do not needan extra manifold On the other hand as we show in (15)the dimension of the algebra determines the dimension ofthe field-strengthThis means that for a Lie-119873-algebra field-strengths should have at most 119873 indices For example for a
Advances in High Energy Physics 7
manifoldwith 11 dimensions the algebra can be of order threeas predicted in recent papers [18 19] and field-strengths mayhave three indices In fact in above equation we have shownthat the physics of an 11-dimensional spacing manifold plusa 3-dimensional algebraic manifold is equal to the physicsof 14-dimensional manifold This helps us to understandwhy 119872-theory with Lie-three algebra is a true theory andsolves many problems in physics In fact119872-theory with Lie-three algebra lives on 14-dimensional manifold where a 3-dimensional part of it corresponds to Lie-3-algebra and theother 11-dimensional part is related to space-time Also weshow that 119872-theory on 11-dimensional manifold could bethe anomaly-free if a three-dimensional algebraic manifoldis added to it or its spacial time is increased to 14 dimensionsThis 14-dimensional manifold can be broken to two parallel11-dimensional manifold which are connected by a three-dimensional Chern-Simons manifold
4 A Chern-Simons Manifold between Two11-Dimensional Manifolds in an 11 + 3Dimensional Space-Time
In the previous section we have found that for an eleven-dimensional manifold the suitable algebra which removesthe anomaly in Chern-Simons gravity is a three-dimensionalLie algebra This means that the rank of the fields can beof order two or three However (2) shows that the rank ofthe fields may be higher than three in eleven-dimensionalsupergravity Thus in Chern-Simons gravity theory whichlives on an eleven-dimensional manifold some extra anoma-lies are expected to show up To remove them we assumethat there is another 11-dimensional manifold in the 14-dimensional space-time which interacts with the first oneby exchanging Chern-Simons fields These fields producea Chern-Simons manifold that connects these two eleven-dimensional manifolds (see Figure 1)Thus in this model wehave two 119866119866 terms which live on 11-dimensional manifolds(see (2)) and two 119862119866119866 terms in the bulk so that each of theminteracts with one of the 11-dimensional manifolds
We can write the supergravity in 14-dimensional space-time as follows
119878SUGRA-14 = int119872119873=3
(int11987211
119866119866 + int11987211
119862119866119866+ int11987211
119862119866119866 + int11987211
119866119866) = (int11987214
119862119866119866+ int11987214
119862119866119866 + int11987214
119862119866119866 + int11987214
119862119866119866) (22)
In this equation119862119866119866 and119862119866119866 are related to the Chern-Simons gravities which live on the two eleven-dimensionalmanifolds and are extracted from 119866119866 and 119866119866 terms Also119862119866119866 and119862119866119866 correspond to the Chern-Simons fields whichare exchanged between the two manifolds in 14-dimensionalspace-time By generalizing the results of (3) (6) and (11) weget
119866119868119869119870119871 sim minus (119865119868119869119865119870119871 minus 119877119868119869119877119870119871) + 120597119868120601120597119869120601119877119870119871 sdot sdot sdot
119866119868119869119870119871 sim minus (119865119868119869119865119870119871 minus 119877119868119869119877119870119871) + 120597119868120601120597119869120601119877119870119871 sdot sdot sdot120575119862119868119894119868119895119868119896 sim minus120598119868119896 tr(119865119868119894119868119895 minus 119877119868119894119868119895 + 120597119868119894120601120597119868119895120601
minus 12120598119868119894119868119895119868119896119868119898119877119868119894119868119895119868119896119868119898 + sdot sdot sdot) (23)
Here the 119865rsquos 119877rsquos and 120601rsquos live on one of the supergravitymanifolds as depicted in Figure 1 whereas the 119865rsquos 119877rsquos and 120601rsquosare fields of the other supergravity manifold To obtain theirrelations we should make use of (12) and the gauge variationof the actions (22) in doing so we obtain
These results show that to remove the anomaly in 14-dimensional space-time coordinates and fields in one of theeleven-dimensionalmanifolds should be equal to coordinatesand fields in the othermanifold in addition to one extra 119894Thisimplies that time- or space-like coordinates and fields in onemanifold transmute into space- or time-like coordinates andfields of anothermanifold For example the zeroth coordinatewhich is known as time on one manifold will transmute intoa space coordinate of the other manifold If our universe withone time and three space coordinates is located on one of themanifolds an antiuniverse with one space and three times islocated in the other manifold
Now we shall show that by substituting the results of (24)into the action of (22) the topology of the 14-dimensionalmanifold tends to one This means that the world with allits matter began from a point and it may be thought of asa signature of Big Bang in our proposal To this end using(6) (7) (9) and (12) we rewrite 119862119866119866 terms in terms ofderivatives of scalar strings
On the other hand results in (24) show that (119883119868119894 119883119868119895 =minus119883119868119894 119883119868119895) Thus we can conclude that the action givenabove tends to an action on the three-dimensional manifold
119883119868119894 119883119868119895 = minus 119883119868119894 119883119868119895119878SUGRA-14 = 119896int
119872119873=3int11987211
(12057511 (119910)) = 119896int119872119873=3
997888rarr119896412058711987733
(28)
Advances in High Energy Physics 9
14-dimensionalspace-time
Chern-Simons manifold
11-dimensionalmanifold11-dimensional
manifold
Point
Figure 2 14-dimensional manifold shrinks to one point
where119877 is the radius ofmanifold By redefining scalar stringswe are able to show that the action of supergravity shrinks toone
This equation yields results that deserve our commentsIn fact two eleven-dimensional manifolds and the bulkinteract with each other via different types of 119862 119866 andChern-Simons-fields When we sum up supergravities thatlive on these manifolds and consider fields in the spacebetween them we get supergravity in 14-dimensional spaceBy canceling the anomaly in this new supergravity we canobtain the relations between fields By substituting theserelations into the action of the 14-dimensional supergravitywe simply obtain one This means that the 14-dimensionalmanifold with all its matter content can be topologicallyshrunk to one point (see Figure 2) In fact the system ofthe world began from this point and then expanded and
constructed a 14-dimensional world similar to what happensin a Big Bang theory
Another interesting result that comes out of our theoryconcerns the reduction of the action of the world to oneThismeans that the origin of allmatter or fields is the same In factat the first stage of the world we have only a constant energywhich lives on 14-dimensional manifold Then by internalinteractions two 11-dimensional manifolds are createdThesemanifolds interact with each other via an extraChern-Simonsmanifold By exchanging the energy between 11-dimensionalmanifolds their energies are changed and two differentactions are produced Also as due to interaction betweenstrings their shapes become different and different types offields like spinors and bosons are emerged
5 Summary and Final Considerations
In this paper we have shown that the Chern-Simons termsof modified gravity may be understood as due to the inter-action between two 11-dimensional manifolds in an 11 + 3-dimensional space-time where 3 is the dimension of a Lie-type-algebra We also argue that there is a direct relationbetween the dimension of the algebra and the dimensionof the manifold For example for 11-dimensional world thedimension of the Lie-algebra is three If the rank of the fieldswhich live in one manifold becomes larger than the rankof the algebra there emerges an anomaly This anomaly isproduced as an effect of connecting fields in the bulk tofields which live in the manifold To cancel this anomaly we
10 Advances in High Energy Physics
need to introduce another 11-dimensional manifold in the11+3-dimensional space-timewhich interacts with the initialmanifold by exchanging Chern-Simons terms These Chern-Simons terms produce an extra manifold If we sum up thetopology of the 11-dimensional manifolds and the topologyof the Chern-Simons manifold we can show that the totaltopology shrinks to one which is consistent with predictionsof the Big Bang theory
Our proposal is actually another version of the 119866-theory in [15 16] This approach removes anomalies in 11-dimensional 119872-theory by a generalization of dimension to14 In this research contribution we have shown that thephysics of an 11-dimensional spacing manifold plus a 3-dimensional algebraic manifold is the same as the physics of14-dimensional manifold In fact 11-dimensional 119872-theoryis a free-anomaly theory whenever a three-dimensionalalgebraic manifold is added to it This 14-dimensional man-ifold can be broken to two 11-dimensional manifolds whichare connected by a 3-dimensional Chern-Simons theoryAnother interesting result that comes out is the reduction ofthe action of the world to one This means that the originof all matter or fields is the same In fact at the first stageof the world we have only a constant energy which lives on14-dimensional manifold Then by internal interactions two11-dimensional manifolds come outThesemanifolds interactwith each other via one extra Chern-Simons manifold Byexchanging the energy between 11-dimensional manifoldstheir energies are changed and two different actions areproduced Also as due to interaction between strings theirshapes differ and diverse types of fields like spinors andbosons emerge
To conclude we would like to point out that that all ourtreatment has been restricted to the purely bosonic sectorof 11-dimensional supergravity whose on-shell multipletbesides the metric tensor and the 3-form gauge potentialincludes the gravitino fieldThe latter has not been consideredhere we have restricted ourselves to the bosonic sectorHowever it would be a further task to inspect how theinclusion of the gravitino would affect our developmentspossibly changing the dimension of the Lie algebra to beadded to the 11-dimensional manifold By including thefermion it is no longer ensured that the local supersymmetryof the 11 + 3-dimensional supergravity action remains validThe change in the dimension of the Lie algebra could in turngive rise to new terms so that the Chern-Simons modifiedgravity would be further extended as a result of including thegravitino We intend to pursue an investigation on this issueand to report on it elsewhere
In our model the three-Lie algebra that we add to passfrom 11 to 14119863 is geometrical It is shown that all anomalieswhich are produced by axial bosonic fields in 10- and 11-dimensional theories can be removed on 14-dimensionalmanifold Maybe a question arises ldquowhich is the effect offermionic anomalies on the number of dimensionsrdquo It mustbe clear that this paper is the first step for solving anomaliesin 11-dimensional theories In the next step by applyingthe model of this paper for fermions the exact number ofdimensions of the world can be estimated
Appendix
Details of Calculations
In this section we will bring details of calculations First letus disclose more aspects of (6)
1205981198681198941198681015840119894 (1205971198681015840119894119860119868119895 minus 1205971198681015840119894 (120598119868119895119868119896Γ120572120572119868119896) + 120597119868119894120601120597119868119895120601 + sdot sdot sdot)= 119865119868119894119868119895 minus 119877119868119894119868119895 + 120597119868119894120601120597119868119895120601 minus 12120598119868119894119868119895119868119896119868119898120601119877119868119894119868119895119868119896119868119898
+ sdot sdot sdot
(A1)
where 120601 is the Chern-Simons scalar field119860119868 is the gauge fieldΓ is related to the curvature (119877) and 119868 is a unit vector in thedirection of the coordinate which can be expanded in termsof derivatives of metric
Also it is needed to reobtain the results of (10) with doingsome mathematical calculations
In the equation above the first integration is in agreementwith previous predictions of Chern-Simons gravity in [7ndash9] and can be reduced to the four-dimensional Chern-Simonsmodified gravity of (1) Also the second integration isrelated to the interaction of gauge fields with Chern-Simonsfields Thus this model not only produces the Chern-Simonsmodified gravity but also exhibits some modifications to it
Now we can explain the mathematical model for obtain-ing results of (12)
The first line of this equation removes the anomaly on the 11-dimensional manifold of (4) however the second and thirdlines show that extra anomalies can emerge due to the Chern-Simons fields
At this stage we can consider the process of obtaining(24)
119878SUGRA-14 = 120575 (int11987214
119862119866119866 + int11987214
119862119866119866 + int11987214
119862119866119866+ int11987214
119862119866119866) = int11987214
5sum119899=1
(tr119865119899 minus tr119877119899 + tr (1198651198991198775minus119899))+ int11987214
5sum119899=1
119899sum119869=0
119899minus119869sum119898=0
(tr119865119869119865119898 minus tr119877119869119877119898 + tr (1198651198691198775minus1198691198651198981198775minus119898))+ int11987214
5sum119899=1
119899sum119869=0
119899minus119869sum119898=0
(tr119865119898119865119869 minus tr119877119898119877119869 + tr (1198651198981198775minus1198981198651198691198775minus119869))+ int11987214
5sum119899=1
(tr119865119899 minus tr119877119899 + tr (1198651198991198775minus119899))+ int11987210+119872119873=3
The authors declare that they have no conflicts of interest
Acknowledgments
The work of Alireza Sepehri has been supported financiallyby the Research Institute for Astronomy and Astrophysics ofMaragha (RIAAM) Iran under Research Project no 15237-75
References
[1] R Jackiw and S-Y Pi ldquoChern-Simons modification of generalrelativityrdquo Physical Review D Particles Fields Gravitation andCosmology vol 68 Article ID 104012 2003
[2] B Pereira-Dias C A Hernaski and J A Helayel-Neto ldquoProb-ing the effects of lorentz-symmetry violating chern-simons andricci-cotton terms in higher derivative gravityrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 83 no 82011
[3] K Konno T Matsuyama and S Tanda ldquoDoes a black holerotate in Chern-Simons modified gravityrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 76 no 2Article ID 024009 2007
[4] D Guarrera and A J Hariton ldquoPapapetrou energy-momentumtensor for Chern-Simons modified gravityrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 76 no 4 2007
Advances in High Energy Physics 13
[5] K K Nandi I R Kizirgulov O V Mikolaychuk N P Miko-laychuk and A A Potapov ldquoQuantum phase shift in Chern-Simons modified gravityrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 79 no 8 Article ID 0830062009
[6] P J Porfırio J B Fonseca-Neto J R Nascimento A Petrov JRicardo and A Santos ldquoChern-Simons modified gravity andclosed timelike curvesrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 94 no 4 2016
[7] S Alexander and N Yunes ldquoChernndashSimons modified generalrelativityrdquo Physics Reports vol 480 no 1-2 pp 1ndash55 2009
[8] S Chen and J Jing ldquoGeodetic precession and strong gravita-tional lensing in dynamical Chern-Simons-modified gravityrdquoClassical and QuantumGravity vol 27 no 22 225006 16 pages2010
[9] N Yunes and C F Sopuerta ldquoPerturbations of Schwarzschildblack holes in Chern-Simons modified gravityrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 77 no 62008
[10] F Canfora A Giacomini and A R Zerwekh ldquoKaluza-Kleintheory in the limit of large number of extra dimensionsrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 80 no 8 Article ID 084039 2009
[11] S Wu ldquoGeneral rotating charged Kaluza-Klein AdS black holesin higher dimensionsrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 83 no 12 2011
[12] T Houri and K Yamamoto ldquoKilling-Yano symmetry of Kaluza-KLEin black holes in five dimensionsrdquo Classical and QuantumGravity vol 30 no 7 075013 21 pages 2013
[13] T Tatsuoka H Ishihara M Kimura and K MatsunoldquoExtremal charged black holes with a twisted extra dimensionrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 85 no 4 2012
[14] P Horava and E Witten ldquoHeterotic and Type I string dynamicsfrom eleven dimensionsrdquo Nuclear Physics B vol 460 no 3Article ID 9510209 pp 506ndash524 1996
[15] A Sepehri and R Pincak ldquoThe birth of the universe in a newG-theory approachrdquo Modern Physics Letters A vol 32 no 5Article ID 1750033 1750033 32 pages 2017
[16] A Sepehri A Pradhan R Pincak F Rahaman A Beeshamand T Ghaffary ldquoBirth of the GUP and its effect on theentropy of the universe in Lie-N-algebrardquo International Journalof Geometric Methods in Modern Physics vol 14 no 9 ArticleID 1750130 2017
[17] P Horava and E Witten ldquoEleven-dimensional supergravity ona manifold with boundaryrdquo Nuclear Physics B vol 475 pp 94ndash114 1996
[18] J Bagger and N Lambert ldquoGauge symmetry and supersymme-try of multiple M2-branesrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 77 no 6 065008 6 pages 2008
[19] P Ho and Y Matsuo ldquoM5 from M2rdquo Journal of High EnergyPhysics vol 2008 no 06 p 105 2008
manifoldwith 11 dimensions the algebra can be of order threeas predicted in recent papers [18 19] and field-strengths mayhave three indices In fact in above equation we have shownthat the physics of an 11-dimensional spacing manifold plusa 3-dimensional algebraic manifold is equal to the physicsof 14-dimensional manifold This helps us to understandwhy 119872-theory with Lie-three algebra is a true theory andsolves many problems in physics In fact119872-theory with Lie-three algebra lives on 14-dimensional manifold where a 3-dimensional part of it corresponds to Lie-3-algebra and theother 11-dimensional part is related to space-time Also weshow that 119872-theory on 11-dimensional manifold could bethe anomaly-free if a three-dimensional algebraic manifoldis added to it or its spacial time is increased to 14 dimensionsThis 14-dimensional manifold can be broken to two parallel11-dimensional manifold which are connected by a three-dimensional Chern-Simons manifold
4 A Chern-Simons Manifold between Two11-Dimensional Manifolds in an 11 + 3Dimensional Space-Time
In the previous section we have found that for an eleven-dimensional manifold the suitable algebra which removesthe anomaly in Chern-Simons gravity is a three-dimensionalLie algebra This means that the rank of the fields can beof order two or three However (2) shows that the rank ofthe fields may be higher than three in eleven-dimensionalsupergravity Thus in Chern-Simons gravity theory whichlives on an eleven-dimensional manifold some extra anoma-lies are expected to show up To remove them we assumethat there is another 11-dimensional manifold in the 14-dimensional space-time which interacts with the first oneby exchanging Chern-Simons fields These fields producea Chern-Simons manifold that connects these two eleven-dimensional manifolds (see Figure 1)Thus in this model wehave two 119866119866 terms which live on 11-dimensional manifolds(see (2)) and two 119862119866119866 terms in the bulk so that each of theminteracts with one of the 11-dimensional manifolds
We can write the supergravity in 14-dimensional space-time as follows
119878SUGRA-14 = int119872119873=3
(int11987211
119866119866 + int11987211
119862119866119866+ int11987211
119862119866119866 + int11987211
119866119866) = (int11987214
119862119866119866+ int11987214
119862119866119866 + int11987214
119862119866119866 + int11987214
119862119866119866) (22)
In this equation119862119866119866 and119862119866119866 are related to the Chern-Simons gravities which live on the two eleven-dimensionalmanifolds and are extracted from 119866119866 and 119866119866 terms Also119862119866119866 and119862119866119866 correspond to the Chern-Simons fields whichare exchanged between the two manifolds in 14-dimensionalspace-time By generalizing the results of (3) (6) and (11) weget
119866119868119869119870119871 sim minus (119865119868119869119865119870119871 minus 119877119868119869119877119870119871) + 120597119868120601120597119869120601119877119870119871 sdot sdot sdot
119866119868119869119870119871 sim minus (119865119868119869119865119870119871 minus 119877119868119869119877119870119871) + 120597119868120601120597119869120601119877119870119871 sdot sdot sdot120575119862119868119894119868119895119868119896 sim minus120598119868119896 tr(119865119868119894119868119895 minus 119877119868119894119868119895 + 120597119868119894120601120597119868119895120601
minus 12120598119868119894119868119895119868119896119868119898119877119868119894119868119895119868119896119868119898 + sdot sdot sdot) (23)
Here the 119865rsquos 119877rsquos and 120601rsquos live on one of the supergravitymanifolds as depicted in Figure 1 whereas the 119865rsquos 119877rsquos and 120601rsquosare fields of the other supergravity manifold To obtain theirrelations we should make use of (12) and the gauge variationof the actions (22) in doing so we obtain
These results show that to remove the anomaly in 14-dimensional space-time coordinates and fields in one of theeleven-dimensionalmanifolds should be equal to coordinatesand fields in the othermanifold in addition to one extra 119894Thisimplies that time- or space-like coordinates and fields in onemanifold transmute into space- or time-like coordinates andfields of anothermanifold For example the zeroth coordinatewhich is known as time on one manifold will transmute intoa space coordinate of the other manifold If our universe withone time and three space coordinates is located on one of themanifolds an antiuniverse with one space and three times islocated in the other manifold
Now we shall show that by substituting the results of (24)into the action of (22) the topology of the 14-dimensionalmanifold tends to one This means that the world with allits matter began from a point and it may be thought of asa signature of Big Bang in our proposal To this end using(6) (7) (9) and (12) we rewrite 119862119866119866 terms in terms ofderivatives of scalar strings
On the other hand results in (24) show that (119883119868119894 119883119868119895 =minus119883119868119894 119883119868119895) Thus we can conclude that the action givenabove tends to an action on the three-dimensional manifold
119883119868119894 119883119868119895 = minus 119883119868119894 119883119868119895119878SUGRA-14 = 119896int
119872119873=3int11987211
(12057511 (119910)) = 119896int119872119873=3
997888rarr119896412058711987733
(28)
Advances in High Energy Physics 9
14-dimensionalspace-time
Chern-Simons manifold
11-dimensionalmanifold11-dimensional
manifold
Point
Figure 2 14-dimensional manifold shrinks to one point
where119877 is the radius ofmanifold By redefining scalar stringswe are able to show that the action of supergravity shrinks toone
This equation yields results that deserve our commentsIn fact two eleven-dimensional manifolds and the bulkinteract with each other via different types of 119862 119866 andChern-Simons-fields When we sum up supergravities thatlive on these manifolds and consider fields in the spacebetween them we get supergravity in 14-dimensional spaceBy canceling the anomaly in this new supergravity we canobtain the relations between fields By substituting theserelations into the action of the 14-dimensional supergravitywe simply obtain one This means that the 14-dimensionalmanifold with all its matter content can be topologicallyshrunk to one point (see Figure 2) In fact the system ofthe world began from this point and then expanded and
constructed a 14-dimensional world similar to what happensin a Big Bang theory
Another interesting result that comes out of our theoryconcerns the reduction of the action of the world to oneThismeans that the origin of allmatter or fields is the same In factat the first stage of the world we have only a constant energywhich lives on 14-dimensional manifold Then by internalinteractions two 11-dimensional manifolds are createdThesemanifolds interact with each other via an extraChern-Simonsmanifold By exchanging the energy between 11-dimensionalmanifolds their energies are changed and two differentactions are produced Also as due to interaction betweenstrings their shapes become different and different types offields like spinors and bosons are emerged
5 Summary and Final Considerations
In this paper we have shown that the Chern-Simons termsof modified gravity may be understood as due to the inter-action between two 11-dimensional manifolds in an 11 + 3-dimensional space-time where 3 is the dimension of a Lie-type-algebra We also argue that there is a direct relationbetween the dimension of the algebra and the dimensionof the manifold For example for 11-dimensional world thedimension of the Lie-algebra is three If the rank of the fieldswhich live in one manifold becomes larger than the rankof the algebra there emerges an anomaly This anomaly isproduced as an effect of connecting fields in the bulk tofields which live in the manifold To cancel this anomaly we
10 Advances in High Energy Physics
need to introduce another 11-dimensional manifold in the11+3-dimensional space-timewhich interacts with the initialmanifold by exchanging Chern-Simons terms These Chern-Simons terms produce an extra manifold If we sum up thetopology of the 11-dimensional manifolds and the topologyof the Chern-Simons manifold we can show that the totaltopology shrinks to one which is consistent with predictionsof the Big Bang theory
Our proposal is actually another version of the 119866-theory in [15 16] This approach removes anomalies in 11-dimensional 119872-theory by a generalization of dimension to14 In this research contribution we have shown that thephysics of an 11-dimensional spacing manifold plus a 3-dimensional algebraic manifold is the same as the physics of14-dimensional manifold In fact 11-dimensional 119872-theoryis a free-anomaly theory whenever a three-dimensionalalgebraic manifold is added to it This 14-dimensional man-ifold can be broken to two 11-dimensional manifolds whichare connected by a 3-dimensional Chern-Simons theoryAnother interesting result that comes out is the reduction ofthe action of the world to one This means that the originof all matter or fields is the same In fact at the first stageof the world we have only a constant energy which lives on14-dimensional manifold Then by internal interactions two11-dimensional manifolds come outThesemanifolds interactwith each other via one extra Chern-Simons manifold Byexchanging the energy between 11-dimensional manifoldstheir energies are changed and two different actions areproduced Also as due to interaction between strings theirshapes differ and diverse types of fields like spinors andbosons emerge
To conclude we would like to point out that that all ourtreatment has been restricted to the purely bosonic sectorof 11-dimensional supergravity whose on-shell multipletbesides the metric tensor and the 3-form gauge potentialincludes the gravitino fieldThe latter has not been consideredhere we have restricted ourselves to the bosonic sectorHowever it would be a further task to inspect how theinclusion of the gravitino would affect our developmentspossibly changing the dimension of the Lie algebra to beadded to the 11-dimensional manifold By including thefermion it is no longer ensured that the local supersymmetryof the 11 + 3-dimensional supergravity action remains validThe change in the dimension of the Lie algebra could in turngive rise to new terms so that the Chern-Simons modifiedgravity would be further extended as a result of including thegravitino We intend to pursue an investigation on this issueand to report on it elsewhere
In our model the three-Lie algebra that we add to passfrom 11 to 14119863 is geometrical It is shown that all anomalieswhich are produced by axial bosonic fields in 10- and 11-dimensional theories can be removed on 14-dimensionalmanifold Maybe a question arises ldquowhich is the effect offermionic anomalies on the number of dimensionsrdquo It mustbe clear that this paper is the first step for solving anomaliesin 11-dimensional theories In the next step by applyingthe model of this paper for fermions the exact number ofdimensions of the world can be estimated
Appendix
Details of Calculations
In this section we will bring details of calculations First letus disclose more aspects of (6)
1205981198681198941198681015840119894 (1205971198681015840119894119860119868119895 minus 1205971198681015840119894 (120598119868119895119868119896Γ120572120572119868119896) + 120597119868119894120601120597119868119895120601 + sdot sdot sdot)= 119865119868119894119868119895 minus 119877119868119894119868119895 + 120597119868119894120601120597119868119895120601 minus 12120598119868119894119868119895119868119896119868119898120601119877119868119894119868119895119868119896119868119898
+ sdot sdot sdot
(A1)
where 120601 is the Chern-Simons scalar field119860119868 is the gauge fieldΓ is related to the curvature (119877) and 119868 is a unit vector in thedirection of the coordinate which can be expanded in termsof derivatives of metric
Also it is needed to reobtain the results of (10) with doingsome mathematical calculations
In the equation above the first integration is in agreementwith previous predictions of Chern-Simons gravity in [7ndash9] and can be reduced to the four-dimensional Chern-Simonsmodified gravity of (1) Also the second integration isrelated to the interaction of gauge fields with Chern-Simonsfields Thus this model not only produces the Chern-Simonsmodified gravity but also exhibits some modifications to it
Now we can explain the mathematical model for obtain-ing results of (12)
The first line of this equation removes the anomaly on the 11-dimensional manifold of (4) however the second and thirdlines show that extra anomalies can emerge due to the Chern-Simons fields
At this stage we can consider the process of obtaining(24)
119878SUGRA-14 = 120575 (int11987214
119862119866119866 + int11987214
119862119866119866 + int11987214
119862119866119866+ int11987214
119862119866119866) = int11987214
5sum119899=1
(tr119865119899 minus tr119877119899 + tr (1198651198991198775minus119899))+ int11987214
5sum119899=1
119899sum119869=0
119899minus119869sum119898=0
(tr119865119869119865119898 minus tr119877119869119877119898 + tr (1198651198691198775minus1198691198651198981198775minus119898))+ int11987214
5sum119899=1
119899sum119869=0
119899minus119869sum119898=0
(tr119865119898119865119869 minus tr119877119898119877119869 + tr (1198651198981198775minus1198981198651198691198775minus119869))+ int11987214
5sum119899=1
(tr119865119899 minus tr119877119899 + tr (1198651198991198775minus119899))+ int11987210+119872119873=3
The authors declare that they have no conflicts of interest
Acknowledgments
The work of Alireza Sepehri has been supported financiallyby the Research Institute for Astronomy and Astrophysics ofMaragha (RIAAM) Iran under Research Project no 15237-75
References
[1] R Jackiw and S-Y Pi ldquoChern-Simons modification of generalrelativityrdquo Physical Review D Particles Fields Gravitation andCosmology vol 68 Article ID 104012 2003
[2] B Pereira-Dias C A Hernaski and J A Helayel-Neto ldquoProb-ing the effects of lorentz-symmetry violating chern-simons andricci-cotton terms in higher derivative gravityrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 83 no 82011
[3] K Konno T Matsuyama and S Tanda ldquoDoes a black holerotate in Chern-Simons modified gravityrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 76 no 2Article ID 024009 2007
[4] D Guarrera and A J Hariton ldquoPapapetrou energy-momentumtensor for Chern-Simons modified gravityrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 76 no 4 2007
Advances in High Energy Physics 13
[5] K K Nandi I R Kizirgulov O V Mikolaychuk N P Miko-laychuk and A A Potapov ldquoQuantum phase shift in Chern-Simons modified gravityrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 79 no 8 Article ID 0830062009
[6] P J Porfırio J B Fonseca-Neto J R Nascimento A Petrov JRicardo and A Santos ldquoChern-Simons modified gravity andclosed timelike curvesrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 94 no 4 2016
[7] S Alexander and N Yunes ldquoChernndashSimons modified generalrelativityrdquo Physics Reports vol 480 no 1-2 pp 1ndash55 2009
[8] S Chen and J Jing ldquoGeodetic precession and strong gravita-tional lensing in dynamical Chern-Simons-modified gravityrdquoClassical and QuantumGravity vol 27 no 22 225006 16 pages2010
[9] N Yunes and C F Sopuerta ldquoPerturbations of Schwarzschildblack holes in Chern-Simons modified gravityrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 77 no 62008
[10] F Canfora A Giacomini and A R Zerwekh ldquoKaluza-Kleintheory in the limit of large number of extra dimensionsrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 80 no 8 Article ID 084039 2009
[11] S Wu ldquoGeneral rotating charged Kaluza-Klein AdS black holesin higher dimensionsrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 83 no 12 2011
[12] T Houri and K Yamamoto ldquoKilling-Yano symmetry of Kaluza-KLEin black holes in five dimensionsrdquo Classical and QuantumGravity vol 30 no 7 075013 21 pages 2013
[13] T Tatsuoka H Ishihara M Kimura and K MatsunoldquoExtremal charged black holes with a twisted extra dimensionrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 85 no 4 2012
[14] P Horava and E Witten ldquoHeterotic and Type I string dynamicsfrom eleven dimensionsrdquo Nuclear Physics B vol 460 no 3Article ID 9510209 pp 506ndash524 1996
[15] A Sepehri and R Pincak ldquoThe birth of the universe in a newG-theory approachrdquo Modern Physics Letters A vol 32 no 5Article ID 1750033 1750033 32 pages 2017
[16] A Sepehri A Pradhan R Pincak F Rahaman A Beeshamand T Ghaffary ldquoBirth of the GUP and its effect on theentropy of the universe in Lie-N-algebrardquo International Journalof Geometric Methods in Modern Physics vol 14 no 9 ArticleID 1750130 2017
[17] P Horava and E Witten ldquoEleven-dimensional supergravity ona manifold with boundaryrdquo Nuclear Physics B vol 475 pp 94ndash114 1996
[18] J Bagger and N Lambert ldquoGauge symmetry and supersymme-try of multiple M2-branesrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 77 no 6 065008 6 pages 2008
[19] P Ho and Y Matsuo ldquoM5 from M2rdquo Journal of High EnergyPhysics vol 2008 no 06 p 105 2008
On the other hand results in (24) show that (119883119868119894 119883119868119895 =minus119883119868119894 119883119868119895) Thus we can conclude that the action givenabove tends to an action on the three-dimensional manifold
119883119868119894 119883119868119895 = minus 119883119868119894 119883119868119895119878SUGRA-14 = 119896int
119872119873=3int11987211
(12057511 (119910)) = 119896int119872119873=3
997888rarr119896412058711987733
(28)
Advances in High Energy Physics 9
14-dimensionalspace-time
Chern-Simons manifold
11-dimensionalmanifold11-dimensional
manifold
Point
Figure 2 14-dimensional manifold shrinks to one point
where119877 is the radius ofmanifold By redefining scalar stringswe are able to show that the action of supergravity shrinks toone
This equation yields results that deserve our commentsIn fact two eleven-dimensional manifolds and the bulkinteract with each other via different types of 119862 119866 andChern-Simons-fields When we sum up supergravities thatlive on these manifolds and consider fields in the spacebetween them we get supergravity in 14-dimensional spaceBy canceling the anomaly in this new supergravity we canobtain the relations between fields By substituting theserelations into the action of the 14-dimensional supergravitywe simply obtain one This means that the 14-dimensionalmanifold with all its matter content can be topologicallyshrunk to one point (see Figure 2) In fact the system ofthe world began from this point and then expanded and
constructed a 14-dimensional world similar to what happensin a Big Bang theory
Another interesting result that comes out of our theoryconcerns the reduction of the action of the world to oneThismeans that the origin of allmatter or fields is the same In factat the first stage of the world we have only a constant energywhich lives on 14-dimensional manifold Then by internalinteractions two 11-dimensional manifolds are createdThesemanifolds interact with each other via an extraChern-Simonsmanifold By exchanging the energy between 11-dimensionalmanifolds their energies are changed and two differentactions are produced Also as due to interaction betweenstrings their shapes become different and different types offields like spinors and bosons are emerged
5 Summary and Final Considerations
In this paper we have shown that the Chern-Simons termsof modified gravity may be understood as due to the inter-action between two 11-dimensional manifolds in an 11 + 3-dimensional space-time where 3 is the dimension of a Lie-type-algebra We also argue that there is a direct relationbetween the dimension of the algebra and the dimensionof the manifold For example for 11-dimensional world thedimension of the Lie-algebra is three If the rank of the fieldswhich live in one manifold becomes larger than the rankof the algebra there emerges an anomaly This anomaly isproduced as an effect of connecting fields in the bulk tofields which live in the manifold To cancel this anomaly we
10 Advances in High Energy Physics
need to introduce another 11-dimensional manifold in the11+3-dimensional space-timewhich interacts with the initialmanifold by exchanging Chern-Simons terms These Chern-Simons terms produce an extra manifold If we sum up thetopology of the 11-dimensional manifolds and the topologyof the Chern-Simons manifold we can show that the totaltopology shrinks to one which is consistent with predictionsof the Big Bang theory
Our proposal is actually another version of the 119866-theory in [15 16] This approach removes anomalies in 11-dimensional 119872-theory by a generalization of dimension to14 In this research contribution we have shown that thephysics of an 11-dimensional spacing manifold plus a 3-dimensional algebraic manifold is the same as the physics of14-dimensional manifold In fact 11-dimensional 119872-theoryis a free-anomaly theory whenever a three-dimensionalalgebraic manifold is added to it This 14-dimensional man-ifold can be broken to two 11-dimensional manifolds whichare connected by a 3-dimensional Chern-Simons theoryAnother interesting result that comes out is the reduction ofthe action of the world to one This means that the originof all matter or fields is the same In fact at the first stageof the world we have only a constant energy which lives on14-dimensional manifold Then by internal interactions two11-dimensional manifolds come outThesemanifolds interactwith each other via one extra Chern-Simons manifold Byexchanging the energy between 11-dimensional manifoldstheir energies are changed and two different actions areproduced Also as due to interaction between strings theirshapes differ and diverse types of fields like spinors andbosons emerge
To conclude we would like to point out that that all ourtreatment has been restricted to the purely bosonic sectorof 11-dimensional supergravity whose on-shell multipletbesides the metric tensor and the 3-form gauge potentialincludes the gravitino fieldThe latter has not been consideredhere we have restricted ourselves to the bosonic sectorHowever it would be a further task to inspect how theinclusion of the gravitino would affect our developmentspossibly changing the dimension of the Lie algebra to beadded to the 11-dimensional manifold By including thefermion it is no longer ensured that the local supersymmetryof the 11 + 3-dimensional supergravity action remains validThe change in the dimension of the Lie algebra could in turngive rise to new terms so that the Chern-Simons modifiedgravity would be further extended as a result of including thegravitino We intend to pursue an investigation on this issueand to report on it elsewhere
In our model the three-Lie algebra that we add to passfrom 11 to 14119863 is geometrical It is shown that all anomalieswhich are produced by axial bosonic fields in 10- and 11-dimensional theories can be removed on 14-dimensionalmanifold Maybe a question arises ldquowhich is the effect offermionic anomalies on the number of dimensionsrdquo It mustbe clear that this paper is the first step for solving anomaliesin 11-dimensional theories In the next step by applyingthe model of this paper for fermions the exact number ofdimensions of the world can be estimated
Appendix
Details of Calculations
In this section we will bring details of calculations First letus disclose more aspects of (6)
1205981198681198941198681015840119894 (1205971198681015840119894119860119868119895 minus 1205971198681015840119894 (120598119868119895119868119896Γ120572120572119868119896) + 120597119868119894120601120597119868119895120601 + sdot sdot sdot)= 119865119868119894119868119895 minus 119877119868119894119868119895 + 120597119868119894120601120597119868119895120601 minus 12120598119868119894119868119895119868119896119868119898120601119877119868119894119868119895119868119896119868119898
+ sdot sdot sdot
(A1)
where 120601 is the Chern-Simons scalar field119860119868 is the gauge fieldΓ is related to the curvature (119877) and 119868 is a unit vector in thedirection of the coordinate which can be expanded in termsof derivatives of metric
Also it is needed to reobtain the results of (10) with doingsome mathematical calculations
In the equation above the first integration is in agreementwith previous predictions of Chern-Simons gravity in [7ndash9] and can be reduced to the four-dimensional Chern-Simonsmodified gravity of (1) Also the second integration isrelated to the interaction of gauge fields with Chern-Simonsfields Thus this model not only produces the Chern-Simonsmodified gravity but also exhibits some modifications to it
Now we can explain the mathematical model for obtain-ing results of (12)
The first line of this equation removes the anomaly on the 11-dimensional manifold of (4) however the second and thirdlines show that extra anomalies can emerge due to the Chern-Simons fields
At this stage we can consider the process of obtaining(24)
119878SUGRA-14 = 120575 (int11987214
119862119866119866 + int11987214
119862119866119866 + int11987214
119862119866119866+ int11987214
119862119866119866) = int11987214
5sum119899=1
(tr119865119899 minus tr119877119899 + tr (1198651198991198775minus119899))+ int11987214
5sum119899=1
119899sum119869=0
119899minus119869sum119898=0
(tr119865119869119865119898 minus tr119877119869119877119898 + tr (1198651198691198775minus1198691198651198981198775minus119898))+ int11987214
5sum119899=1
119899sum119869=0
119899minus119869sum119898=0
(tr119865119898119865119869 minus tr119877119898119877119869 + tr (1198651198981198775minus1198981198651198691198775minus119869))+ int11987214
5sum119899=1
(tr119865119899 minus tr119877119899 + tr (1198651198991198775minus119899))+ int11987210+119872119873=3
The authors declare that they have no conflicts of interest
Acknowledgments
The work of Alireza Sepehri has been supported financiallyby the Research Institute for Astronomy and Astrophysics ofMaragha (RIAAM) Iran under Research Project no 15237-75
References
[1] R Jackiw and S-Y Pi ldquoChern-Simons modification of generalrelativityrdquo Physical Review D Particles Fields Gravitation andCosmology vol 68 Article ID 104012 2003
[2] B Pereira-Dias C A Hernaski and J A Helayel-Neto ldquoProb-ing the effects of lorentz-symmetry violating chern-simons andricci-cotton terms in higher derivative gravityrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 83 no 82011
[3] K Konno T Matsuyama and S Tanda ldquoDoes a black holerotate in Chern-Simons modified gravityrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 76 no 2Article ID 024009 2007
[4] D Guarrera and A J Hariton ldquoPapapetrou energy-momentumtensor for Chern-Simons modified gravityrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 76 no 4 2007
Advances in High Energy Physics 13
[5] K K Nandi I R Kizirgulov O V Mikolaychuk N P Miko-laychuk and A A Potapov ldquoQuantum phase shift in Chern-Simons modified gravityrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 79 no 8 Article ID 0830062009
[6] P J Porfırio J B Fonseca-Neto J R Nascimento A Petrov JRicardo and A Santos ldquoChern-Simons modified gravity andclosed timelike curvesrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 94 no 4 2016
[7] S Alexander and N Yunes ldquoChernndashSimons modified generalrelativityrdquo Physics Reports vol 480 no 1-2 pp 1ndash55 2009
[8] S Chen and J Jing ldquoGeodetic precession and strong gravita-tional lensing in dynamical Chern-Simons-modified gravityrdquoClassical and QuantumGravity vol 27 no 22 225006 16 pages2010
[9] N Yunes and C F Sopuerta ldquoPerturbations of Schwarzschildblack holes in Chern-Simons modified gravityrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 77 no 62008
[10] F Canfora A Giacomini and A R Zerwekh ldquoKaluza-Kleintheory in the limit of large number of extra dimensionsrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 80 no 8 Article ID 084039 2009
[11] S Wu ldquoGeneral rotating charged Kaluza-Klein AdS black holesin higher dimensionsrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 83 no 12 2011
[12] T Houri and K Yamamoto ldquoKilling-Yano symmetry of Kaluza-KLEin black holes in five dimensionsrdquo Classical and QuantumGravity vol 30 no 7 075013 21 pages 2013
[13] T Tatsuoka H Ishihara M Kimura and K MatsunoldquoExtremal charged black holes with a twisted extra dimensionrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 85 no 4 2012
[14] P Horava and E Witten ldquoHeterotic and Type I string dynamicsfrom eleven dimensionsrdquo Nuclear Physics B vol 460 no 3Article ID 9510209 pp 506ndash524 1996
[15] A Sepehri and R Pincak ldquoThe birth of the universe in a newG-theory approachrdquo Modern Physics Letters A vol 32 no 5Article ID 1750033 1750033 32 pages 2017
[16] A Sepehri A Pradhan R Pincak F Rahaman A Beeshamand T Ghaffary ldquoBirth of the GUP and its effect on theentropy of the universe in Lie-N-algebrardquo International Journalof Geometric Methods in Modern Physics vol 14 no 9 ArticleID 1750130 2017
[17] P Horava and E Witten ldquoEleven-dimensional supergravity ona manifold with boundaryrdquo Nuclear Physics B vol 475 pp 94ndash114 1996
[18] J Bagger and N Lambert ldquoGauge symmetry and supersymme-try of multiple M2-branesrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 77 no 6 065008 6 pages 2008
[19] P Ho and Y Matsuo ldquoM5 from M2rdquo Journal of High EnergyPhysics vol 2008 no 06 p 105 2008
This equation yields results that deserve our commentsIn fact two eleven-dimensional manifolds and the bulkinteract with each other via different types of 119862 119866 andChern-Simons-fields When we sum up supergravities thatlive on these manifolds and consider fields in the spacebetween them we get supergravity in 14-dimensional spaceBy canceling the anomaly in this new supergravity we canobtain the relations between fields By substituting theserelations into the action of the 14-dimensional supergravitywe simply obtain one This means that the 14-dimensionalmanifold with all its matter content can be topologicallyshrunk to one point (see Figure 2) In fact the system ofthe world began from this point and then expanded and
constructed a 14-dimensional world similar to what happensin a Big Bang theory
Another interesting result that comes out of our theoryconcerns the reduction of the action of the world to oneThismeans that the origin of allmatter or fields is the same In factat the first stage of the world we have only a constant energywhich lives on 14-dimensional manifold Then by internalinteractions two 11-dimensional manifolds are createdThesemanifolds interact with each other via an extraChern-Simonsmanifold By exchanging the energy between 11-dimensionalmanifolds their energies are changed and two differentactions are produced Also as due to interaction betweenstrings their shapes become different and different types offields like spinors and bosons are emerged
5 Summary and Final Considerations
In this paper we have shown that the Chern-Simons termsof modified gravity may be understood as due to the inter-action between two 11-dimensional manifolds in an 11 + 3-dimensional space-time where 3 is the dimension of a Lie-type-algebra We also argue that there is a direct relationbetween the dimension of the algebra and the dimensionof the manifold For example for 11-dimensional world thedimension of the Lie-algebra is three If the rank of the fieldswhich live in one manifold becomes larger than the rankof the algebra there emerges an anomaly This anomaly isproduced as an effect of connecting fields in the bulk tofields which live in the manifold To cancel this anomaly we
10 Advances in High Energy Physics
need to introduce another 11-dimensional manifold in the11+3-dimensional space-timewhich interacts with the initialmanifold by exchanging Chern-Simons terms These Chern-Simons terms produce an extra manifold If we sum up thetopology of the 11-dimensional manifolds and the topologyof the Chern-Simons manifold we can show that the totaltopology shrinks to one which is consistent with predictionsof the Big Bang theory
Our proposal is actually another version of the 119866-theory in [15 16] This approach removes anomalies in 11-dimensional 119872-theory by a generalization of dimension to14 In this research contribution we have shown that thephysics of an 11-dimensional spacing manifold plus a 3-dimensional algebraic manifold is the same as the physics of14-dimensional manifold In fact 11-dimensional 119872-theoryis a free-anomaly theory whenever a three-dimensionalalgebraic manifold is added to it This 14-dimensional man-ifold can be broken to two 11-dimensional manifolds whichare connected by a 3-dimensional Chern-Simons theoryAnother interesting result that comes out is the reduction ofthe action of the world to one This means that the originof all matter or fields is the same In fact at the first stageof the world we have only a constant energy which lives on14-dimensional manifold Then by internal interactions two11-dimensional manifolds come outThesemanifolds interactwith each other via one extra Chern-Simons manifold Byexchanging the energy between 11-dimensional manifoldstheir energies are changed and two different actions areproduced Also as due to interaction between strings theirshapes differ and diverse types of fields like spinors andbosons emerge
To conclude we would like to point out that that all ourtreatment has been restricted to the purely bosonic sectorof 11-dimensional supergravity whose on-shell multipletbesides the metric tensor and the 3-form gauge potentialincludes the gravitino fieldThe latter has not been consideredhere we have restricted ourselves to the bosonic sectorHowever it would be a further task to inspect how theinclusion of the gravitino would affect our developmentspossibly changing the dimension of the Lie algebra to beadded to the 11-dimensional manifold By including thefermion it is no longer ensured that the local supersymmetryof the 11 + 3-dimensional supergravity action remains validThe change in the dimension of the Lie algebra could in turngive rise to new terms so that the Chern-Simons modifiedgravity would be further extended as a result of including thegravitino We intend to pursue an investigation on this issueand to report on it elsewhere
In our model the three-Lie algebra that we add to passfrom 11 to 14119863 is geometrical It is shown that all anomalieswhich are produced by axial bosonic fields in 10- and 11-dimensional theories can be removed on 14-dimensionalmanifold Maybe a question arises ldquowhich is the effect offermionic anomalies on the number of dimensionsrdquo It mustbe clear that this paper is the first step for solving anomaliesin 11-dimensional theories In the next step by applyingthe model of this paper for fermions the exact number ofdimensions of the world can be estimated
Appendix
Details of Calculations
In this section we will bring details of calculations First letus disclose more aspects of (6)
1205981198681198941198681015840119894 (1205971198681015840119894119860119868119895 minus 1205971198681015840119894 (120598119868119895119868119896Γ120572120572119868119896) + 120597119868119894120601120597119868119895120601 + sdot sdot sdot)= 119865119868119894119868119895 minus 119877119868119894119868119895 + 120597119868119894120601120597119868119895120601 minus 12120598119868119894119868119895119868119896119868119898120601119877119868119894119868119895119868119896119868119898
+ sdot sdot sdot
(A1)
where 120601 is the Chern-Simons scalar field119860119868 is the gauge fieldΓ is related to the curvature (119877) and 119868 is a unit vector in thedirection of the coordinate which can be expanded in termsof derivatives of metric
Also it is needed to reobtain the results of (10) with doingsome mathematical calculations
In the equation above the first integration is in agreementwith previous predictions of Chern-Simons gravity in [7ndash9] and can be reduced to the four-dimensional Chern-Simonsmodified gravity of (1) Also the second integration isrelated to the interaction of gauge fields with Chern-Simonsfields Thus this model not only produces the Chern-Simonsmodified gravity but also exhibits some modifications to it
Now we can explain the mathematical model for obtain-ing results of (12)
The first line of this equation removes the anomaly on the 11-dimensional manifold of (4) however the second and thirdlines show that extra anomalies can emerge due to the Chern-Simons fields
At this stage we can consider the process of obtaining(24)
119878SUGRA-14 = 120575 (int11987214
119862119866119866 + int11987214
119862119866119866 + int11987214
119862119866119866+ int11987214
119862119866119866) = int11987214
5sum119899=1
(tr119865119899 minus tr119877119899 + tr (1198651198991198775minus119899))+ int11987214
5sum119899=1
119899sum119869=0
119899minus119869sum119898=0
(tr119865119869119865119898 minus tr119877119869119877119898 + tr (1198651198691198775minus1198691198651198981198775minus119898))+ int11987214
5sum119899=1
119899sum119869=0
119899minus119869sum119898=0
(tr119865119898119865119869 minus tr119877119898119877119869 + tr (1198651198981198775minus1198981198651198691198775minus119869))+ int11987214
5sum119899=1
(tr119865119899 minus tr119877119899 + tr (1198651198991198775minus119899))+ int11987210+119872119873=3
The authors declare that they have no conflicts of interest
Acknowledgments
The work of Alireza Sepehri has been supported financiallyby the Research Institute for Astronomy and Astrophysics ofMaragha (RIAAM) Iran under Research Project no 15237-75
References
[1] R Jackiw and S-Y Pi ldquoChern-Simons modification of generalrelativityrdquo Physical Review D Particles Fields Gravitation andCosmology vol 68 Article ID 104012 2003
[2] B Pereira-Dias C A Hernaski and J A Helayel-Neto ldquoProb-ing the effects of lorentz-symmetry violating chern-simons andricci-cotton terms in higher derivative gravityrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 83 no 82011
[3] K Konno T Matsuyama and S Tanda ldquoDoes a black holerotate in Chern-Simons modified gravityrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 76 no 2Article ID 024009 2007
[4] D Guarrera and A J Hariton ldquoPapapetrou energy-momentumtensor for Chern-Simons modified gravityrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 76 no 4 2007
Advances in High Energy Physics 13
[5] K K Nandi I R Kizirgulov O V Mikolaychuk N P Miko-laychuk and A A Potapov ldquoQuantum phase shift in Chern-Simons modified gravityrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 79 no 8 Article ID 0830062009
[6] P J Porfırio J B Fonseca-Neto J R Nascimento A Petrov JRicardo and A Santos ldquoChern-Simons modified gravity andclosed timelike curvesrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 94 no 4 2016
[7] S Alexander and N Yunes ldquoChernndashSimons modified generalrelativityrdquo Physics Reports vol 480 no 1-2 pp 1ndash55 2009
[8] S Chen and J Jing ldquoGeodetic precession and strong gravita-tional lensing in dynamical Chern-Simons-modified gravityrdquoClassical and QuantumGravity vol 27 no 22 225006 16 pages2010
[9] N Yunes and C F Sopuerta ldquoPerturbations of Schwarzschildblack holes in Chern-Simons modified gravityrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 77 no 62008
[10] F Canfora A Giacomini and A R Zerwekh ldquoKaluza-Kleintheory in the limit of large number of extra dimensionsrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 80 no 8 Article ID 084039 2009
[11] S Wu ldquoGeneral rotating charged Kaluza-Klein AdS black holesin higher dimensionsrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 83 no 12 2011
[12] T Houri and K Yamamoto ldquoKilling-Yano symmetry of Kaluza-KLEin black holes in five dimensionsrdquo Classical and QuantumGravity vol 30 no 7 075013 21 pages 2013
[13] T Tatsuoka H Ishihara M Kimura and K MatsunoldquoExtremal charged black holes with a twisted extra dimensionrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 85 no 4 2012
[14] P Horava and E Witten ldquoHeterotic and Type I string dynamicsfrom eleven dimensionsrdquo Nuclear Physics B vol 460 no 3Article ID 9510209 pp 506ndash524 1996
[15] A Sepehri and R Pincak ldquoThe birth of the universe in a newG-theory approachrdquo Modern Physics Letters A vol 32 no 5Article ID 1750033 1750033 32 pages 2017
[16] A Sepehri A Pradhan R Pincak F Rahaman A Beeshamand T Ghaffary ldquoBirth of the GUP and its effect on theentropy of the universe in Lie-N-algebrardquo International Journalof Geometric Methods in Modern Physics vol 14 no 9 ArticleID 1750130 2017
[17] P Horava and E Witten ldquoEleven-dimensional supergravity ona manifold with boundaryrdquo Nuclear Physics B vol 475 pp 94ndash114 1996
[18] J Bagger and N Lambert ldquoGauge symmetry and supersymme-try of multiple M2-branesrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 77 no 6 065008 6 pages 2008
[19] P Ho and Y Matsuo ldquoM5 from M2rdquo Journal of High EnergyPhysics vol 2008 no 06 p 105 2008
need to introduce another 11-dimensional manifold in the11+3-dimensional space-timewhich interacts with the initialmanifold by exchanging Chern-Simons terms These Chern-Simons terms produce an extra manifold If we sum up thetopology of the 11-dimensional manifolds and the topologyof the Chern-Simons manifold we can show that the totaltopology shrinks to one which is consistent with predictionsof the Big Bang theory
Our proposal is actually another version of the 119866-theory in [15 16] This approach removes anomalies in 11-dimensional 119872-theory by a generalization of dimension to14 In this research contribution we have shown that thephysics of an 11-dimensional spacing manifold plus a 3-dimensional algebraic manifold is the same as the physics of14-dimensional manifold In fact 11-dimensional 119872-theoryis a free-anomaly theory whenever a three-dimensionalalgebraic manifold is added to it This 14-dimensional man-ifold can be broken to two 11-dimensional manifolds whichare connected by a 3-dimensional Chern-Simons theoryAnother interesting result that comes out is the reduction ofthe action of the world to one This means that the originof all matter or fields is the same In fact at the first stageof the world we have only a constant energy which lives on14-dimensional manifold Then by internal interactions two11-dimensional manifolds come outThesemanifolds interactwith each other via one extra Chern-Simons manifold Byexchanging the energy between 11-dimensional manifoldstheir energies are changed and two different actions areproduced Also as due to interaction between strings theirshapes differ and diverse types of fields like spinors andbosons emerge
To conclude we would like to point out that that all ourtreatment has been restricted to the purely bosonic sectorof 11-dimensional supergravity whose on-shell multipletbesides the metric tensor and the 3-form gauge potentialincludes the gravitino fieldThe latter has not been consideredhere we have restricted ourselves to the bosonic sectorHowever it would be a further task to inspect how theinclusion of the gravitino would affect our developmentspossibly changing the dimension of the Lie algebra to beadded to the 11-dimensional manifold By including thefermion it is no longer ensured that the local supersymmetryof the 11 + 3-dimensional supergravity action remains validThe change in the dimension of the Lie algebra could in turngive rise to new terms so that the Chern-Simons modifiedgravity would be further extended as a result of including thegravitino We intend to pursue an investigation on this issueand to report on it elsewhere
In our model the three-Lie algebra that we add to passfrom 11 to 14119863 is geometrical It is shown that all anomalieswhich are produced by axial bosonic fields in 10- and 11-dimensional theories can be removed on 14-dimensionalmanifold Maybe a question arises ldquowhich is the effect offermionic anomalies on the number of dimensionsrdquo It mustbe clear that this paper is the first step for solving anomaliesin 11-dimensional theories In the next step by applyingthe model of this paper for fermions the exact number ofdimensions of the world can be estimated
Appendix
Details of Calculations
In this section we will bring details of calculations First letus disclose more aspects of (6)
1205981198681198941198681015840119894 (1205971198681015840119894119860119868119895 minus 1205971198681015840119894 (120598119868119895119868119896Γ120572120572119868119896) + 120597119868119894120601120597119868119895120601 + sdot sdot sdot)= 119865119868119894119868119895 minus 119877119868119894119868119895 + 120597119868119894120601120597119868119895120601 minus 12120598119868119894119868119895119868119896119868119898120601119877119868119894119868119895119868119896119868119898
+ sdot sdot sdot
(A1)
where 120601 is the Chern-Simons scalar field119860119868 is the gauge fieldΓ is related to the curvature (119877) and 119868 is a unit vector in thedirection of the coordinate which can be expanded in termsof derivatives of metric
Also it is needed to reobtain the results of (10) with doingsome mathematical calculations
In the equation above the first integration is in agreementwith previous predictions of Chern-Simons gravity in [7ndash9] and can be reduced to the four-dimensional Chern-Simonsmodified gravity of (1) Also the second integration isrelated to the interaction of gauge fields with Chern-Simonsfields Thus this model not only produces the Chern-Simonsmodified gravity but also exhibits some modifications to it
Now we can explain the mathematical model for obtain-ing results of (12)
The first line of this equation removes the anomaly on the 11-dimensional manifold of (4) however the second and thirdlines show that extra anomalies can emerge due to the Chern-Simons fields
At this stage we can consider the process of obtaining(24)
119878SUGRA-14 = 120575 (int11987214
119862119866119866 + int11987214
119862119866119866 + int11987214
119862119866119866+ int11987214
119862119866119866) = int11987214
5sum119899=1
(tr119865119899 minus tr119877119899 + tr (1198651198991198775minus119899))+ int11987214
5sum119899=1
119899sum119869=0
119899minus119869sum119898=0
(tr119865119869119865119898 minus tr119877119869119877119898 + tr (1198651198691198775minus1198691198651198981198775minus119898))+ int11987214
5sum119899=1
119899sum119869=0
119899minus119869sum119898=0
(tr119865119898119865119869 minus tr119877119898119877119869 + tr (1198651198981198775minus1198981198651198691198775minus119869))+ int11987214
5sum119899=1
(tr119865119899 minus tr119877119899 + tr (1198651198991198775minus119899))+ int11987210+119872119873=3
The authors declare that they have no conflicts of interest
Acknowledgments
The work of Alireza Sepehri has been supported financiallyby the Research Institute for Astronomy and Astrophysics ofMaragha (RIAAM) Iran under Research Project no 15237-75
References
[1] R Jackiw and S-Y Pi ldquoChern-Simons modification of generalrelativityrdquo Physical Review D Particles Fields Gravitation andCosmology vol 68 Article ID 104012 2003
[2] B Pereira-Dias C A Hernaski and J A Helayel-Neto ldquoProb-ing the effects of lorentz-symmetry violating chern-simons andricci-cotton terms in higher derivative gravityrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 83 no 82011
[3] K Konno T Matsuyama and S Tanda ldquoDoes a black holerotate in Chern-Simons modified gravityrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 76 no 2Article ID 024009 2007
[4] D Guarrera and A J Hariton ldquoPapapetrou energy-momentumtensor for Chern-Simons modified gravityrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 76 no 4 2007
Advances in High Energy Physics 13
[5] K K Nandi I R Kizirgulov O V Mikolaychuk N P Miko-laychuk and A A Potapov ldquoQuantum phase shift in Chern-Simons modified gravityrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 79 no 8 Article ID 0830062009
[6] P J Porfırio J B Fonseca-Neto J R Nascimento A Petrov JRicardo and A Santos ldquoChern-Simons modified gravity andclosed timelike curvesrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 94 no 4 2016
[7] S Alexander and N Yunes ldquoChernndashSimons modified generalrelativityrdquo Physics Reports vol 480 no 1-2 pp 1ndash55 2009
[8] S Chen and J Jing ldquoGeodetic precession and strong gravita-tional lensing in dynamical Chern-Simons-modified gravityrdquoClassical and QuantumGravity vol 27 no 22 225006 16 pages2010
[9] N Yunes and C F Sopuerta ldquoPerturbations of Schwarzschildblack holes in Chern-Simons modified gravityrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 77 no 62008
[10] F Canfora A Giacomini and A R Zerwekh ldquoKaluza-Kleintheory in the limit of large number of extra dimensionsrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 80 no 8 Article ID 084039 2009
[11] S Wu ldquoGeneral rotating charged Kaluza-Klein AdS black holesin higher dimensionsrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 83 no 12 2011
[12] T Houri and K Yamamoto ldquoKilling-Yano symmetry of Kaluza-KLEin black holes in five dimensionsrdquo Classical and QuantumGravity vol 30 no 7 075013 21 pages 2013
[13] T Tatsuoka H Ishihara M Kimura and K MatsunoldquoExtremal charged black holes with a twisted extra dimensionrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 85 no 4 2012
[14] P Horava and E Witten ldquoHeterotic and Type I string dynamicsfrom eleven dimensionsrdquo Nuclear Physics B vol 460 no 3Article ID 9510209 pp 506ndash524 1996
[15] A Sepehri and R Pincak ldquoThe birth of the universe in a newG-theory approachrdquo Modern Physics Letters A vol 32 no 5Article ID 1750033 1750033 32 pages 2017
[16] A Sepehri A Pradhan R Pincak F Rahaman A Beeshamand T Ghaffary ldquoBirth of the GUP and its effect on theentropy of the universe in Lie-N-algebrardquo International Journalof Geometric Methods in Modern Physics vol 14 no 9 ArticleID 1750130 2017
[17] P Horava and E Witten ldquoEleven-dimensional supergravity ona manifold with boundaryrdquo Nuclear Physics B vol 475 pp 94ndash114 1996
[18] J Bagger and N Lambert ldquoGauge symmetry and supersymme-try of multiple M2-branesrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 77 no 6 065008 6 pages 2008
[19] P Ho and Y Matsuo ldquoM5 from M2rdquo Journal of High EnergyPhysics vol 2008 no 06 p 105 2008
In the equation above the first integration is in agreementwith previous predictions of Chern-Simons gravity in [7ndash9] and can be reduced to the four-dimensional Chern-Simonsmodified gravity of (1) Also the second integration isrelated to the interaction of gauge fields with Chern-Simonsfields Thus this model not only produces the Chern-Simonsmodified gravity but also exhibits some modifications to it
Now we can explain the mathematical model for obtain-ing results of (12)
The first line of this equation removes the anomaly on the 11-dimensional manifold of (4) however the second and thirdlines show that extra anomalies can emerge due to the Chern-Simons fields
At this stage we can consider the process of obtaining(24)
119878SUGRA-14 = 120575 (int11987214
119862119866119866 + int11987214
119862119866119866 + int11987214
119862119866119866+ int11987214
119862119866119866) = int11987214
5sum119899=1
(tr119865119899 minus tr119877119899 + tr (1198651198991198775minus119899))+ int11987214
5sum119899=1
119899sum119869=0
119899minus119869sum119898=0
(tr119865119869119865119898 minus tr119877119869119877119898 + tr (1198651198691198775minus1198691198651198981198775minus119898))+ int11987214
5sum119899=1
119899sum119869=0
119899minus119869sum119898=0
(tr119865119898119865119869 minus tr119877119898119877119869 + tr (1198651198981198775minus1198981198651198691198775minus119869))+ int11987214
5sum119899=1
(tr119865119899 minus tr119877119899 + tr (1198651198991198775minus119899))+ int11987210+119872119873=3
The authors declare that they have no conflicts of interest
Acknowledgments
The work of Alireza Sepehri has been supported financiallyby the Research Institute for Astronomy and Astrophysics ofMaragha (RIAAM) Iran under Research Project no 15237-75
References
[1] R Jackiw and S-Y Pi ldquoChern-Simons modification of generalrelativityrdquo Physical Review D Particles Fields Gravitation andCosmology vol 68 Article ID 104012 2003
[2] B Pereira-Dias C A Hernaski and J A Helayel-Neto ldquoProb-ing the effects of lorentz-symmetry violating chern-simons andricci-cotton terms in higher derivative gravityrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 83 no 82011
[3] K Konno T Matsuyama and S Tanda ldquoDoes a black holerotate in Chern-Simons modified gravityrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 76 no 2Article ID 024009 2007
[4] D Guarrera and A J Hariton ldquoPapapetrou energy-momentumtensor for Chern-Simons modified gravityrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 76 no 4 2007
Advances in High Energy Physics 13
[5] K K Nandi I R Kizirgulov O V Mikolaychuk N P Miko-laychuk and A A Potapov ldquoQuantum phase shift in Chern-Simons modified gravityrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 79 no 8 Article ID 0830062009
[6] P J Porfırio J B Fonseca-Neto J R Nascimento A Petrov JRicardo and A Santos ldquoChern-Simons modified gravity andclosed timelike curvesrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 94 no 4 2016
[7] S Alexander and N Yunes ldquoChernndashSimons modified generalrelativityrdquo Physics Reports vol 480 no 1-2 pp 1ndash55 2009
[8] S Chen and J Jing ldquoGeodetic precession and strong gravita-tional lensing in dynamical Chern-Simons-modified gravityrdquoClassical and QuantumGravity vol 27 no 22 225006 16 pages2010
[9] N Yunes and C F Sopuerta ldquoPerturbations of Schwarzschildblack holes in Chern-Simons modified gravityrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 77 no 62008
[10] F Canfora A Giacomini and A R Zerwekh ldquoKaluza-Kleintheory in the limit of large number of extra dimensionsrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 80 no 8 Article ID 084039 2009
[11] S Wu ldquoGeneral rotating charged Kaluza-Klein AdS black holesin higher dimensionsrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 83 no 12 2011
[12] T Houri and K Yamamoto ldquoKilling-Yano symmetry of Kaluza-KLEin black holes in five dimensionsrdquo Classical and QuantumGravity vol 30 no 7 075013 21 pages 2013
[13] T Tatsuoka H Ishihara M Kimura and K MatsunoldquoExtremal charged black holes with a twisted extra dimensionrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 85 no 4 2012
[14] P Horava and E Witten ldquoHeterotic and Type I string dynamicsfrom eleven dimensionsrdquo Nuclear Physics B vol 460 no 3Article ID 9510209 pp 506ndash524 1996
[15] A Sepehri and R Pincak ldquoThe birth of the universe in a newG-theory approachrdquo Modern Physics Letters A vol 32 no 5Article ID 1750033 1750033 32 pages 2017
[16] A Sepehri A Pradhan R Pincak F Rahaman A Beeshamand T Ghaffary ldquoBirth of the GUP and its effect on theentropy of the universe in Lie-N-algebrardquo International Journalof Geometric Methods in Modern Physics vol 14 no 9 ArticleID 1750130 2017
[17] P Horava and E Witten ldquoEleven-dimensional supergravity ona manifold with boundaryrdquo Nuclear Physics B vol 475 pp 94ndash114 1996
[18] J Bagger and N Lambert ldquoGauge symmetry and supersymme-try of multiple M2-branesrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 77 no 6 065008 6 pages 2008
[19] P Ho and Y Matsuo ldquoM5 from M2rdquo Journal of High EnergyPhysics vol 2008 no 06 p 105 2008
The first line of this equation removes the anomaly on the 11-dimensional manifold of (4) however the second and thirdlines show that extra anomalies can emerge due to the Chern-Simons fields
At this stage we can consider the process of obtaining(24)
119878SUGRA-14 = 120575 (int11987214
119862119866119866 + int11987214
119862119866119866 + int11987214
119862119866119866+ int11987214
119862119866119866) = int11987214
5sum119899=1
(tr119865119899 minus tr119877119899 + tr (1198651198991198775minus119899))+ int11987214
5sum119899=1
119899sum119869=0
119899minus119869sum119898=0
(tr119865119869119865119898 minus tr119877119869119877119898 + tr (1198651198691198775minus1198691198651198981198775minus119898))+ int11987214
5sum119899=1
119899sum119869=0
119899minus119869sum119898=0
(tr119865119898119865119869 minus tr119877119898119877119869 + tr (1198651198981198775minus1198981198651198691198775minus119869))+ int11987214
5sum119899=1
(tr119865119899 minus tr119877119899 + tr (1198651198991198775minus119899))+ int11987210+119872119873=3
The authors declare that they have no conflicts of interest
Acknowledgments
The work of Alireza Sepehri has been supported financiallyby the Research Institute for Astronomy and Astrophysics ofMaragha (RIAAM) Iran under Research Project no 15237-75
References
[1] R Jackiw and S-Y Pi ldquoChern-Simons modification of generalrelativityrdquo Physical Review D Particles Fields Gravitation andCosmology vol 68 Article ID 104012 2003
[2] B Pereira-Dias C A Hernaski and J A Helayel-Neto ldquoProb-ing the effects of lorentz-symmetry violating chern-simons andricci-cotton terms in higher derivative gravityrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 83 no 82011
[3] K Konno T Matsuyama and S Tanda ldquoDoes a black holerotate in Chern-Simons modified gravityrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 76 no 2Article ID 024009 2007
[4] D Guarrera and A J Hariton ldquoPapapetrou energy-momentumtensor for Chern-Simons modified gravityrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 76 no 4 2007
Advances in High Energy Physics 13
[5] K K Nandi I R Kizirgulov O V Mikolaychuk N P Miko-laychuk and A A Potapov ldquoQuantum phase shift in Chern-Simons modified gravityrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 79 no 8 Article ID 0830062009
[6] P J Porfırio J B Fonseca-Neto J R Nascimento A Petrov JRicardo and A Santos ldquoChern-Simons modified gravity andclosed timelike curvesrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 94 no 4 2016
[7] S Alexander and N Yunes ldquoChernndashSimons modified generalrelativityrdquo Physics Reports vol 480 no 1-2 pp 1ndash55 2009
[8] S Chen and J Jing ldquoGeodetic precession and strong gravita-tional lensing in dynamical Chern-Simons-modified gravityrdquoClassical and QuantumGravity vol 27 no 22 225006 16 pages2010
[9] N Yunes and C F Sopuerta ldquoPerturbations of Schwarzschildblack holes in Chern-Simons modified gravityrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 77 no 62008
[10] F Canfora A Giacomini and A R Zerwekh ldquoKaluza-Kleintheory in the limit of large number of extra dimensionsrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 80 no 8 Article ID 084039 2009
[11] S Wu ldquoGeneral rotating charged Kaluza-Klein AdS black holesin higher dimensionsrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 83 no 12 2011
[12] T Houri and K Yamamoto ldquoKilling-Yano symmetry of Kaluza-KLEin black holes in five dimensionsrdquo Classical and QuantumGravity vol 30 no 7 075013 21 pages 2013
[13] T Tatsuoka H Ishihara M Kimura and K MatsunoldquoExtremal charged black holes with a twisted extra dimensionrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 85 no 4 2012
[14] P Horava and E Witten ldquoHeterotic and Type I string dynamicsfrom eleven dimensionsrdquo Nuclear Physics B vol 460 no 3Article ID 9510209 pp 506ndash524 1996
[15] A Sepehri and R Pincak ldquoThe birth of the universe in a newG-theory approachrdquo Modern Physics Letters A vol 32 no 5Article ID 1750033 1750033 32 pages 2017
[16] A Sepehri A Pradhan R Pincak F Rahaman A Beeshamand T Ghaffary ldquoBirth of the GUP and its effect on theentropy of the universe in Lie-N-algebrardquo International Journalof Geometric Methods in Modern Physics vol 14 no 9 ArticleID 1750130 2017
[17] P Horava and E Witten ldquoEleven-dimensional supergravity ona manifold with boundaryrdquo Nuclear Physics B vol 475 pp 94ndash114 1996
[18] J Bagger and N Lambert ldquoGauge symmetry and supersymme-try of multiple M2-branesrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 77 no 6 065008 6 pages 2008
[19] P Ho and Y Matsuo ldquoM5 from M2rdquo Journal of High EnergyPhysics vol 2008 no 06 p 105 2008
[5] K K Nandi I R Kizirgulov O V Mikolaychuk N P Miko-laychuk and A A Potapov ldquoQuantum phase shift in Chern-Simons modified gravityrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 79 no 8 Article ID 0830062009
[6] P J Porfırio J B Fonseca-Neto J R Nascimento A Petrov JRicardo and A Santos ldquoChern-Simons modified gravity andclosed timelike curvesrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 94 no 4 2016
[7] S Alexander and N Yunes ldquoChernndashSimons modified generalrelativityrdquo Physics Reports vol 480 no 1-2 pp 1ndash55 2009
[8] S Chen and J Jing ldquoGeodetic precession and strong gravita-tional lensing in dynamical Chern-Simons-modified gravityrdquoClassical and QuantumGravity vol 27 no 22 225006 16 pages2010
[9] N Yunes and C F Sopuerta ldquoPerturbations of Schwarzschildblack holes in Chern-Simons modified gravityrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 77 no 62008
[10] F Canfora A Giacomini and A R Zerwekh ldquoKaluza-Kleintheory in the limit of large number of extra dimensionsrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 80 no 8 Article ID 084039 2009
[11] S Wu ldquoGeneral rotating charged Kaluza-Klein AdS black holesin higher dimensionsrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 83 no 12 2011
[12] T Houri and K Yamamoto ldquoKilling-Yano symmetry of Kaluza-KLEin black holes in five dimensionsrdquo Classical and QuantumGravity vol 30 no 7 075013 21 pages 2013
[13] T Tatsuoka H Ishihara M Kimura and K MatsunoldquoExtremal charged black holes with a twisted extra dimensionrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 85 no 4 2012
[14] P Horava and E Witten ldquoHeterotic and Type I string dynamicsfrom eleven dimensionsrdquo Nuclear Physics B vol 460 no 3Article ID 9510209 pp 506ndash524 1996
[15] A Sepehri and R Pincak ldquoThe birth of the universe in a newG-theory approachrdquo Modern Physics Letters A vol 32 no 5Article ID 1750033 1750033 32 pages 2017
[16] A Sepehri A Pradhan R Pincak F Rahaman A Beeshamand T Ghaffary ldquoBirth of the GUP and its effect on theentropy of the universe in Lie-N-algebrardquo International Journalof Geometric Methods in Modern Physics vol 14 no 9 ArticleID 1750130 2017
[17] P Horava and E Witten ldquoEleven-dimensional supergravity ona manifold with boundaryrdquo Nuclear Physics B vol 475 pp 94ndash114 1996
[18] J Bagger and N Lambert ldquoGauge symmetry and supersymme-try of multiple M2-branesrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 77 no 6 065008 6 pages 2008
[19] P Ho and Y Matsuo ldquoM5 from M2rdquo Journal of High EnergyPhysics vol 2008 no 06 p 105 2008
