eTheory Division, CERN, 1211 Geneva 23, Switsedand, and Department of Physics, University of Patras, 26110 Patras, Greece
Duality tzandormstioas with respect to rotational isometrics relate supersymmetric with non-supersymmetrlc backgrounds in string theory. We find that non-local world-sheet effects have to be taken into account in order to restore supersymmetry at the string level. The underlying supe~conformal algebra remsins the same, but in this cue T-duality relates local with non-local real~atio-- of the algebra in terms of parsfermionJ. This is another example where stringy effects resolve paradoxes of the effective field theory.
I . Ou t l ine and s u m m a r y
These notes, based on two lectures given at the Trieste conference on "S-Duality and Mirror Sym- merry", summarise some of our recent results on supersymmetry and duality [I, 2, 3]. The prob- lem that arises in this context concerns the su- persymmetr/c properties of the lowest order elfec- tive theory and their behaviour under T-duality (and more generally U-duality) trm~sformations (see also [4]). We will describe the geometric con- ditions for the J~illing vector fields that lead to the preservation or the loss of manifest space.time su- persymmetry under duality, and classify them as translational versus rotational respectively. An- alysing this problem in detail we find that there is no contradiction at the level of string theory. This is part of the standard lore that stringy ef- fects manifest as paradoxes of the effective field theory, and supersymmetry is no exception to it.
In toroidal compactifications of superstring theory T-duality is a symmetry that interchanges momentum with winding modes, and it manifests geometrically as a amal/-I~ge re~cl/ns equivalence in the effective theory of the lmckground mnssless fields. This equivalence has been genezalised to arbitrary string backgrounds with Abel/an/some- tries following Buschez's formula that was origi- na/ly derived for the fl-function equations of the metric G~,v, sntisymmetric tensor field B~,, and dilaton @ to lowest order in the e-model coupling constant (inverse string tension) a I (see for in-
stance 15], and references therein). One might naively expect that if a background is supersym- metric (eg a bosonic solution of effective super- gravity), the dual background will also share the same number of supersymmetries. This is a rea- sonable expectation provided that there are no non-local world-sheet effects, so that the dual f~es of the theory provide a trustworthy low energy appro~mation to string dynamics in the a'-expansion. There are circumstances, however, when the dual geometry has strong curvature sin- guladties and the effective field theory descrip- tion brealm down in their vicinity. This is a typ- ical property of gravitational backgrounds hay- ing }(i]]ing isometrics with fixed points, in which case the T-dual background exhibits curvature singularities. Then, T-duality appears to be in- compatible with space-time supersymmetry, and non-local wo,ld-sheet effects have to be taken into account for resolving this issue consistently. Af- tern11, it is s well-known fact that symmetries not commuting with a given ICilling vector field ~re not manifest in the dual background of the elfec- rive field theory description.
In the first part most of our discussion will con- centr&te on 4-dim space-time backgrounds, think- ing of superstrin g vacua as sa~ing from ten di- mensions by eompactification on a 6-dim inter- nal space K. It is much simpler to expose the main ideas in this case and employ the notion of self-duality (when it is appropriate) to refine
E. Alvarez et aL/Nuclear Physics B (Proc. Suppl.) 46 (1996) 16-29 17
the distinction we want to make. In four dimen- sions, and in the presence of some Killlng isome- tries, it is also possible to intertwine T with S- duality, thus forming a much larger symmetry group known as U-duedity. We will present the essential features of this idea st the level of the ef- fective theory by considering 4-dim backgrounds with (at least) one isometzy, in which case the U-duality corresponds to O(2, 2) tr,Lnfformations. The supersymmetde properties of the effective theory will also be considered under the action of such generaKsed symmetries, where the effect on supezsymmetry can be even more severe.
In the second part we will investigate more elaborately the issue ofsupersymmetry versus du- edify in heterotie ~-models, where we analyse pos- sible anomalies and find some modilication~ of Buscher's rules. We consider the hetezotic string in 10-dim fiat Minkowski space with SO(32) gauge group (though ~ arguments apply to the Es x Es string) sad study the effect of T- duality with respect to rotations in a 2-dlm plane, using a manifest I S O ( 1 , 7) x 50(30) × 50(2) sym- metric formalism. We prove that the number of space-time fermionie symmetries remains un- changed. The conformed field theory analysis sug- gests that the emission vertices of low energy par- tides in the dual theory are represented by non- local operators, which do not admit a straightfor- ward a'-expsasion. In fact the underlying confor- med theory provides a natured explanation of this, where local reedi~tious of the superconformed al- gebra become non-local in terms of parsfezmions. In view of these circumstances, ¢ertaht theorems relating world-sheet with manifest space-time su- persymmetry should be revised. Also, the pow- erful constraints imposed by supersymmetry on the geometry of the target space are not valid for non-local realizations.
Some of the topics of this work may aim be rel- evant for supezstring phenomenology, where we mainly work in terms of the lowest order effective theory. The issue of space-time supe~ymmetry versus duality, which ~ to lowest order in a e, demonstrates explicitly that sn apparently non- supersymmetrie background can qualify u a vac- uum solution of superstring theory, in contrary to the "standard wisdom" that has been consid-
ered so far. So, whether supersymmetry is bro- ken or not in various phenomenologieal applica- tions cannot be decided, unless one knows how to incorporate the appropriate non-local world- sheet effects that might lead to its restoration at the string level. Also, various gravitational solu- tions, like black holes, might enjoy some super- symmetric properties in the string context. This could also provide & better understanding of the way that string theory, through its world-sheet ef- fects, c u resolve the fundamental problems of the quantum theory of black holes. Finally, from the low energy point of view the fact that T-duality relates supersymmetrie with non-supersymmetric backgrounds could provide examples of the mech- ,Lniam advocated by Witten to shed new light in the cosmological constant problem [6]. Hence, we view T-duality at this moment as a method for probing some of these possibilities, sad leave to future investigation the applicability of these ideas to the physical problems of black holes sad cosmology within supezstring theory.
2. S u p e r s y m m e t r y and dual i t ies : a first d a s h
Consider the class of 4-dlm backgrounds with one isometry generated by a Killing vector field K = 8 /8r . The ~ t u r e is taken to be Eu- clidean, but most of the resultl geuerediz¢ to the phydca] Minkowskiaa dgnature. The metric can be written locally in the form
ds 2 = V(dl" + w, dz~) ~ + V - 1 7 ~ j d z ~ d ~ , (2.1)
where z I = z, z 2 = y, z # = z are coordinates on the space of non-trivinl orbits of 8/81", and V, wi, 7ij are all independent of~'. Of course, wi are not unique, since they are defined up to a gauge trans- formation w~ --, ~i - &~ that amounts to the co- ordinate shift r ~ ~ + ~(z'). These badtgrounds provide gravitational mlutions of the ~-funetion equations with ¢onstaat dilaton and antisymmet- de tearer fields to lowest order in ~'. Although this ~ is rather restdetive, it is & good strating point for exploring the clash of supersymmetry with dualities in the effective theory.
Performing a T-duality transformation to the gravitational background (2.1) we obtain the fol-
18 E. Alvarez et al./Nuclear Physics B (Proc. Suppl.) 46 (1996) 16-29
lowing configuration in the o-model frame, (000) 1 0 (2.2) G ~ = ~ 0 "f~j
0
with non-trivial dilaton and antisymmetric tensor fields
1 log V B,~ = ~ . (2.3) ~=-~ A necessary condition for having manifest space- time supersymmetry in the new background is given by the dilatino variation 6A = 0,
= 0 , (2.4) G
where H~.~ is the field strength of Bjw. Substi- tuting the expressions (2.2) and (2.3) we arrive at the differential equations
o, j - oi , ± , i k = o (2.5)
which select only s subclass of solutions (2.1) of the vacuum Einstein equations.
It is useful at this point to introduce a non-local quantity b,
O~b = 1 V a d~ /~ eiJ~(Ojw~ - Ok~j) , (2.6)
where e~f. is the fully antisymmetrie tensor with respect to the metric T with exu = 1. It is then clear that the dilatino variation of the dual back- ground is sero provided that b 4- Y is constant. This constraint can be reformulated as an axionic instanton condition on the dual background, be- cause V = e -2@, and b coincides with the axion field associated with B,~ = ~,. Another useful interpretation of b in the initial frame (2.1) is given by the nut potential of the ori~nsl metric [7]. We note that b exists only on-shell. Purther- more, the identification of the nut potential of the original metric with the axion field of the T- dual background is/mportant for describing not only the necessary condition for having manifest space-t/me supersymmetry, but also later for un- derstanding the concept of U-dmdity at work.
The original background being purely gravi- tational trivially satisfies the dilatino equation.
MAnifest space-time supersymmetry requires the existence of a Killing spinor ~/, which in turns im- plies the existence of a complex structure as a bilinear form in T/and ~ in the usual way. But a Ks~erisn Ricci-flat metric is also hyper-Kahler, and hence self-dual (or anti-self-dual),
R~ = ±1 deV~T--G e~'~R~.~ • (2.7)
Consequently, exploring the behaviour of space. time supersymmetry for T-duality transforma- tions of purely gravitational backgrounds forces us to consider self-dual solutions of the lowest or- der effective theory. In this case the original back- grounds exhibit N = 4 extended superconformal symmetry associated with the three underlying complex structures. According to the standard rules of manifest supersymmetry it is natural to expect that the dual background (2.2) and (2.3) will also ,.Thlbit N = 4 world-sheet supersymme. try, and its metric will be eonformally equivalent to a hyper-Kalder metric G~v (in the presence of non-trivial Bj, v and @) so that
G = f iG ' , CYf = 0 . (2.8)
It will be demonstrated later that this is possible only for b 4- V = 0 (up to a constant). Other- wise manifest space.t/me supersymmetry is lost, and non-local realisations of the N = 4 super- conforms] algebra have to be introduced for re- solving this paradox of the effective theory at the string level. In this case the d a d background ad- mits no Killing spinors, thus providing a vacuum solution of superstring theory that is apparently non-supersymmetric.
A potential trouble with supersymmetry has already been spotted in the adapted coordinate system (2.1) for the Killing isometry 0/01". The covariant description of the criterion for having manifest space-time supersymmetry after dual- ity can be formulated as follows: we divide the Killing vector fields K# into two classes, the tramdational and the rotational. The first class consists of Killing vector fields with self-dual (re- spectively anti-self-dual) ¢ovarlant derivatives
V,K = (2.9)
E. Alvarez et al. /Nuclear Physics B (Proc. Suppl.) 46 (1996) 16-29 19
while the second class encompasses all the rest. Consider now the conjugate fields S~ = b 4- V and introduce the quadratic qnsat i ty
= a,s a s , (?.1o)
which is clearly _~ 0. It is a well-known theo- rem that AS~ = 0 for trunslational isometries (in which case S+ is constant), while A S , > 0 for rotational isometries [8]. In the latter case it is al- ways possible to choose the coordinates z, ~ sad z so that $ I = z. This unalyJis provides the covari- ant distinction between those isometries that pre- serve manifest space-time supersymmetry after duality and those that do not. The result can be easily extended from purely gravitational back- grounds to more general solutions with N = 4 superconformal symmetry having sntisymmetric tensor and dilaton fields. Indeed, if the original background has a metric which is hypcr-Kabler up to a conformed factor, as in (2.8), the T-dual background will aim be manifestly supersymmet- tie if the Killing vector field i, translational with zespect to G ~.
Next, we investigate in detail the specific form of hypex-Kaldez metrics with one isometry, and use it to determine the dependence of the cor- responding complex structures and the Killing spinors on the Killing coordinate 1". It turns out that for rotational isometries these quantities de- pend explicitly on r , which is the key for having non-local realisatious of supe~ymmetry after T- duality. This problem does not arise for trmuda~ tional isometries as nothing depends explicitly on r; for this reason trunslational isometrics are also known us tri-holomorphic.
Translational isometries : In this case the adapted coordinate system (2.1) can be chosen so that
and so self-dual metrics with translational isom- etry are determined by solutions of the 3-dim Laplace equation
(a; + a ; + a, )v = o. (2.12)
The localised solutions are of the general form [7, 9]
n
m~... (2.13) v - t = i i , / = 1
where • is a constant that determines the asymp- totic behaviour of the metric sad n~, z ~ are moduli parameters. The apparent singnbLrlties of the metric are removable if all n~ = M and I" is taken to be periodic with range 0 < r < 47rM/n. For e = 0 the resulting metrics are the multi- center Gibbons-Hawking metrics, with n = 2 be- ing the simplest non-trivial example known as the Eguchi-Hmmon instsaton. For • ~ 0 (in which case its value is normali=ed to 1) one has the multi-Tanb-NUT metrics, with n = 1 being the ordinary self-dual Tanb-NUT metric.
The three independent complex structures are known to be r-independent. Choosing wa = 0 for convenience, we have the following result for the corresponding Kskler forms in the special frame (2.11) [10]:
F~ = (dr + ~2dy) ^ d~ - V - t dy ^ dz ,
F2 = (dr + ~ld:c) A dy + V - a d z A d z ,
F. = (dr + ~xdz + ~3dy) ^ d z -
V-X dz A dy . (2.14)
As for the Kilting spinors of the background (2.1), (2.11) one ena easily check that they are the con- stant, independent of say coordinates.
Rotational isometries : In this case the self- duality condition reduces to a non-linear equa- tion in three dimensions involving a function • (~, I/, z). The m e t ~ (2.1) can always be cho- sen so that [8]
~ t = q : g e W , ~ 2 = ± 8 , ~ , ~ a = 0 , V - t = a , e , "Y~i = d/ag (e t , • e , 1) , (2.15)
where • satisfies the continual Toda equation
(a: + a ; ) t + ° = o. (2.1o)
It can be verified directly that A S , = 1, sad hence S~- = z indicating the anomalous beh&viouz of m~nifest space-time supezsymmetry under ro- tational T-duality transformations.
20 E. Alvarez et al./Nuclear Physics B (Proc. Suppl.) 46 (1996) 16-29
The Kalder forms that describe the three inde- pendent complex structures in this case axe not all y-independent; one of them is a SO(2)-singiet, while the otl~er two form a SO(2)-doublet. We have explicitly the following result [2] for the dou- blet,
where
fx = ( d r + ~ 2 d r ) ^ d z - V - I d z ^ d ~ ,
f2 = (dr + ^ dr + V - l d z ^ d= (2.1S)
and for the singlet,
Fa = (dr + w ldz + w2dr) ^ dz+
V - l e V dz ^ d r . (2.19)
It is straightforward to verify that they axe eo- vaxiantly constant on-shall and uLtisfy the $U(2) Clifford algebra, as required. As for the Killing spinors in this case one finds that they axe con- stant spinors up to an overall phase • +iT/4 that depends explicitly on r .
Rotational isometries axe more rare them trans- lational isometries in 4-dlm hyper-Kabler geom- etry. The only example known to thiffi date with only rotational isometdes is the &tiyab-Hit~hin metric, whereas other metrics like the Eguchi- Hanson and Taub-NUT ,Thlbit both (see for in- stance [10]). The fiat space also *Thibits both type of isometrles. Choosing ~ = log z as the ~i~plest solution of the continual Toda equation we obtain the metric
d,' = zd.' + d e + de + dz, (2.20) z
which is the fiat space metric written in t e r m of the coordinates 2v~eos( r /2 ) and 2 v ~ d a ( r / 2 ) . Introducing r 2 = 4z and 0 = r / 2 we obtain the metric in standard polal coordinates,
da 2 = d:: ~ + dy 2 + dr 2 + r2dO ~ . (2.21)
This example is rather instructive because it pro- rides a good approximation of target space met- rics around the fixed points (located at r = 0) of a Killing isomettT. It is dear that the dual back- ground will have a curvature dnguladty at r = 0,
where the ~-model formulation of strings is ex- pected to break down. If one insists on exploring the dual metric from the low energy point of view by determining the corresponding dilaton field to lowest order in a ' , space-tlme supersymmetry will appear to be lost. Clearly more powerful tech- nlques should be used in order to understand the behaviour of strings dose to this point.
One might think that any isometry with fixed points could lead to anomalous behavlour of space-time supersymmetry under duality, as it is seen from the effective theory viewpoint. This is however not true in general. The relevant analysis of this hmue is rather simple for 4-dlm spaces M. At a fixed point p the action of K gives rise to an isometry Tp(M) --* Tp(M) on the tangent space, which is generated by the untisymmetric 4 x 4 matrix VI, Kv. Any such matrix can have rank 2 or 4, provided that the Killing vector tleld K is not zero everywhere. For rank 2 the fixed points form a 2-dim sn l~pace called bolt. The metric dose to a bolt can be appzoTimated by (2.21) and the apparent ~ugularity at r = 0 is nothing but a coordinate dugulaxity in the fiat polar coordi- nate system on R :t. For rank 4, p is an isolated fixed point called nat after the fixed point at the center of the self-dual Tanb-NUT metric. In this case the metric has a removable singularity and it is apprnrimated using the ~ t polar coordinate system on R a centered at the nut. Conmder now for general M (not necessarily hyper-Kahler) the decomposition
V, Kv = K+~ + K;~ (2.22)
into seif-dual and anti-self-dual parts. A nut is ~ id to be ,~it'-dual (or anti-self-dual) if K ~ is zero (in., K is tr~nilp, tional) at p. If M is hyper- Knldex, as it is the case of interest here, then K will be tranalational everywhere. Therefore we can have a Killing immetry with a fixed point (self-dual nut) which preserves manifest space- time supersymmetry under T-duality. For bolts, however, we always have
--~'--?(+ T¢+~" = K~. . ,K -~j' , (2.23)
u d so K l, cannot be t ru t l a t ionn l since other- wise both K ~ = 0 (ie., K, will be constant ev- erywhere if M is hyper-Kalder). Only in this
E. Alvarez et al./Nuclear Physics B (Proc. Suppl.) 46 (I996) 16-29 21
case manifest supersymmetry behaves anoma- lously under duality. For this reason we may consider the bolt-type metric (2.21) as a charac- teristic example of rotations] isometrics. In the next section we will study the 10-dim analogue of this in heterotic string theory and perform dual- ity transformations with respect to rotations in a 2-dim plane.
We return now to the general situation where the T-dus]ity transformation is performed on a purely gravitations] background (2.1). The start- dard description of T-dus]ity as t canonical trans- formation in I" and its conjugate momentum [11] amounts to
T --, f ( V - 1 0 ~ - ~18z - ,o2ay)dz-
+ + (2.24)
where we have set wa = 0. Applying this formula to the frame with translations] isometries we find that the complex structures (2.14) become in the dual background [2]
1 = V-X(±dr ^ dz - e jkdx' ^ dz . (2.25)
Here 4- refers to the right or left structures, which are not the same becsnse the duality generates non-trivis] torsion (2.3). We also note in this case that the dual metric (2.2) is conforms]]y fiat (and hence hyper-Kahler) with a conforms] fac- tor f~ = V - I that atisfies the Laplace equation (2.12) in agreement with (2.8). Hence, the dual background ~Thlbits N = 4 superconforms] sym- metry which is locally res]med with the aid of the dual complex structures, and space-time super- symmetry is msnlfest.
For rotations] isometrics the duality transfor- marion (2.24) acts differently on the complex structures and yields the dual forms
F~ = V-X(4-dr A d z % e t d z ^ dy) , (2.26)
and
= A d z - d z A r i l l ) ,
f ~ = V - X ( + d 1 " ^ d y + d z A d z ) . (2.27)
Then, we see explicitly that F+ become non- 1,2 locs]ly res]~ed in terms of the target space fields
[2]. These non-locs] vurisbles csa be used explic- itly to construct a new renl~ation of the N -- 4 superconforms] algebra, which is preserved under duality. Indeed only the renliation chanses form, since otherwise T-duality would not be a string symmetry. Hence, we conclude that non-local world-sheet effects have to be taken into account in order to restore the space-time supersymmetry that is apparently lost in this case. If one takes seriously the low energy effective theory will face the paradox that the dual metric (2.2) is not con- formally hyper-Knlder anymore. In fact, even if we have one local complex structure (provided in this example by Fs) we do not seem to have man- ifest A r = I space-time supersymmetry, although the converse is s]wsys true. Further details on the resolution of this issue will be presented in the next section.
The non-locs] renfiation of the N = 4 su- perconforms] algebra that arises in this context is reminiRcent of the psrafermionlc renliations in conforms] field theory. Unfortunately there is no exact conforms] field theory description of the backgrounds (2.2), (2.3) and so to strengthen this analogy we appes] to another example that a similar problem arises after T-dus]ity. Namely, consider the pair of cruet modeb SU(2) x U(1) and ( s v ( 2 ) / u ( 1 ) ) x u 0 ) ' , which are r ted by T-dus]ity with respect to a rotational immetry. Both models have N = 4 world-sheet supersym- metry, but in the latter it is non-locs]ly reaiiaed in term- of the SU(2)/U(1) pazsfennions [12]. The relevant analysis for the trsadormstion of the un- derlying complex structures has been performed in this case [2], in exact analogy with the above discussion, but we omit the details here. The background that arises to lowest order in a ~ for the second coset turns out not to be manifestly space-time supersymmetric.
A note of general value is that Killing spinors with explicit dependence on the Killing coordi- nate ~ are not maintained after duality.
Finally, we discuss briefly the behaviour of su- persymmetry under U-dus]ity transformations. For 4-dim string backgrounds with one isometry it is possible to intertwine T with S-duMity non- trivially and produce new symmetry generators. The key point in this investigation is provided by
22 E. Abarez et al /Nuclear Physics B (Proc. Suppl..) 46 (1996.) 16--29
the Elders transform, which for pure gravity is a continuous SL(2) symmetry acting on the space of vacuum solutions [13, 7]. It acts non-locally, as its formulation requires the notion of the nut po- tential b. More precisely, the dimensional reduc- tion of 4-dlm gravity for the metrics (2.1) reads as follows,
1 V - ' ( O , Vi~iV - OibSib)) , (2.28)
and so b± V form a conjugate pair of SL(2) /U(1) ~-model variables. The celebrated Elders trans- form is
Y V'- -
(Cb + D) 2 - C 2 v ~ '
b' ( A D + BC)b + AC(b 2 - V 2) + B D (2.29) = (Cb + D) 2 - C2V 2
where AD - BC = 1. Recall now the observa-
tion that the nut potential coincides with the ax- ion field of the T-dual background (2.2), (2.3). Therefore the Elders transform behaves as an S- duality transformation, where one starts from a purely gravitational solution and reaches the new one (2.29) via a sequence of T-S-T trAnlforms- tion within the context of the string effective the- ory (switching on and then off again non-trivial torsion and di~ton fields). Straightforward gen- eralisation of this argument to the full massless sector leads to an enlarged symmetry of the ~- function equations that is called U-duality (given in its continuous form). For 4-dlm backgrounds S and T-S-T are two SL(2) symmetries that com- bine into an 0(2, 2) group [1], while for the 10-dim heterotic string compact~ed on a 7-dlm torus this procedure yields the bigger group 0(8, 24) [14].
U-duality transformations have a more severe effect on supersymmetries, provided that the in- tertwining of S with T is performed using rots- tional isometries. It is rather instructive for this purpose to consider the class of self-dual met- tics and ask whether the Elders transform always preserves the self-duality. The answer is yes for translational isometries and no for rotational. A simple example to demonstrate this is provided
by the SL(2) action (2.29) on the fiat space met- ric (2.20) written in polar coordinates. Choosing A = D = I and B = 0 we may verify that the new metric fails to be self-dual [1]. On the contrary, the same group element acts in the tranRlational frmne by a simple shift
V -* --, V -x + 2 C , (2.30)
and while it preserves the self-duality it has a non- trivial effect on the boundary conditions; starting from (2.13) with • = 0, the parameter C generates solutions with e # 0. The main point we wish to make here is that rotational isometries, from the space-time point of view in the T-S-T dual face, apparently destroy all three complex structures; otherwise Ricci-flatness would imply self-duality for the transformed metric (2.29).
It will be interesting to explore the pomibKity to have non-local re~ligations of supersymmetry in this case by reformulating the Elders symme- try as a non-local transformation on the target space coordinates. This way we hope to extend the results of our investigation on "supersymme- try versus duality" to the most general situation for U-duality symmetries that arise by compactili- cation to three or even two space-time dimensions [15].
8. Dua l i t y in he te ro t i e s t r ing t h e o r y
We begin this section by working out the T- duality transformation for a general (1, 0) her- erotic c-model with arbitrary connection and background gauge field. We find that if one does not want to have a non-local dual world-sheet ac- tion, due to anomalies which appear when im- plementiug the duality transformation, one has to transform under the isometry the right-moving fermions. This yields a non-trivial transformation of the background gauge field under T-duality [3]. We also find that if in the original model the gauge and the spin connections match, so that there is anomaly cancellation in the effective the- ory, the change in the gauge field under T-duality ensures the same matching condition in the dual theory. Furthermore, if the original theory had (2, O) or (2, 2) superconformal invariance, the dual theory also has these properties.
E. Alvarez et al./Nuclear Physics B (Proc. Suppl.) 46 (1996) 16-29 23
We consider a manifold M with metric Gig, an- tisymmetric tensor field Bij u d a background gauge connection V/~tB associated to a gauge group G E 0(32) (for ~implicity we consider the 0(32) heterotic string). We introduce the (1, O) superfields
where zi(0.) are the fields embedding the world- sheet in the target space, F ~ are an~li~ry fields, and the fermions A and ~ have opposite world- sheet chlrality. Using light-cone coordinates on the world-sheet, 0.~ = (0.°:t:0.~)/v/2, and defining the operator
o is o D ~ = i 8 + (3.2) D = ~ + 80"+ '
and
2)q~.4 __ D ~ + ~ A 8 ( X ) D X i ~ S , (3.3)
we consider the Lagranglan density
= / d O ( - i ( G i i + B i i ) D X i g _ X i -
~ v ~ ~) . (3.4)
Ellmln~ting the anxqllary fields one obtains [16]
+ 2 F ~ j ~ s ~ ' ~ ¢ ~ ¢ ~ , (3.~) i¢~tD+¢A
with world-sheet supercurrent of type (1,0),
¢+ = (2G~ + ~ ) 8 + ~ ~ - ; H , ~ , ~ ' . (3.6)
We will carry out a duality transformation with respect to an isometry of the metric that leaves (3.4) and (3.~) inva~-iut. Following the proce- dure outlined in [17], we gauge the isometry, with some gauge fields A±, and add an extra term with a Lagrange multiplier making the gauge super- field strength vanish. If we integrate out the La- grange multiplier, the gauge superfields become pure gauge. Using the invaxiance of the action we can change variables to remove all presence of gauge fields and recover the original action. If instead we integrate first over A± and then fix the gauge, we obtain the dual theory. In our case
we will perform these steps in a manifestly (1, 0)- invariant formalism. Recall that the necessary condition for gauging u isometry generated by a Killing vector field K ~ of the metric is [18]
KiH~j~ = 0iUk -- 8,Uj , (3.7)
and
5KB, j = 8,(KIBlj + Uj) - 8j(K1Bu) , (3.8)
for some vector U. Then, the conserved (1,0)- supercurrent for the first term of (3.4) is
.7._ = ( K , - r h ) 8 _ x ~, J+ = (K~+~r~)DX' ,(3.9)
so that D f f _ + 8_ if+ = 0. We now introduce (1,0) gauge fields A _ =
A_ + 0 X_ and .4 = X + igA+ of bosonic and fermionic character respectively. If e(~, 0) is the gauge parameter, we can take 6~.A._ = - 8 _ e and 6,A = -De , and with the variation 6~X ~ : eKe(X) the gauge invariant Lagranglan (assum- ing that KiU, is constant) is
(G~i + B~i)DX'8_X~ + Y+.~_ + .7_.4 + K 2 A _ A . (3.10)
The left-moving part W(DW + V~DXi)~ is in- variant under this global transformtion when the isometry variation can be compensated by s gauge transformation
6X ~ = e K ~ ( X ) , 6~ = - , ~ ,
6Kv, = v , , , = 8,,, + IV,.,,]. (3.11)
which implies
KiF, j = Dip ; ~ = ~ - K'V~ . (3.12)
Making e a function of (1, 0) superspace one ob- tains after some algebra 6 . (~Ti)~) -- De~T/~W, and hence, adding the coupling . 4 ~ T p ~ we achieve gauge invariance. The full gauge invaxi- ant Lagrangiaa reads
L = - i ((G,j + B , j ) D X i O _ X j + ?+.&_+
i f_ .4+ K2A_.4) _ (~TD~ + .4~T#~) . (3.13)
Add now the Ls4~ange multiplier superfield term iA(D.4_ - 8.4) and integrate out A and .4_
24 E. Alvarez et al./Nuclear Physics B (Proc. Suppl.) 46 (1996) 16-29
to obtain the dual Lagranginn
Ll = - i ((O,i + ~,i)DX'a-X~ + (y+ + DA).
(3 .14)
In coordinates adapted to the Killing vector the dual values for G~j, B~i, ~ are
1 1 ~oo = K2 ' ~o= = ~-~-V',, ,
Or# = Ga# - K=K~ - U=~# K a
1 [~o= = - - k ' ~ K o ,
B=# = B=# + KaBo# - K#Boa K ~
1
1 #=An = v=. ,n - ~ . ( K , , + rY=,)pAs • (3.15)
These results axe equivalent to Buscher's formnlae (~ee for ~ t a n c e [5]), but in this case we find a change in the background gauge field as well.
The preceding formnl&e were obtained using only classical manipnlations. In general, how- ever, there will be anomalies and the dual action may not have the same properties =us the origi- hal one. Depending on the choice for/= and the gauge group G E 0(32), the dual theory may be afflicted with anomalies, in which case the two theories axe not equivalent. Equivalence would follow provided we include some Wess-Zumino- Witten terms generated by the quantum measure. I f we want the two local Lagrangians L and L to be equivalent, we must find the conditions on G, B, V and # in order to cancel the anomnlies. Using standard techniques in anomaly computa- tions, the vaxiation of the effective action is ex- pressible in the form (see [3] for details)
6Pry/ = 1 ?
[ dZSTr(w,O_X ~ - A_fi)D(e(KJcoj + fI)) G J
+T,(V;VX' - .m)O_(~(K~ + . ) ) (316)
where daZ = d2¢d0, and we have introduced the effective gauge field
V_ "b = w- "~ - A- f ] " b • (3.17)
Note that the dual theory will contain non- local contributions unless we cancel the anomaly, which is a miTture between U(1) and ~r-model anomalies [19]. The simplest way to cancel it is to assume that the spin and gauge connection match in the original theory, a condition that also makes the space.time anomalies to cancel (see for instance [20]). In this case if p = f~ with match- ing quadratic Casimizs, the anomalotm variation 6P,I ! can be cancelled by a local counterterm. A test of the vnlidity of the generalised duality transformation is that if we staxt with s theory with matching spin and gauge connection w = V, the dual theory is also guaxanteed to lutve ~ = # . It is straightforward to verify this condition in our case. This is also important for the consistency of the model with respect to global world-sheet and taxget-space anomalies u d it implies that if the original theory is conformally invaxiant to O(a ' ) , so is the dual theory.
There is yet one more possible source of anoms- lles under duality if the original model is (2, 0) or (2, 2)-superconformal invadmtt. In the (2, 2) ease for instance we have a U(1)~ x U(1)R current al- gebra. The manifold has a covaxiantly constant complex structure, V i 3 i i = 0, J i k 3 k j = --5 i j . The R-symmetry is generated by the to- tation 6A i = ~J' ~.,~ with current ,7+ = i3~jAiA j . This current has an anomaly
a_y+ = -~--;P~i. z& 'o+='o_z~ , (3.1g)
which can be removed only if the right-hand side is cohomologicaHy trivial. Prom [17] we know that T-duality preserves N = 2 global supersym- merry, hence, we should be able to improve the dual R-current so that the U(1)L x U(I)R current algebra is preserved, s= needed for the appllcs- tion of the theorem in [21]. Since 8enerica~y a T- duality transformation generates a non-constant dilaton, the energy-momentum tensor of the dual theory contains an improvement term due to the dilaton @ of the form 0~ @. As a consequence of N = 2 global supersymmetry there should aim be an improvement term in the fezmiouic cur- rents and in the U(1) currents. Since the one- loop ~-function implies (in complex coordinates) Ra/J ,., 0=0~@, we can improve the U(1)r. x U(1)s
E. Alvarez et al./Nuclear Physics B (Proc. Suppl.) 46 (1996) 16-29 25
currents so that they me chizally conserved. In the (2,2) case the improvements are
Ay+ = ~o~a+z" - aaca+z a ,
~ y _ = - ( 0 o ~ a _ z o - a ~ ¢ a _ z ~ ) . (3.10)
With these improvements the currents are chi- rally conserved to order a ' (and presumably to all orders, since the higher loop counterterms are cohomologically trivial for a (2, 2) supersymmet- tic e-model); hence we conclude that under du- ality the (2, 0) or (2, 2) supereonfozmal algebra is preserved, as was also pointed out earlier. We meet the conditions to apply the Banks et al the- orem [21] implying that the theory is space-time supersymmetric.
In the following we give a concrete mmwer to the puzzles raised in section 2 for space.time su- persymmetry in the context of 10-dim heterotic string theory. We consider the motion in fiat Minkowski space
ds' = dr 2 + r'dO' + (dz~) ' - (dr°) 2 , (3.20)
(with i = 1, . - . , 7) and revisit the problem of "space-time supersymmetry versus duality" first from the effective action point of view, sad then within the framework of conformnl field theory. The frame (3.20) provides the natural general- i~ation of the bolt-type coordinates (2.21). Be- fore duality we have the full IS0(1,9) Lorents invuriance and 0(32) gauge symmetry. Since we perform duality in (3.20) with respect to ro- tations in a 2 -d lm plane, only the subgroup of ISO(1, 9) commuting with them will be a mani- fest local symmetry of the effective action. SimS- laxly if we preserve manifest (1,0) supersymmetry on the world-sheet sad avoid anomalies, we em- bed the isometry group SO(2) C G = SO(32). The subgroup of G commuting with SO(2) is SO(30) x SO(2) sad once again this will be a manifest symmetry in the low energy theory. It is well known that under T-duality, symmetries not commuting with the ones generating dual- ity are generally realized non-locally. Hence al- though the dual background still contains all the original symmetries from the CFT point of view, the low energy theory does not seem to exhibit them. The theory will be explicitly symmetric
under IS0(1, 7) x $0(30) x S0(2) only. We want to make sure nevertheless that the origins/space- time supersymmetry is preserved, although not in a muifes t O(1, 9)-covariaat formalism. For this we e u consider the variation of the fermlonic de- grees of freedom in a formaImm adapted to the Lqo(1, 7) x SO(30) x SO(2) symmetry, and look for which combination of the 0(1, 9) fermions ,,re =.nlhilt, ted by lupersymmetry.
The low energy approximation to the heterotic string is given by N = 1 supergravity coupled to N = I super Ysag-Milis in d = I0. In ten di- mensions we can impose simnltsaeomdy the Ms- jorana sad Weyl conditions [22]; then, in terms of SO(l, 7), a M&jorsas-Weyl spinor of $0(1,9) becomes a Weyl spinor. Write the Dirae algebra (in an orthonormnl frame) as
F j , = ~ ® ' y ~ ; p = O , l , . . . , 7 ,
F T + i = i ~ @ l ; i = 1 , 2 ,
(3.21)
where 3'e is the ualogous of the 4-dlm 3'i in eight dimensions. Ten dimensional indices will be hat- ted. The supersymmetric variation of the ten- dlmensional fermions is given by
6~ = ( r ~ a ~ - ] n . t r # ~ ' ) ~ ,
6X at = _ I F X ~ o r ~ (3.22)
for the gravitino, dilatino sad giuino, respectively. The dual background is
= dr 2 + ~ d 0 3 - (dffi°) 2 + (dz~) 2 ,
¢ = - l o g , , V~d~ = -~d~M, (3.23)
where M is the matrix describing the embedding of the spin connection in the gauge group, which we take to be the standaxd one acting only on two of the right-moving fermions. The background gauge field strength is
• ' = - ~ d , . ^ ~ , (3.~4) T "
26 E. Alvarez et al./Nuclear Physics B (Proc, Suppl.) 46 (1996) 16-29
and so decomposing the above variations with re- spect to SO(I , 7) we find: for the gravitino
i
for the dilatino
6A - --;(Tt ® l )e , (3.26)
and for the g]nlno
i ~X A = 0 , 6X = - ~ ( ~ , ® I)~ (3.27)
for A G SO(32) and along the embedded SO(2) respectively. In the preeeding formulas • is an $ 0 ( 1 , 7 ) Weyl splnor with the same number of independent components as a 10-dlm Majorana- Weyl spinor.
If we define now
~ = ~ , , ~ ' { , ) = ~ { , ) ,
it is easy to see that the new fie]& transform as 6~'~. -- a~e. Similarly,
-- A + ~e-*(T= ® 1)X ,
-- X - / e * ( ~ ® I)A (3.29)
have vanishing variation under space-time super- symmetry. Furthermore, they have the correct ch/ralities as d/crated by the ten-dimensional mnl- tiplet. Thus if we use a formalism covariant only under the explicit $0(1 ,7) x S0(30) x $0(2) symmetry of the dual background we recover the full number of supersymmetric charges. Prom the low-energy effective action this is the most we eonld expect since at the level of the world- sheet CFT the full symmetry SO( l , 9) x SO(32) is only real~ed non-locally. If we want to consider the complete symmetry and the complete mass- less spectrum in the dual theory it seems that the only reasonable thing to do is to go back to the two-dlmensional point of view.
In the remaining part we will see how to ohtnin in principle the vertex operators for the full mass- less spectrum in the dual theory by investigating
in detail the way the full symmetry is realised from the conformal field theory point of view. The main point is to show explicitly that there are indeed world-sheet operators in the dual the- ory associated to the space-time supersymmetry charges, although some world-sheet non-locallty is generated. We will find at this end an interest- ing interplay between the picture-changing oper- ator and T-duality.
We are considering the effect of rotational du- ality in s 2-dlm plane, and so the relevant part of the free heterotic Lagrangian is
L = + i,#Aa+ A +.. .(3.30)
where the vector quantities are two-dlmensional. The isometry is ~ --, exp(~s3)~, but for the time being we shall work in an unadapted frame. Hence for (3.30) we can perform duality only in the bosonic sector. The world-sheet supercurrent is
g+ = ~. a+~ = ~ . /~+ , (3.31)
where if+ is a ehlral current generating transla- tions in the target space. It is convenient to work in canonical pictures [23] ( - { for fermion vertices, - 1 for boson vertices). The space-time supersym- re®try charge is
qo (- i ) = / e - C a n , (3.32)
where ~b is the scalar which bosonlses the super- conformal ghost current and Sa is the spin-field associated to the ~-fermions. The translation op- erator in the - 1 picture is
p# (-t) _. / e-÷A# . (3.33)
Note that in these definitions only the space-time fermion and the (fl, 7)-ghosts appear. Hence
{0° (-t), Qa (-t)} = r#P (-t) (3.34)
is satisfied, and if we choose to perform duality for the bosonic part of the Lsgrangian only, the same relstionshipl should still hold.
From this point of view there is dearly no prob- lem with space-time supersymmetry. However, in constructing scattering amplitudes we need to use
E. Alvarez et al./Nuclear Physics B (Proc. Suppl.) 46 (1996) 16-29 27
vertex operators in different pictures. Hence any problem should come from the interplay with the picture changing operator 9 . The picture chang- ing operator acting on a vertex operator Yg(z) in the q-picture can be represented as [23]
WV,(z) - - | i m ~ _ . . , , e 4 ' ( u ' ) Q + ( w ) V e ( z ) . ( 3 . 3 5 )
The only poesible difficulties may appear in anomalies in the world-sheet supercurrent under duality. Since g+ does not commute with purely bosonic rotations, after duality ~+ will become non-local in the world-sheet. To guarantee that there are no problems with g+ we want to make sure that the dual world-sheet supercurrent still
has the form ~- P+, where JS+ is the representa- tion of the translation current in the duul theory, and it is here that the non-locallty resides. In fact the full theory in (3.30) can be constructed out of the knowledge that P+ ~ (i = I, 2) is chiral]y conserved and that its operator product expan- siou (OPE) is P'(z)PJ(w) ,'-, 6" ~=~-~. It is hard to believe that the existence of the chiral currents is going to be lost under duality. To make sure that this is not the case, the simplest thing to do is to include sources for these currents and then follow their transformation under duality.
Following [17] we gauge the symmetry ~ --, exp(/aez)£ and concentrate only on the bosonlc part of (3.30), the only one relevant due to the previous arguments. Thus our starting point is
L = D+zTD_z +AF+_ , (3.36)
where D~z = 8±z +/¢2zA±, F+_ = 8+A_ -
8_ A+ and A is a Lagrunge multiplier. Here and in the following we drop for convenience the vec- torlal notation for • and ~, and introduce • =/¢2. Using the A-equation of motion, A+ = 8±~, D + z = e-a '8±(e" 'z) , and changing variables z -* e -a ' z , the original theory is recovered. It proves convenient to pazametrise locally
A+ = 8+at. , A_ = 8 _ a s . (3.37)
Then (3.36) has the symmetries
&z = e - ' a Z a a , &A = - z T e e - ' a ~ a a , (3.38)
and
6z = e-ea~aL , 6A = zT ~e-'a~aL , (3.39)
yielding respectively the conserved currents
8 _ ( e m a D + z ) = 0 , 8+(em~D_z) = 0 . (3.40)
When aR = aL = a, we recover the original cur- rents 01(e°az). Acting on z, these symmetries c o m m u t e .
The sources to be added to (3.36) should be gauge invariant,
J_e '÷X D + z + J+e "~L D _ . (3.41)
The exponents are non-local in A~, and they make the coupling gauge invarianL This also guarantees that the currents in (3.41) satisfy the OPE of the original theory as expected. In our example, let us take for simplicity J+ = 0. The most straightforward way to integrate out the gauge fields is to work in the light-cone gauge A_ = 0. Then the integral over A+ becomes a 6-function which can be solved in two ways. If we choose to solve it in order to write the Lagrange multiplier A as a function of the other fields, we recover the o~ginal theory.
On the other hand if we choose to solve the adapted coordinate to the isometry in terms of A and the other variables, we obtain the dual the- ory. Furthermore we also obtain a determinant, which when properly evaluated ([5] and references therein) yields the transformation of the dflaton. Using polar coordinates r, 8 for the 2-dlm plane we obtain the equation
i " 3_" ~ ) (3.42) = - -
where w = e ~e and 3"_ is the complex conjugate of 3_. This is a Riccatti equation, which can be solved order by order in 3_. Restoring powers of ote w e h a v e
o~ s 8 _ , o = , o ; i g _ A - - ( 3 . 4 3 )
The lowest order solution is
w = exp i ~ - i0_Ad¢- , (3.44)
and to this order the current looks like
28 E. Alvarez et al./Nuclear Physics B (Proc. Suppl.) 46 (1996) 16-29
The extra terms depending on J_, .T*_ are re- quired to guarantee the equality between the cor- relation functions before and after the duality transformation. To leading order the currents a r e :
0 f , o , - . ) Note however that in solving (3.43) there will be correctious to all orders in a I in order to ob- tain the correct OPE's for the dual currents/3+ • These currents can be used to write the emis- sion vertex operators in the dual theory and they are almost always non-local. Since the OPE's of /~± are preserved, the spectrum of the original and the dual theories are equivalent. Neverthe- less, we have to be careful regarding the operator mapping.
In conclusion, there is no problem with space- time supersymmetry from the point of view of CFT, but the correct operators that needed to be used to represent the emlLqlon vertices of low energy particles in the dual theory are often non- local, and do not admit a straightforward a ' ex- pansion unless we write the dual states in terms of those which follow from the correspondence as dictated by the duality transformation. When there are curvature singularities the approach based on the effective low energy theory has many limitations and to obtain reliable information we should go back to the underlying string theory. Finally to find the graviton, gravitino, etc vertex operators in the dual picture we could have solved the anomalous dimension operators in the dual background (including the dilaton and the back- ground gauge fields), as was done for tachyous in [24]. We believe that the two approaches are equivalent.
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