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STRING FIELD THEORY EFFECTIVE ACTIONSTRING FIELD THEORY EFFECTIVE ACTION FORFOR THE TACHYON AND GAUGE FIELDSTHE TACHYON AND GAUGE FIELDS
secondo incontro del P.R.I.N.secondo incontro del P.R.I.N.““TEORIA DEI CAMPI SUPERSTRINGHE E TEORIA DEI CAMPI SUPERSTRINGHE E
GRAVITA`”GRAVITA`”Capri, October 2003
Marta OrselliMarta Orselli
Based on:Based on:Phys. Lett. B543 (2002) 127, Phys. Lett. B543 (2002) 127, in collaboration with:in collaboration with:G. Grignani G. Grignani (Perugia University)(Perugia University) , M. Laidlaw (UBC), , M. Laidlaw (UBC), and G. W. Semenoff (UBC),and G. W. Semenoff (UBC),andandhep-th/0311xxx, hep-th/0311xxx, in collaboration with:in collaboration with:E. Coletti (MIT), V. Forini, G. Grignani (Perugia University) E. Coletti (MIT), V. Forini, G. Grignani (Perugia University) and G. Nardelli (Trento University)and G. Nardelli (Trento University)
•
•
PLAN OF THE TALKPLAN OF THE TALK
• Witten-Shatashvili String Field Theory Witten-Shatashvili String Field Theory (BIOSFiT)(BIOSFiT)
• MotivationsMotivations
• non-linear non-linear -function-function
• RGRG String dynamicsString dynamics
=0=0 scattering amplitudes scattering amplitudes
• Tachyon and Abelian gauge fieldsTachyon and Abelian gauge fields
• BIOSFiTBIOSFiT Cubic SFT Cubic SFT
• ConclusionsConclusions
MOTIVATIONSMOTIVATIONStwo formulationsCUBIC STRING FIELD THEORYCUBIC STRING FIELD THEORY BOUNDARY STRING FIELD THEORYBOUNDARY STRING FIELD THEORY
Abstract definition, complicated star product. Can be quantized and reproduce perturbative on-shell amplitudes.
or Background Independent Background Independent open String Field Theoryopen String Field Theory. Directly tied to world-sheet RG picture. Exact results for tachyon condensation.
Witten 1986
Witten-Shatashvili 1992
lead toan effective actioneffective action for the field representing the bosonic open string modes and• provide a solution to the problem of what is the configuration space of
string theory. • provide a non-perturbative formulation of string theory.motivations for our motivations for our
workworkEstablish a relationship between the effective actions of Cubic SFT and Witten-Shatashvili SFT.Correct the result found by Kutasov, Marino, Moore (hep-th/0009148)
linear -functionwrong integralStudy on the disk the relation between string dynamics and RG flow:
how the on-shell scattering amplitudes emerge from the fixed points of the theory.
Find a correct formulation for the effective action that could be extended to the non-abelian case should lead to derivative corrections to the BI action
Calculation beyond II order are very complicated. We arrive at the III order in BIOSFiT.
WITTEN-SHATASHVILI STRING FIELD THEORYWITTEN-SHATASHVILI STRING FIELD THEORY
Sen’s conjectures on tachyon condensation (A. Sen 1999):
An open bosonic string in 26 dim. contains a tachyon T, a massless gauge field A and an infinite tower of massive fields.
tachyon The theory is unstable
1:the form of the tachyon potential is: TMfTU
mass of the D-brane
universal function
2:there are soliton configurations of the tachyon field on unstable Dp-branes – lower dim. branes - .3:at the new vacuum there are no open string states; it describes the closed string vacuum.To demonstrate the validity of these conjectures one can use the Witten-Shatashvili string field theory. In this theory, the configuration space of the open string field is seen as the “space of all 2-dim. world-sheet field theories” on the disk. The world-sheet action and correlation functions are given by:
XVdSSws
2
00 2 .... .... wsSedX
free action defining an open + closed conformal background
general boundary boundary perturbationperturbation of ghost number 0
ObV 1Usually V is defined in terms of a ghost number 1 operator O cVO If V is constructed out of matter fields alone,
then
XVdSSws
2
00 2
From the action
The boundary term modifies the b.c. on X from the Neumann b.c. (follows from S0) 01 rrX to “arbitrary” non-linear condition
Dr XVX
X
The space-time action S(O) is formally independent of the choice of a particular open string background (Witten ’92) and it is defined trough its derivative
OQdOddKOdS ,222
2
0
2
0
Q is the BRST operator
DC BRSTJQ
Since dO is an arbitrary operator, all solutions of the eq.n dS=0 correspond to boundary deformations with {Q, O}=0 2dim. theory is conformal (scale invariant, =0) valid string background.
...2
XXCXXXBXXAXTXV
V(X) can be expanded into “Taylor series” in the derivatives of X
,..., XAXTSS the action becomes the functional of the coefficients
GoalGoal: write S as an integral over the space-time (constant mode of X()) of some local functional of T(X), A(X),…
XX 0with the condition
The action is a kind of field theory in space-timeMore generally we can parametrize the space of boundary
perturbations V by couplings gi
i
iiVgV
The coefficients gi are couplings on the world-sheet theory and are regarded as fields from the space-time point of view.
At the origin, gi=0, the theory is un-deformed0SSws
and in linear approximation the deformation is given by the integral of Vi
000
gii
ii
ws gVg
gSVgSS
For arbitrary perturbation the theory is non-renormalizable, because the Taylor expansion of V contains an infinite number of massive fields.But for the case of the tachyontachyon and gauge fieldsgauge fields only, the theory is renormalizablerenormalizable (perturbatively).
In this parametrization, the expression of the action is (Shatashvili ’93)
gZg
ggS ii
1Witten-Shatashvili actionWitten-Shatashvili action
The derivative of the action with respect to the coupling has a zero exactly where the theory is conformal
0 ggGggS j
iji this means 0gj
because the metric G has to be invertible and non-degenerate, otherwise we would have an extra zero which cannot be interpreted as conformal field theory on the world-sheet
at the fixed point gZgS 0
This action seems to be only formally background independent. In the world-sheet formalism background independence is manifest, it is lost once we compute the action S perturbatively. If the relation between the action S and the partition function Z is true to all orders in coupling constant, then we recover the background independence. It seems to depend on the choice of coordinates in the space of boundary interactions (choice of contact terms). If we ignore contact terms, then the -function is linear
Q depends on the couplingsQ depends on the couplings
The way to fix the structure of contact terms is that, since dS is a one-form, whatever choice of contact terms we made in the computation, d of dS should be zero
0,222
2
0
2
0
OQdOddKddSd
This leads to the formula with all non-linear terms for the -function
lkjijkl
kjijk
ii
i gggggg 1
where is the anomalous dimension of the operator corresponding to the coupling gi, is the contribution of the 3-point function and so on.
ijk
Only relevant coefficients in the formula for the -function are those which satisfiy the “resonant condtion”
It means that the -function cannot be reduced to the linear part of it by a field redefinition and the non-linear terms cannot be removed.
It also means that in the expansion of S, coordinates should be chosen in such a way that the corresponding metric G is invertible and non-degenerate.
1 ikj
Do not ignore contact terms (Shatashvili ’93)
ATZA
AT
TS ,1
TZT
TS
1
XSdXZ ws exp][ Partition function
perturbatively super-renormalizable
bulk action interactions
2
0 2,,
41 XTdXXddXSws
where the action is
1
The bulk excitations can be integrated out to get an effective non-local field theory which lives on the boundary
qucl XXX field on the bulk 02 clX bdrycl XX 0quX
2
0 21
2][XJXTXiXd
jdir
jj
edXZZ
2
0ˆ
21
2][XikXTXiXd
j
jj
edX
The absolute value of the derivative operator is defined by the Fourier transform
n
inen
i
2
2
0 2ˆ jj XdX
zero mode
WITTEN-SHATASHVILI ACTIONWITTEN-SHATASHVILI ACTION
First order
2
0 11ˆ
21
22
0 11
1
1
2
][XikXikXiXd
dir
jj
ekTddkdXZ
kZ
The functional integral over the non-zero modes of X() gives
XkkiGk
jdir
ekTdkXdZ
kZ ˆ02
11
11
21
ˆ
Green function
2
sin4log 21221
Gambiguity in subtracting the divergent termsWhen the Green function is not
defined21
Introducing a cut-off , we set log20 G
11
2
2 kD
dir
kTZ
kZ
The integrals over the zero modes give a D dim. function and the result is
From this expression we can identify the renormalized T in terms of the bare coupling to the lowest order
12 kR kTkT
21 k anomalous dimension of the tachyon
All the integrals are well defined even for in the convergence region, so we choose to regularize by analytic continuation.
21
Second order
XikXikXik
dir
eeekTkTdkdkdddXZ
kZ ˆ2
0 212121
22211
24][
21
2
0212
21212121
2
2
sin4224
kk
RRD
dir
kTkTkkkdkdkddZ
kZ
21
221
212121
2
1212
21
kkkkkTkTkkkdkdk
ZkZ
RRD
dir
From this expression we can identify the renormalized T in terms of the bare coupling to the second order in perturbation theory
21
22121
2121211
1212
21
212
kkkkkTkTkkkdkdkkTkT kkDk
R
for021 21 kk
The functional integral over X gives
where 12 ik
iiR kTkT
Third order
all the expressions are understood to be completely symmetrized in the indices 1,2,3The renormalized T in terms of the bare coupling to the third order is
),,()()()()(2!3
1
1212
21
32112
321321321
212
2121212121
1
313221
212
kkkIkTkTkTkkkkdkdkdk
kkkkkTkTkkkdkdkkTkT
kkkkkkD
kkDkR
where I is the integral
313221313221
2sin
2sin
2sin
22,, 312322212
3213
222
321
kkkkkkkkkkkk
dddkkkI
),,()()()()(2!3
1)(321321321321
)3(
kkkIkTkTkTkkkkdkdkdkZ
kZ D
dir
The computation of I is highly non-trivial and the result is
)1()1()1()1()1()1()21()21()21()1(
312131323221313221
313221313221
kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkI
The convergence is for 01 313221 kkkkkk 021 31 kk
The computation of I is highly non-trivial and the result is
The convergence is for 01 313221 kkkkkk 021 31 kk
RGRG STRING DYNAMICSSTRING DYNAMICSTo compute the Witten-Shatashvili action we need the expression for the -function of the tachyon field.One of the most interesting topics of string theory is the relation between RG and string dynamics.
log
ii gThe RG -function is defined
as
The Witten-Shatashvili action provides a prescription for the metric G in the space of couplings. Then one needs the correct -function.We managed to prove also a weaker form of the relationship between the RG and string dynamics: the solutions of the RG fixed point eq.s can the solutions of the RG fixed point eq.s can be used to generate the open string scattering amplitudes.be used to generate the open string scattering amplitudes.
It is very hard to construct the metric G
A practical approach to off-shell string structure would be to obtain the e.o.m. for the particle fields associated with the string modes and then to reconstruct the corresponding action.This action could be an appropriate tree-level action in a field theory formulation of string theory. However, in general one has
jiji G
gS where G is some metric.
The solution of this equations can be written as
000000 lkjijkl
kj
ikj
ijkttiti gggbggeegetg ikji
),,()()()()(2!3
1
1221
2
12
22112
lkjkkkkkk
lkjlkjiD
lkj
kj
kjkji
kkkj
Dkji
kiR
kkkIkTkTkTkkkkdkdkdk
kkkk
kkkkTkTdkdkkTkT
ljlkkj
kji
t
mlkimj
mkl
ijm
ilkj
tijkl
mlk
mkl
ijm
ilkj
tijkl
imj
mkl
ijmi
jkl
mj
ikji
e
eeb
2
22where
lkjijkl
kjijk
ii
ii gggggg
dtdg
The most general RG eq.s for a set of couplings is
logt
bare coupling
21 ii k
ikjikjkjikj kkkkkkkk 222222 11101021
kji
kj
kjijk kkk
kkkk
12
222
lkji
lk
lk
kjkj
ljkjlkljkj
ijkl kkkkcycl
kkkk
kkkkkkkk
Ikkkkkk
1
211
22212
!31
22
We find
cyclkkkk
kkkkkkkk
Ikkkkkk
kTkTkTkkkkdkdkdk
kkkkkTkTkkkdkdkkTkk
lk
lk
kjkj
ljkjlkljkj
D
DT
121
1222
12
)()()()(2!3
11
222211
22
321321321
212
21212121
2
NON-LINEAR NON-LINEAR -FUNCTION-FUNCTION
)1()1()1()1()1()1()21()21()21()1(
312131323221313221
313221313221
kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkI
where
FROM FROM =0=0 SCATTERING AMPLITUDESSCATTERING AMPLITUDESLowest order equation
01 02 kTk
Next order kTkTkT 10
2
212
12
2
2
2010212121 kkkTkTkkkdkdk
kkT D
1k 2k
k
The residue of the pole is the scattering amplitude for 3 on-shell tachyons. In our notation is 1/2One more order kTkTkTkT 210
),,(1)()()(
)(213
1
321313221302010
32132122
kkkIkkkkkkkTkTkT
kkkkdkdkdkk
kT D
The residue of the pole is the 4 tachyon scattering amplitude ),,(131
321313221 kkkIkkkkkk
1k 3k
k
2k
1k 3k
k
2k
1k 3k
k
2k
The 4 tachyon amplitude is the sum of a contact graph and a tachyon exchange graph
Using the on-shell condition we recover the scattering amplitude for 4 on-shell tachyons
.21,212
132212 cyclkkkkB
Veneziano Veneziano AmplitudeAmplitude
.)1()1(
)221()22()222()21(),,()1(4
)()()()(2!3
1)1()22()()(21
31212
32
312132312132321313221
32132132122
2
cyclkkkkkk
kkkkkkkkkkkkkkkIkkkkkk
kkkkTkTkTdkdkdkkkkTkTdkKS DD
pTK normalization constant proportional to the tension of the Dp-brane
WITTEN-SHATASHVILI ACTIONWITTEN-SHATASHVILI ACTION TZT
TS
1
cyclkkkk
kkkkkkkk
Ikkkkkk
kTkTkTkkkkdkdkdk
kkkkkTkTkkkdkdkkTkk
lk
lk
kjkj
ljkjlkljkj
D
DT
121
1222
12
)()()()(2!3
11
222211
22
321321321
212
21212121
2
)1()( TeTU T Exact tachyon Exact tachyon potentialpotential
(Kutasov, Marino, Moore,hep-th/0009148 Gerasimov, Shatashvili, hep-th/0009103)Near the perturbative vacuum, T=0 ...
31
211)( 32 TTTU
For k=0
32
321)( TTKTU The ratio of the cubic and quadratic
term is precisely the one that comes from the expansion of the exact potential
BIOSFiTBIOSFiT CUBIC SFT CUBIC SFT
)()()()(2
31)()(12
2 3213213212 kkkkTkTkTdkdkdkkTkTkdkKS DD
Near on-shell
has a zero for 12 k on-shell constant
This can be compared with the Cubic String Field TheoryCubic String Field Theory result. Near on-shell
)()()()(2
31)()(12
212 321321321
225
2 kkkkkkdkdkdkkkkdkTS DD
The required matching of the quadratic and cubic term implies
225TK kTk
21
This is in agreement with all the conjectures involving tachyon condensationProvides a further verification of the validity of our expression for the non-linear -function and the Witten-Shatashvili action.
the field redefinition is non-singular on-shell
Moreover this shows that, as expected, the Cubic String Field Theory provides an effective action for the tachyon to which corresponds a non-linear -function
We can compare the Witten-Shatashvili action obtained up to the third order in the tachyon field to the cubic string field theory action. We have found the off-shell field redefinition which relates the two formulations. Here I only show how they are related on-shell.
TACHYON AND GAUGE FIELDSTACHYON AND GAUGE FIELDS
12
1212
221
21
21212121
2121
2121
212
211
2121
21
2
kkkkkkkAkAkTkT
kkkkk
kkdkdkkTkT
kkkk
kk
DkR
At the second order
2121
212
21
212121
2121
1212
2
21
212
kkkk
kkkkk
kkkkkAkTdkdkkAkA
kk
kkDkR
)1()1()1()1()1()1(
)21()21()21()1(
312131323221313221
313221313221
kkkkkkkkkkkkkkkkkkkkkkkkkkkkkk
313221313221
2sin
2sin
2sin
22,, 312322212
3213
222
321
kkkkkkkkkkkk
dddkkkI
At the third order we have been able to show that all the integrals (except for the momentum dependence) can be expressed in terms of I
2121212121
2121
212
212
21
1212
2211
21
kkkkkAkAkTkTkk
kkkkk
kkdkdkkTkk
kk
DT
21
1212
2
21212121
2121
212
21212
21
kkkkkAkTkk
kkkk
kkkkkdkdkkAkk
kk
D
A
In the expressions for the -functions there are, as expected, only the terms consistent with the twist symmetry.
no T-A term
no T-T and A-A term
CONCLUSIONSCONCLUSIONS
We computed the Witten-Shatashvili action for the We computed the Witten-Shatashvili action for the tachyon and the Abelian gauge field up to the third tachyon and the Abelian gauge field up to the third order.order.
The Witten-Shatashvili and the Cubic SFT formulations The Witten-Shatashvili and the Cubic SFT formulations are shown to be equivalent (at least up to the third are shown to be equivalent (at least up to the third order in the tachyon and gauge fields) up to a field order in the tachyon and gauge fields) up to a field redefinition.redefinition. Our result can be extended to the study of the non-Our result can be extended to the study of the non-Abelian case.Abelian case.
We obtained the non-linear expression for the We obtained the non-linear expression for the --function of the couplings.function of the couplings.
The string dymanics emerges from the The string dymanics emerges from the -function -function fixed points reproducing the open bosonic string fixed points reproducing the open bosonic string scattering amplitudes.scattering amplitudes.