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.