STRING FIELD THEORY EFFECTIVE ACTION FOR THE TACHYON AND GAUGE FIELDS

1. STRING FIELD THEORY EFFECTIVE ACTION FORTHE TACHYON AND GAUGE FIELDS Marta Orselli Based on: Phys. Lett. B543 (2002) 127, in collaboration with: G. Grignani (Perugia University), M. Laidlaw (UBC), and G. W. Semenoff (UBC), and hep-th/0311xxx, in collaboration with: E. Coletti (MIT), V. Forini, G. Grignani (Perugia University) and G. Nardelli (Trento University) secondo incontro del P.R.I.N. “TEORIA DEI CAMPI SUPERSTRINGHE E GRAVITA`” Capri, October 2003

2. RG String dynamics • BIOSFiT Cubic SFT • =0 scattering amplitudes PLAN OF THE TALK • Motivations • Witten-Shatashvili String Field Theory (BIOSFiT) • non-linear -function • Tachyon and Abelian gauge fields • Conclusions

3. lead to Witten 1986 an effective action for the field representing the bosonic open string modes and Witten-Shatashvili 1992 • 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 work • Establish a relationship between the effective actions of Cubic SFT and Witten-Shatashvili SFT. linear -function Correct the result found by Kutasov, Marino, Moore (hep-th/0009148) wrong integral • Study 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. • Calculation beyond II order are very complicated. We arrive at the III order in BIOSFiT. • 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 MOTIVATIONS two formulations CUBIC STRING FIELD THEORY BOUNDARY STRING FIELD THEORY Abstract definition, complicated star product. Can be quantized and reproduce perturbative on-shell amplitudes. or Background Independent open String Field Theory. Directly tied to world-sheet RG picture. Exact results for tachyon condensation.

4. universal function mass of the D-brane general boundary perturbation of ghost number 0 free action defining an open + closed conformal background Usually V is defined in terms of a ghost number 1 operator O If V is constructed out of matter fields alone, then WITTEN-SHATASHVILI STRING FIELD THEORY 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 Sen’s conjectures on tachyon condensation (A. Sen 1999): 1:the form of the tachyon potential is: 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:

5. From the action 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. V(X) can be expanded into “Taylor series” in the derivatives of X the action becomes the functional of the coefficients The boundary term modifies the b.c. on X from the Neumann b.c. (follows from S0) to “arbitrary” non-linear condition 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 Q is the BRST operator

6. The action is a kind of field theory in space-time More generally we can parametrize the space of boundary perturbations V by couplings gi 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-deformed and in linear approximation the deformation is given by the integral of Vi 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 tachyon and gauge fields only, the theory is renormalizable (perturbatively). Goal: write S as an integral over the space-time (constant mode of X()) of some local functional of T(X), A(X),… with the condition

7. The derivative of the action with respect to the coupling has a zero exactly where the theory is conformal this means 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 • 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 In this parametrization, the expression of the action is (Shatashvili ’93) Witten-Shatashvili action

8. This leads to the formula with all non-linear terms for the -function where  is the anomalous dimension of the operator corresponding to the coupling gi, is the contribution of the 3-point function and so on. 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. Do not ignore contact terms (Shatashvili ’93) Q 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

9. Partition function where the action is bulk action interactions The bulk excitations can be integrated out to get an effective non-local field theory which lives on the boundary field on the bulk The absolute value of the derivative operator is defined by the Fourier transform zero mode WITTEN-SHATASHVILI ACTION perturbatively super-renormalizable

10. When the Green function is not defined ambiguity in subtracting the divergent terms All the integrals are well defined even for in the convergence region, so we choose to regularize by analytic continuation. 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 anomalous dimension of the tachyon First order The functional integral over the non-zero modes of X() gives Green function Introducing a cut-off , we set

11. From this expression we can identifytherenormalized T in terms of the bare coupling to the second order in perturbation theory Second order The functional integral over X gives where for

12. Third order where I is the integral The computation of I is highly non-trivial and the result is The computation of I is highly non-trivial and the result is The convergence is for The convergence is for all the expressions are understood to be completely symmetrized in the indices 1,2,3 The renormalized T in terms of the bare coupling to the third order is

13. RG STRING DYNAMICS 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 where G is some metric. It is very hard to construct the metric G To 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. The 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 be used to generate the open string scattering amplitudes.

14. The solution of this equations can be written as bare coupling where The most general RG eq.s for a set of couplings is

15. NON-LINEAR -FUNCTION We find where

16. FROM =0SCATTERING AMPLITUDES Next order  One more order The residue of the pole is the 4 tachyon scattering amplitude Using the on-shell condition we recover the scattering amplitude for 4 on-shell tachyons Veneziano Amplitude    The 4 tachyon amplitude is the sum of a contact graph and a tachyon exchange graph Lowest order equation The residue of the pole is the scattering amplitude for 3 on-shell tachyons. In our notation is 1/2

17. normalization constant proportional to the tension of the Dp-brane (Kutasov, Marino, Moore,hep-th/0009148 Gerasimov, Shatashvili, hep-th/0009103) Exact tachyon potential Near the perturbative vacuum, T=0 The ratio of the cubic and quadratic term is precisely the one that comes from the expansion of the exact potential For k=0 WITTEN-SHATASHVILI ACTION

18. BIOSFiT CUBIC SFT Near on-shell has a zero for on-shell This can be compared with the Cubic String Field Theory result. Near on-shell The required matching of the quadratic and cubic term implies the field redefinition is non-singular on-shell This is in agreement with all the conjectures involving tachyon condensation Provides a further verification of the validity of our expression for the non-linear -function and the Witten-Shatashvili action. 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.

19. TACHYON AND GAUGE FIELDS At the second order

20. no T-A term no T-T and A-A term 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 In the expressions for the -functions there are, as expected, only the terms consistent with the twist symmetry.

21. CONCLUSIONS • We obtained the non-linear expression for the -function of the couplings. • The string dymanics emerges from the -function fixed points reproducing the open bosonic string scattering amplitudes. • We computed the Witten-Shatashvili action for the tachyon and the Abelian gauge field up to the third order. • The Witten-Shatashvili and the Cubic SFT formulations are shown to be equivalent (at least up to the third order in the tachyon and gauge fields) up to a field redefinition. • Our result can be extended to the study of the non-Abelian case.