On string theory on ${\mathit{AdS}}_3\times S^3\times T^4$ with mixed 3-form flux: Tree-level S-matrix

Abstract

We consider superstring theory on ${\mathit{AdS}}_3\times S^3\times T^4$ supported by a combination of RR and NSNS 3-form fluxes (with parameter of the NSNS 3-form q). This theory interpolates between the pure RR flux model ($q=0$) whose spectrum is expected to be described by a (thermodynamic) Bethe ansatz and the pure NSNS flux model ($q=1$) which is described by the supersymmetric extension of the $\mathit{SL}(2,R)\times \mathit{SU}(2)$ WZW model. As a first step towards the solution of this integrable theory for generic value of q we compute the corresponding tree-level S-matrix for massive BMN-type excitations. We find that this S-matrix has a surprisingly simple dependence on q: the diagonal amplitudes have exactly the same structure as in the $q=0$ case but with the BMN dispersion relation $e^2=p^2+1$ replaced by the one with shifted momentum and mass, $e^2={(p\pm q)}^2+1-q^2$. The off-diagonal amplitudes are then determined from the classical Yang–Baxter equation. We also construct the Pohlmeyer-reduced model corresponding to this superstring theory and find that it depends on q only through the rescaled mass parameter, $\mu \to \sqrt{1-q^2}\mu$, implying that its relativistic S-matrix is q-independent.

Introduction

The subject of this paper is the superstring theory on ${\mathit{AdS}}_3\times S^3\times T^4$ space–time supported by a combination of RR and NSNS 3-form fluxes. The corresponding type IIB supergravity background is the near-horizon limit of the mixed NS5–NS1 + D5–D1 solution that is the basis for an interesting example of AdS/CFT duality [1], [2]. In the near-horizon limit the dilaton is constant, the 3-form fluxes through ${\mathit{AdS}}_3$ and $S^3$ have related coefficients and the radii of the two spaces are equal.

The S-duality symmetry of type IIB supergravity transforms the NSNS 3-form into the RR 3-form, so that if the coefficients of the NSNS and RR fluxes are chosen as q and $q^{\prime }$, respectively, then they enter symmetrically into the supergravity equations, e.g., as $q^2+q^{\prime 2}=1$ (we set the curvature radius to 1). The perturbative fundamental superstring theory is not invariant under the S-duality and should thus depend non-trivially on the parameter q.²

We shall assume that $0⩽q⩽1$, with $q=0$ corresponding to the ${\mathit{AdS}}_3\times S^3\times T^4$ theory with pure RR flux and $q=1$ – to the ${\mathit{AdS}}_3\times S^3\times T^4$ theory with pure NSNS flux. The NSNS flux theory is given by the superstring generalization of the $\mathit{SL}(2)\times \mathit{SU}(2)$ WZW model while the RR flux theory has a Green–Schwarz (GS) formulation [3], [4] similar to the one [5] in the ${\mathit{AdS}}_5\times S^5$ case. The “mixed” theory for generic value of q (first discussed in GS formulation in [3]) has a nice $\frac{\mathit{PSU}(1,1|2)\times \mathit{PSU}(1,1|2)}{\mathit{SU}(1,1)\times \mathit{SU}(2)}$ supercoset formulation [6], exposing its classical integrability and UV finiteness.³

The free-string spectrum of the NSNS ($q=1$) theory can be found using the chiral decomposition property of the WZW model [7] while the apparently more complicated spectral problem of the RR ($q=0$) theory is expected to be solved, as in the ${\mathit{AdS}}_5\times S^5$ case [8], by a thermodynamic Bethe ansatz (for recent progress towards its construction see [9], [10], [11], [12], [13], [14], [15], [16], [17]).

Solving the “interpolating” theory with $0<q<1$ is thus a very interesting problem as that may help to understand the relation between the more standard CFT approach in the $q=1$ case and the integrability-based TBA approach in the $q\ne 1$ case.⁴

The first step towards constructing the $q\ne 0$ generalization of the Bethe ansatz for the string spectrum is to find the corresponding S-matrix for the elementary (BMN-like) massive excitations. Our aim here will be to determine the string tree-level term in this S-matrix following the approach used in the ${\mathit{AdS}}_5\times S^5$ case in [23], [24], [25], [26], [27], i.e. computing it directly from the gauge-fixed string action.

We shall start in Section 2 with an explicit description of the bosonic part of the string action in the sector corresponding to the string moving on $R\times S^3$. In conformal gauge it is described by the $\mathit{SU}(2)$ principal chiral model with a WZ term (with coefficient proportional to q). Fixing a gauge corresponding the BMN vacuum in which the center of mass of the string moves along a circle in $S^3$ we expand the action to quartic order in the fields, sufficient to compute the two-particle tree-level S-matrix.

The computation of this S-matrix for the bosonic string on $S^3$ with B-field flux is the subject of Section 3. This S-matrix has a direct generalization to the full bosonic ${\mathit{AdS}}_3\times S^3$ sector found by using the expression for the relevant gauge-fixed action given in Appendix A.1.

The simplicity of the bosonic result and the requirements of integrability (symmetry factorization and the Yang–Baxter equation) suggest a natural generalization to the fermionic sector, i.e. leading to the full tree-level S-matrix of the GS superstring theory on ${\mathit{AdS}}_3\times S^3\times T^4$ with mixed RR–NSNS flux, which we present in the Section 4.

In Appendix A.2 we explain how the quadratic fermionic action that reproduces the non-trivial BBFF part of this S-matrix should follow from the ${\mathit{AdS}}_3\times S^3\times T^4$ superstring action upon light-cone gauge fixing and expansion in powers of bosons. A candidate for the symmetry algebra of this S-matrix is discussed in Appendix B where we also comment on the symmetry of the S-matrix in the $q=0$ case. Section 5 contains some concluding remarks.

Imposing the conformal gauge, one may solve the Virasoro conditions explicitly and reformulate the classical string theory in terms of current field variables. One way to do this is the Pohlmeyer reduction and another is the Faddeev–Reshetikhin construction (that applies in the bosonic $\mathit{SU}(2)$ case). In Appendix C we construct the Faddeev–Reshetikhin model corresponding to the bosonic string on $R\times S^3$ with B-flux, generalizing the discussion in [28], and present the $q\ne 0$ expression for the corresponding tree-level S-matrix (which is different from the bosonic string sigma model one).

In Appendix D we give a detailed construction of the Pohlmeyer-reduced model corresponding to the superstring theory on ${\mathit{AdS}}_3\times S^3\times T^4$ with mixed 3-form flux. Somewhat unexpectedly, we find that the reduced action is the same as in the $q=0$ case [29], [30] but has the rescaled mass scale parameter, $\mu \to \sqrt{1-q^2}\mu$. As a result, the corresponding relativistic S-matrix does not depend on q.