# Integrability of the superstring in AdS(3) x S(2) x S(2) x T(3)

## TL;DR

This paper investigates the conditions under which the superstring theory on the AdS(3) x S(2) x S(2) x T(3) background is classically integrable, identifying flux parameter constraints and relating it to known integrable backgrounds via T-duality.

## Contribution

It determines the flux parameter constraints for classical integrability of the superstring in AdS(3) x S(2) x S(2) x T(3) and connects these backgrounds to known integrable models through T-duality.

## Key findings

- Vanishing of two flux parameters for integrability.
- Reduction to supersymmetric or non-supersymmetric backgrounds.
- Relation to AdS(3) x S(3) x S(3) x S(1) via T-duality.

## Abstract

Type II supergravity admits an AdS(3) x S(2) x S(2) x T(3) solution with fluxes depending on several free parameters. We determine the constraints on these parameters imposed by the requirement of (classical) integrability of the superstring sigma model. To do this we analyze the low-energy effective action for the spinning GKP string. The absence of particle production in the tree-level S-matrix of bosonic excitations is shown to imply the vanishing of two of the four parameters in the NSNS three-form flux. This reduces the supergravity background to either the one-parameter AdS(3) x S(2) x S(2) x T(3) background preserving eight supersymmetries, or a non-supersymmetric branch, which differs only by flipping a sign in the RR flux. We show that both these branches can be obtained from AdS(3) x S(3) x S(3) x S(1) by T-dualities on the (Hopf) circle fibers of the three-spheres and therefore the integrability of the string in these backgrounds follows.