On integrability of strings on symmetric spaces

TL;DR

This paper demonstrates that the classical integrability of bosonic and superstrings on symmetric spaces extends to cases with non-zero NSNS flux, including certain supersymmetric backgrounds, by constructing appropriate Lax connections.

Contribution

It proves the extension of integrability to strings with NSNS flux and full fermionic contributions, generalizing known supercoset models to more complex flux configurations.

Findings

Integrability extends to symmetric space strings with NSNS flux under specific conditions.

Constructed Lax connections including fermionic terms to quadratic order.

Proved integrability of strings in certain supersymmetric backgrounds with RR flux.

Abstract

In the absence of NSNS three-form flux the bosonic string on a symmetric space is described by a symmetric space coset sigma-model. Such models are known to be classically integrable. We show that the integrability extends also to cases with non-zero NSNS flux (respecting the isometries) provided that the flux satisfies a condition of the form H_{abc}H^{cde}~R_{ab}^{de}. We then turn our attention to the type II Green-Schwarz superstring on a symmetric space. We prove that if the space preserves some supersymmetry there exists a truncation of the full superspace to a supercoset space and derive the general form of the superisometry algebra. In the case of vanishing NSNS flux the corresponding supercoset sigma-model for the string is known to be integrable. We prove that the integrability extends to the full string by augmenting the supercoset Lax connection with terms involving the fermions which are not captured by the supercoset model. The construction is carried out to quadratic order in these fermions. This proves the integrability of strings on symmetric spaces supported by RR flux which preserve any non-zero amount of supersymmetry. Finally we also construct Lax connections for some supercoset models with non-zero NSNS flux describing strings in AdS(2,3) x S(2,3) x S(2,3) x T(2,3,4) backgrounds preserving eight supersymmetries.