Superisometries and integrability of superstrings

TL;DR

This work analyzes classical integrability of type II superstrings in backgrounds with superisometries by constructing the superisometry transformations and the corresponding Noether currents up to order $\Theta^4$, and then develops a deformation framework to extend a bosonic Lax connection to fermionic orders. For symmetric-space backgrounds with constant fluxes, the author demonstrates the existence of a flat Lax connection $L=L^{(0)}+L^{(1)}+L^{(2)}+O(\Theta^3)$ up to quadratic order in fermions, using a postulated superisometry algebra and flux constraints that ensure consistency of the Killing-spinor structure. The construction is explicitly applied to a set of near-horizon AdS backgrounds with $D(2,1;\alpha)$ symmetry (including $AdS_2$ and $AdS_3$ cases), verifying integrability up to $O(\Theta^2)$ for these backgrounds arising from intersecting branes. The study highlights the role of supersymmetry and kappa symmetry in potential extensions to higher orders, and points to future work on extending integrability to quantum regimes and clarifying dual CFT interpretations.

Abstract

For type II supergravity backgrounds with superisometries the corresponding transformations and conserved currents for the superstring are constructed up to fourth order in $Θ$. It is then shown how, for certain backgrounds related to near horizon geometries of intersecting branes, the components of the superisometry current can be used to construct a Lax connection demonstrating the classical integrability of the string in these backgrounds. This includes examples of $AdS_2$ and $AdS_3$ backgrounds with a $D(2,1;α)$ isometry group which have not previously been studied from an integrability point of view. The construction of the Lax connection is carried out up to second order in $Θ$.