On T-Duality and Integrability for Strings on AdS Backgrounds

TL;DR

The paper investigates how T-duality intertwines with integrability for classical strings on $AdS_5\times S^5$, performing a T-duality along the four AdS$_5$ isometries. It demonstrates that a field-dependent gauge transformation removes explicit coordinate dependence in the Lax connection, allowing a gauge-equivalent Lax construction in terms of dual coordinates and yielding an infinite set of conserved charges in the T-dual model. Crucially, the work shows that local (Noether) charges of the original model map to non-local charges in the T-dual model and vice versa, revealing a deep exchange of local/non-local structures under duality. The analysis is illustrated with toy examples ($S^2$ and AdS$_2$) and extended toward AdS$_5$, highlighting how the duality preserves integrability and provides a framework for understanding dual conformal symmetry and potential connections to Wilson-loop observables in the AdS/CFT correspondence.

Abstract

We discuss an interplay between T-duality and integrability for certain classical non-linear sigma models. In particular, we consider strings on the AdS_5 x S^5 background and perform T-duality along the four isometry directions of AdS_5 in the Poincare patch. The T-dual of the AdS_5 sigma model is again a sigma model on an AdS_5 space. This classical T-duality relation was used in the recently uncovered connection between light-like Wilson loops and MHV gluon scattering amplitudes in the strong coupling limit of the AdS/CFT duality. We show that the explicit coordinate dependence along the T-duality directions of the associated Lax connection (flat current) can be eliminated by means of a field dependent gauge transformation. As a result, the gauge equivalent Lax connection can easily be T-dualized, i.e. written in terms of the dual set of isometric coordinates. The T-dual Lax connection can be used for the derivation of infinitely many conserved charges in the T-dual model. Our construction implies that local (Noether) charges of the original model are mapped to non-local charges of the T-dual model and vice versa.