The Algebra of Transition Matrices for the AdS_5 x S^5 Superstring

TL;DR

This work advances the integrability program for the $AdS_{5}\times S^{5}$ superstring by constructing a one-parameter flat current that respects a generalized $\mathbb{Z}_4$ automorphism, enabling a monodromy-based approach to nonlocal charges and Yangian symmetry. Through a detailed Hamiltonian analysis, the authors derive the Poisson algebra of the flat currents and identify the non-ultralocal structure that arises, outlining regularization strategies necessary for a well-defined transition-matrix algebra. The paper also clarifies the role of the $PSU(2,2|4)$ grading and the WZW-type action, and discusses the challenges posed by $\kappa$-symmetry and Virasoro constraints in this setting. Overall, the results lay groundwork for a systematic treatment of the integrable structure of the $AdS_{5}\times S^{5}$ string and its gauge-theory dual, with several open technical issues left for future work.

Abstract

We consider integrability properties of the superstring on $AdS_{5}\times S^{5}$ background and construct a new one parameter family of currents which satisfies the vanishing curvature condition. We present the Hamiltonian analysis for the sigma model action and determine the Poisson algebra of the transition matrices. We reveal the generalization of the $\mathbb{Z}_{4}$ automorphism analogous to the sigma models defined on a symmetric space coset. A possible regularization scheme for the ambiguities present, which respects the generalized automorphism, is also discussed.