The AdS_5xS^5 superstring worldsheet S-matrix and crossing symmetry

TL;DR

This work develops a Hopf-algebraic framework to implement crossing symmetry for the AdS_5 x S^5 worldsheet S-matrix, addressing the lack of standard relativistic invariance. It identifies the antipode as the algebraic mechanism implementing particle-to-antiparticle transformation and derives the crossed form of the su(2|2) × su(2|2) R-matrix, constraining the scalar dressing factor S0 via a nontrivial crossing function f. To avoid branch cuts, it introduces a generalized rapidity plane realized as a coupling-dependent elliptic curve, on which crossing is a simple translation by a half-period and where x^± transform as αβ/x^±. The paper then states the resulting crossing and unitarity equations for S0 and highlights the remaining challenge of finding a minimal solution, setting the stage for further analysis.

Abstract

An S-matrix satisying the Yang-Baxter equation with symmetries relevant to the AdS_5xS^5 superstring has recently been determined up to an unknown scalar factor. Such scalar factors are typically fixed using crossing relations, however due to the lack of conventional relativistic invariance, in this case its determination remained an open problem. In this paper we propose an algebraic way to implement crossing relations for the AdS_5xS^5 superstring worldsheet S-matrix. We base our construction on a Hopf-algebraic formulation of crossing in terms of the antipode and introduce generalized rapidities living on the universal cover of the parameter space which is constructed through an auxillary, coupling constant dependent, elliptic curve. We determine the crossing transformation and write functional equations for the scalar factor of the S-matrix in the generalized rapidity plane.