Review of AdS/CFT Integrability, Chapter III.7: Hirota Dynamics for Quantum Integrability

TL;DR

This paper reviews how Hirota dynamics and Y-systems are applied to compute the spectrum in planar AdS/CFT, connecting integrable discrete equations with Bethe ansatz methods.

Contribution

It elucidates the use of Hirota equations and Backlund transformations in deriving Bethe ansatz equations within the AdS/CFT integrability framework.

Findings

Hirota equations can be solved via Backlund method.

Analyticity conditions lead to nested Bethe ansatz equations.

Application to asymptotic analysis of long operators in AdS/CFT.

Abstract

We review recent applications of the integrable discrete Hirota dynamics (Y-system) in the context of calculation of the planar AdS/CFT spectrum. We start from the description of solution of Hirota equations by the Backlund method where the requirement of analyticity results in the nested Bethe ansatz equations. Then we discuss applications of the Hirota dynamics for the analysis of the asymptotic limit of long operators in the AdS/CFT Y-system.