Quantum integrability and functional equations

TL;DR

This thesis develops a method to transform Bethe Ansatz equations into Riemann-Hilbert problems, facilitating the analysis of integrable models and verifying integrability in AdS/CFT correspondence up to two loops.

Contribution

It introduces a general procedure to represent integral Bethe Ansatz equations as Riemann-Hilbert problems, simplifying the study of integrable systems and their perturbative expansions.

Findings

Verified integrability conjecture up to two loops in AdS/CFT.

Solved crossing equations in the AdS/CFT spectral problem.

Analyzed large-order behavior of asymptotic expansions in sigma models.

Abstract

In this thesis a general procedure to represent the integral Bethe Ansatz equations in the form of the Reimann-Hilbert problem is given. This allows us to study in simple way integrable spin chains in the thermodynamic limit. Based on the functional equations we give the procedure that allows finding the subleading orders in the solution of various integral equations solved to the leading order by the Wiener-Hopf technics. The integral equations are studied in the context of the AdS/CFT correspondence, where their solution allows verification of the integrability conjecture up to two loops of the strong coupling expansion. In the context of the two-dimensional sigma models we analyze the large-order behavior of the asymptotic perturbative expansion. Obtained experience with the functional representation of the integral equations allowed us also to solve explicitly the crossing equations that appear in the AdS/CFT spectral problem.