Review of AdS/CFT Integrability, Chapter I.1: Spin Chains in N=4 Super Yang-Mills

TL;DR

This chapter reviews how the problem of operator mixing in N=4 Super Yang-Mills theory maps to integrable spin chains, specifically the Heisenberg model, at the one-loop level.

Contribution

It introduces the mapping of gauge theory operator mixing to integrable spin chains, highlighting the role of the Bethe ansatz in solving the spectral problem.

Findings

Operator mixing at one-loop maps to a spin chain model.

The SU(2) sector corresponds to the ferromagnetic Heisenberg spin chain.

Eigenvalues are determined by Bethe equations.

Abstract

In this chapter of "Review of AdS/CFT Integrability" we introduce N=4 Super Yang-Mills. We discuss the global superalagebra PSU(2,2|4) and its action on gauge invariant operators. We then discuss the computation of the correlators of certain gauge invariant operators, the so-called single trace operators in the large N limit. We show that interactions in the gauge theory lead to mixing of the operators. We compute this mixing at the one-loop level and show that the problem maps to a one-dimensional spin chain with nearest neighbor interactions. For operators in the SU(2) sector we show that the spin chain is the ferromagnetic Heisenberg spin chain whose eigenvalues are determined by the Bethe equations.