On matrix models for anomalous dimensions of super Yang-Mills theory

TL;DR

This paper explores a matrix model approach to understanding the anomalous dimensions in $ ext{N}=4$ super Yang-Mills theory, proposing an alternative to spin chain descriptions by analyzing the large $N$ limit and topological genus restrictions.

Contribution

It introduces a matrix model framework for the anomalous dimension matrix, connecting large $N$ classical field theories with spin chain models as an alternative perspective.

Findings

Large $N$ limit yields a classical field theory equivalent to spin chains.

Path integral representation for the anomalous dimension density matrix is constructed.

Small $N$ limit corresponds to diagrams of maximal topological genus.

Abstract

We consider the matrix model approach to the anomalous dimension matrix in $\mathcal{N}=4$ super Yang--Mills theory. We construct the path integral representation for the anomalous dimension density matrix and analyze the resulting action. In particular, we consider the large $N$ limit, which results in a classical field theory. Since the same limit leads to spin chains, we propose to consider the former as an alternative description of the latter. We consider also the limit of small $N$, which corresponds to the restriction to the diagrams of maximal topological genus.