Lattice Super Yang-Mills: A Virial Approach to Operator Dimensions

TL;DR

This paper introduces a virial expansion method for calculating operator dimensions in N=4 super Yang-Mills theory, providing an alternative to Bethe ansatz techniques especially at higher loops and for complex sectors.

Contribution

The authors develop a virial expansion approach to compute operator dimensions, simplifying calculations beyond one loop where Bethe ansatz methods become intractable.

Findings

Successfully applied virial expansion to higher-loop operator dimensions

Achieved results consistent with known numerical predictions near the BMN limit

Provided comparisons with Bethe ansatz predictions to validate the method

Abstract

The task of calculating operator dimensions in the planar limit of N=4 super Yang-Mills theory can be vastly simplified by mapping the dilatation generator to the Hamiltonian of an integrable spin chain. The Bethe ansatz has been used in this context to compute the spectra of spin chains associated with various sectors of the theory which are known to decouple in the planar (large-N_c) limit. These techniques are powerful at leading order in perturbation theory but become increasingly complicated beyond one loop in the 't Hooft parameter lambda=g_YM^2 N_c, where spin chains typically acquire long-range (non-nearest-neighbor) interactions. In certain sectors of the theory, moreover, higher-loop Bethe ansaetze do not even exist. We develop a virial expansion of the spin chain Hamiltonian as an alternative to the Bethe ansatz methodology, a method which simplifies the computation of dimensions of multi-impurity operators at higher loops in lambda. We use these methods to extract previously reported numerical gauge theory predictions near the BMN limit for comparison with corresponding results on the string theory side of the AdS/CFT correspondence. For completeness, we compare our virial results with predictions that can be derived from current Bethe ansatz technology.