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

Curtis G. Callan,, Jonathan Heckman, Tristan McLoughlin, Ian Swanson

TL;DR

The paper develops a virial expansion of the N=4 SYM spin-chain Hamiltonian to compute operator dimensions with multiple impurities, addressing higher-loop complications where Bethe Ansatz is incomplete. It analyzes the su(2), su(1|1), and sl(2) sectors, obtaining explicit one-, two-, and three-loop results via a tractable 1/L expansion and momentum-space formalisms. The findings show near-BMN agreement with string theory up to two loops in the su(2) sector, exact one-loop Bethe compatibility in the su(1|1) subsector, and a known three-loop mismatch with string predictions, while offering cross-checks with long-range Bethe Ansatz proposals. The work provides a practical framework for benchmarking AdS/CFT predictions and guiding future all-loop integrable formulations for N=4 SYM.

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.

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

TL;DR

The paper develops a virial expansion of the N=4 SYM spin-chain Hamiltonian to compute operator dimensions with multiple impurities, addressing higher-loop complications where Bethe Ansatz is incomplete. It analyzes the su(2), su(1|1), and sl(2) sectors, obtaining explicit one-, two-, and three-loop results via a tractable 1/L expansion and momentum-space formalisms. The findings show near-BMN agreement with string theory up to two loops in the su(2) sector, exact one-loop Bethe compatibility in the su(1|1) subsector, and a known three-loop mismatch with string predictions, while offering cross-checks with long-range Bethe Ansatz proposals. The work provides a practical framework for benchmarking AdS/CFT predictions and guiding future all-loop integrable formulations for N=4 SYM.

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.

Paper Structure

This paper contains 7 sections, 82 equations, 1 figure, 8 tables.

Table of Contents

  1. Introduction
  2. The $\mathfrak{su}(2)$ sector
  3. One-loop order
  4. Two and three-loop order
  5. A closed $\mathfrak{su}(1|1)$ subsector of $\mathfrak{su}(2|3)$
  6. The $\mathfrak{sl}(2)$ sector
  7. Conclusions

Figures (1)

  • Figure 1: One-loop $\mathfrak{su}(2)$ spin chain spectrum vs. lattice length $L$ ($6\leq L \leq 40$)