Nonrelativistic Corners of ${\cal N} = 4$ Supersymmetric Yang--Mills Theory

TL;DR

This paper derives a simplified nonrelativistic field theory from ${ m N}=4$ SYM in a near-BPS limit, revealing a lower-dimensional structure that matches known quantum operators and offers insights into strongly coupled gauge dynamics.

Contribution

It constructs a novel nonrelativistic, lower-dimensional field theory from ${ m N}=4$ SYM in a near-BPS limit, demonstrating quantum consistency and potential for richer theories.

Findings

Matching with one-loop dilatation operator

Identification of a 2D complex chiral boson structure

Proof of concept for nonrelativistic limits in supersymmetric theories

Abstract

We show that ${\cal N} = 4$ supersymmetric-Yang-Mills (SYM) theory on $\mathbb{R} \times S^3$ with gauge group $\text{SU}(N)$ is described in a near-BPS limit by a simple lower-dimensional nonrelativistic field theory with $\text{SU}(1,1) \times \text{U}(1)$ invariant interactions. In this limit, a single complex adjoint scalar field survives, and part of its interaction is obtained by exactly integrating out the gauge boson of the SYM theory. Taking into account normal ordering, the interactions match the one-loop dilatation operator of the $\text{SU}(1,1)$ sector, establishing the consistency of the limit at the quantum level. We discover a tantalizing field-theoretic structure, corresponding to a $(1+1)$-dimensional complex chiral boson on a circle coupled to a nondynamical gauge field, both in the adjoint representation of $\text{SU}(N)$. The successful construction of a lower-dimensional nonrelativistic field theory in the $\text{SU}(1,1)$ near-BPS limit provides a proof of concept for other BPS bounds. These are expected to lead to richer field theories in nonrelativistic corners of ${\cal N} = 4$ SYM that include fermions, gauge fields and supersymmetry and can provide a novel path towards understanding strongly coupled finite-$N$ dynamics of gauge theories.