Nonrelativistic near-BPS corners of $\mathcal{N}=4$ super-Yang-Mills with $SU(1,1)$ symmetry

TL;DR

This paper explores nonrelativistic limits of $ =4$ super-Yang-Mills theory near BPS bounds, revealing new Spin Matrix theories with $SU(1,1)$ symmetry, superfield formulations, and simple semi-local models on a circle.

Contribution

It introduces a class of nonrelativistic near-BPS theories with $SU(1,1)$ symmetry derived from $ =4$ SYM, including superfield formulations and semi-local models.

Findings

Consistent quantization with dilatation operator limits.

Superfield formulation for $SU(1,1|1)$ Spin Matrix theory.

Classical interaction expressions for strong coupling analysis.

Abstract

We consider limits of $\mathcal{N}=4$ super Yang-Mills (SYM) theory that approach BPS bounds and for which an $SU(1,1)$ structure is preserved. The resulting near-BPS theories become non-relativistic, with a $U(1)$ symmetry emerging in the limit that implies the conservation of particle number. They are obtained by reducing $\mathcal{N}=4$ SYM on a three-sphere and subsequently integrating out fields that become non-dynamical as the bounds are approached. Upon quantization, and taking into account normal-ordering, they are consistent with taking the appropriate limits of the dilatation operator directly, thereby corresponding to Spin Matrix theories, found previously in the literature. In the particular case of the $SU(1,1|1)$ near-BPS/Spin Matrix theory, we find a superfield formulation that applies to the full interacting theory. Moreover, for all the theories we find tantalizingly simple semi-local formulations as theories living on a circle. Finally, we find positive-definite expressions for the interactions in the classical limit for all the theories, which can be used to explore their strong coupling limits. This paper will have a companion paper in which we explore BPS bounds for which a $SU(2,1)$ structure is preserved.