Nonrelativistic near-BPS corners of N=4 super-Yang-Mills with SU(1,1) symmetry

We consider limits of 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 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.

Automated summary

The study explores the limits of N = 4 super Yang-Mills theory approaching BPS bounds with preserved SU(1,1) structure, resulting in non-relativistic theories with emerging U(1) symmetry. Reduction of N = 4 SYM on a three-sphere and subsequent integration of non-dynamical fields as the bounds are approached lead to these near-BPS theories. Through quantization and normal-ordering, they are consistent with taking appropriate limits of the dilatation operator, corresponding to Spin Matrix theories found in previous literature.