Symmetry structure of the interactions in near-BPS corners of $$ \mathcal{N} $$ = 4 super-Yang-Mills

Abstract

We consider limits of $$ \mathcal{N} $$ N = 4 super-Yang-Mills (SYM) theory that approach BPS bounds. These limits result in non-relativistic near-BPS theories that describe the effective dynamics near the BPS bounds and upon quantization are known as Spin Matrix theories. The near-BPS theories can be obtained by reducing $$ \mathcal{N} $$ N = 4 SYM on a three-sphere and integrating out the fields that become non-dynamical in the limits. We perform the sphere reduction for the near-BPS limit with SU(1, 2|2) symmetry, which has several new features compared to the previously considered cases with SU(1) symmetry, including a dynamical gauge field. We discover a new structure in the classical limit of the interaction term. We show that the interaction term is built from certain blocks that comprise an irreducible representation of the SU(1, 2|2) algebra. Moreover, the full interaction term can be interpreted as a norm in the linear space of this representation, explaining its features including the positive definiteness. This means one can think of the interaction term as a distance squared from saturating the BPS bound. The SU(1, 1|1) near-BPS theory, and its subcases, is seen to inherit these features. These observations point to a way to solve the strong coupling dynamics of these near-BPS theories.

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

“…Finally, we would like to advertise our companion paper [32] in which we explore a class of near-BPS limits with a SU(1, 2) global symmetry present. Among these near-BPS theories are the theory with PSU(1, 2|3) symmetry that captures the behavior of N = 4 SYM near the BPS bound E ≥ S 1 + S 2 + J 1 + J 2 + J 3 , a bound saturated by the supersymmetric black hole in AdS 5 × S 5 [33].…”

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

“…Finally, we would like to advertise our companion paper [31] in which we explore a class of near-BPS limits with a SU(1, 2) global symmetry present. Among these near-BPS theories are the theory with SU(1, 2|3) symmetry that captures the behavior of N = 4 SYM near the BPS bound E ≥ S 1 + S 2 + J 1 + J 2 + J 3 , a bound saturated by the supersymmetric black hole in AdS 5 × S 5 [32].…”

Symmetry structure of the interactions in near-BPS corners of $ \mathcal{N} = 4$ super-Yang-Mills

“…As a subcase it should include the PSU(1, 1|2) near-BPS theory previously considered in [12]. It would be highly interesting to see how the symmetry structure of the interactions found in this work can extend to the full PSU(1, 2|3) case; this will be covered in a related work [23].…”