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

Abstract

We consider limits of 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 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, 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.

Aspects of Nonrelativistic Strings

“…5.28. Finally, a new approach has recently been developed for the explicit construction of Spin Matrix theories using a classical reduction of N 4 followed by a suitable quantization and normal ordering procedure [102][103][104][105]. This construction reproduces earlier results obtained [85] from limits of the one-loop spectrum of N 4 and also leads to a two-dimensional field theory formulation of the SU(1, 1) sectors, suggesting a natural dual description for SMT strings on the three-dimensional SU(1, 1) geometry in Table 1 at large N.…”

Decoding the Spin-Matrix limit of strings on $AdS_5 \times S^5$

“…The conjectured duality between various near BPS corner(s) of N = 4 SYM [1]- [4] and the stringy dynamics over non-Lorentzian manifolds has drawn lots of attention in the recent years. Major breakthrough will come once we understand these nonrelativistic string states in terms of the dual quantum mechanical degrees of freedom known as the Spin-Matrix theory (SMT) [5]- [8]. The first step is therefore to strengthen our current understanding in the strong coupling regime of the correspondence and in particular to find an interpretation for the propagating (nonrelativistic) string states (on AdS 5 × S 5 ) in terms of dual SMT degrees of freedom in the near BPS limit(s) of N = 4 SYM on R × S 3 .…”

“…Any systematic verification of these results from dual Spin-Matrix theory counterpart should be an exciting project to look for. Recently, there has been some progress along this line of investigation which involves a systematic investigation of Spin-Matrix theories with spin groups containing non compact SU (1, 1) factor [6]- [8]. A similar control over other SMT limits at strong coupling and a verification of the corresponding string theory results should be something worthwhile to look for in the future.…”