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

Abstract

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.

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

“…We identify the highest weight of such representations and we derive all the other blocks by acting on it with lowering operators. In retrospective, we observe that there is a natural algebraic derivation of the similar blocks that were discovered in the SU(1, 1) limits [12]. The interactions are written in a manifestly positive-definite form, which also extends the same pattern already observed in the SU(1, 1) limits.…”

“…In this section we show that one can reinterpret the interacting part of the classical Hamiltonian of the SU(1, 1|1) near-BPS theory found in [12] in terms of a norm on the linear space of an infinite-dimensional representation of the SU (1, 1|1) algebra. This provides a natural framework that can explain why the interaction has its particular form, including that it is invariant with respect to any SU(1, 1|1) transformations, as well as the relative normalization of the terms in the interaction and the fact that it is positive definite.…”

“…This provides a natural framework that can explain why the interaction has its particular form, including that it is invariant with respect to any SU(1, 1|1) transformations, as well as the relative normalization of the terms in the interaction and the fact that it is positive definite. We begin by briefly reviewing the SU(1, 1|1) near-BPS theory [12] in Section 2.1, exhibiting its symmetry properties in Section 2.2 and finally interpretating it as a norm in Section 2.3. One can think of this as a warm-up to the SU(1, 2|2) case considered in Section 4, where we show that one can make an analogous interpretation.…”

“…A novel approach to obtain an effective description of the degrees of freedom surviving the near-BPS limits in terms of the Hamiltonian of a lower-dimensional field theory was found in [11], and then pursued in [12] for all the cases containing SU(1, 1) as a subset of the spin group. The technique is the following: we put classical N = 4 SYM on R × S 3 and we take a decoupling limit by zooming in towards a BPS bound of the CFT.…”

“…The main novelty with respect to the cases considered in [12] is the appearance of the rotation generator S 2 , which was instead turned off in the SU(1, 1) limits; its presence will allow the existence of dynamical gauge degrees of freedom in the near-BPS limit, which is performed in the following way:…”