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

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