Spin Matrix Theory in near $\frac{1}{8}$-BPS corners of $\mathcal{N} = 4$ super-Yang-Mills

TL;DR

This paper constructs and analyzes Spin Matrix Theories near the 1/8-BPS bounds in N=4 super-Yang-Mills, deriving Hamiltonians, confirming their match with spin chain results, and proving spectrum positivity in certain sectors.

Contribution

It extends the construction of Spin Matrix Theories to include the PSU(1,1|2) and SU(2|3) sectors, providing explicit Hamiltonians and symmetry analyses.

Findings

Hamiltonians match spin chain loop corrections

Spectrum positivity proven in PSU(1,1|2) sector

Symmetry structures interpreted in terms of fundamental blocks

Abstract

We consider limits of $\mathcal{N} = 4$ super-Yang-Mills (SYM) theory that approach BPS bounds. These limits result in non-relativistic 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 $\mathcal{N}=4$ SYM on a three-sphere and integrating out the fields that become non-dynamical in the limits. In previous works we have considered various SU(1,1) and SU(1,2) types of subsectors in this limit. In the current work, we will construct the remaining Spin Matrix Theories defined near the $\frac{1}{8}$-BPS subsectors, which include the PSU(1,1|2) and SU(2|3) cases. We derive the Hamiltonians by applying the spherical reduction algorithm and show that they match with the spin chain result, coming from the loop corrections to the dilatation operator. In the PSU(1,1|2) case, we prove the positivity of the spectrum by constructing cubic supercharges using the enhanced PSU$(1|1)^2$ symmetry and show that they close to the interacting Hamiltonian. We finally analyse the symmetry structure of the sectors in view of an interpretation of the interactions in terms of fundamental blocks.