Spin Matrix Theory in near 1/8-BPS corners of N=4 super-Yang-Mills

We consider limits of 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 N . = 4 SYM on a three-sphere and integrating out the fields that become non-dynamical in the limits. In the previous works [1-3] 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 1/8-BPS subsectors, which include the PSU(1,1 vertical bar 2) and SU(2 vertical bar 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 vertical bar 2) case, we prove the positivity of the spectrum by constructing cubic supercharges using the enhanced PSU(1 vertical bar 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.

Automated summary

This paper explores the limits of N = 4 super-Yang-Mills theory that approach BPS bounds. The resulting non-relativistic theories, known as Spin Matrix Theories, describe the effective dynamics near the BPS bounds and are obtained by reducing N = 4 SYM on a three-sphere and integrating out non-dynamical fields. Previous works have studied various subsectors in this limit, and this paper constructs the remaining Spin Matrix Theories near the 1/8-BPS subsectors. Hamiltonians are derived using the spherical reduction algorithm and shown to match the results from loop corrections. In the PSU(1,1 | 2) case, the positivity of the spectrum is proven by constructing cubic supercharges and their closure to the interacting Hamiltonian. The symmetry structure of the sectors is also analyzed.