The Panorama of Spin Matrix Theory

TL;DR

This paper constructs the largest Spin Matrix theory for $ ext{N}=4$ SYM, revealing a comprehensive view of the theory's structure, supersymmetry, and positive definiteness, advancing understanding of near-BPS limits and finite N effects.

Contribution

It introduces the PSU(1,2|3) Spin Matrix theory Hamiltonian with a novel division into D-terms and F-terms, providing a unifying framework for various Spin Matrix theories.

Findings

Hamiltonian is positive definite and supersymmetry invariant.

Hamiltonian divided into D-terms and F-terms, both invariant and positive.

Reveals the 'Panorama' of Spin Matrix theories from $ ext{N}=4$ SYM.

Abstract

Spin Matrix theory describes near-BPS limits of $\mathcal{N}=4$ SYM theory, which enables us to probe finite $N$ effects like D-branes and black hole physics. In previous works, we have developed the spherical reduction and spin chain methods to construct Spin Matrix theory for various limits. In this paper, by considering a supercharge $\mathcal{Q}$ which is cubic in terms of the letters, we construct the Hamiltonian of the largest Spin Matrix theory of $\mathcal{N}=4$ SYM, called the PSU$(1,2|3)$ Spin Matrix theory, as $H = \{\mathcal{Q}, \mathcal{Q}^\dagger \}$. We show the resulting Hamiltonian is automatically positive definite and manifestly invariant under supersymmetry. The Hamiltonian is made of basic blocks which transform as supermultiplets. A novel feature of this Hamiltonian is its division into D-terms and F-terms that are separately invariant under PSU$(1,2|3)$ symmetry and positive definite. As all the other Spin Matrix theories arising from $\mathcal{N}=4$ SYM can be acquired by turning off certain letters in the theory, we consider our work as revealing the "Panorama" of Spin Matrix theory.