Zero Energy States of Reduced Super Yang-Mills Theories in $d+1 = 4,6$ and 10 dimensions are necessarily $Spin(d)$ invariant

TL;DR

This paper proves that ground states of reduced Super Yang-Mills theories in dimensions 4, 6, and 10 are necessarily invariant under the Spin(d) group, with specific results for lower dimensions and gauge groups.

Contribution

It provides a rigorous proof of Spin(d) invariance of ground states in certain reduced Super Yang-Mills theories across multiple dimensions.

Findings

Ground states in d=3,5,9 are Spin(d) singlets.

No Spin(d) invariant state exists for odd-dimensional gauge groups in d=2.

An upper bound on total angular momentum is established for d=2.

Abstract

We consider reduced Super Yang-Mills Theory in $d+1$ dimensions, where $d=2,3,5,9$. We present commutators to prove that for $d=3,5$ and 9 a possible ground state must be a $Spin(d)$ singlet. We also discuss the case $d=2$, where we give an upper bound on the total angular momentum and show that for odd dimensional gauge group no $Spin(d)$ invariant state exists in the Hilbert space.