Small Circle Expansion for Adjoint QCD$_2$ with Periodic Boundary Conditions

TL;DR

This paper investigates the quantum mechanics of adjoint QCD$_2$ on a small circle, revealing supersymmetry properties, spontaneous symmetry breaking, and degeneracies, with implications for understanding the model's extended symmetries.

Contribution

It derives an effective action for adjoint QCD$_2$ on a small circle, showing supersymmetry at specific mass values and analyzing symmetry breaking across different sectors.

Findings

Supersymmetry is unbroken in one sector and broken in others.

At zero mass, the Hamiltonian exhibits $2^{N-1}$ degeneracies.

Generalization to other gauge groups shows supersymmetry at specific mass-squared values.

Abstract

We study $1+1$-dimensional $\text{SU}(N)$ gauge theory coupled to one adjoint multiplet of Majorana fermions on a small spatial circle of circumference $L$. Using periodic boundary conditions, we derive the effective action for the quantum mechanics of the holonomy and the fermion zero modes in perturbation theory up to order $(gL)^3$. When the adjoint fermion mass-squared is tuned to $g^2 N/(2\pi)$, the effective action is found to be an example of supersymmetric quantum mechanics with a nontrivial superpotential. We separate the states into the $\mathbb{Z}_N$ center symmetry sectors (universes) labeled by $p=0, \ldots, N-1$ and show that in one of the sectors the supersymmetry is unbroken, while in the others it is broken spontaneously. These results give us new insights into the $(1,1)$ supersymmetry of adjoint QCD$_2$, which has previously been established using light-cone quantization. When the adjoint mass is set to zero, our effective Hamiltonian does not depend on the fermions at all, so that there are $2^{N-1}$ degenerate sectors of the Hilbert space. This construction appears to provide an explicit realization of the extended symmetry of the massless model, where there are $2^{2N-2}$ operators that commute with the Hamiltonian. We also generalize our results to other gauge groups $G$, for which supersymmetry is found at the adjoint mass-squared $g^2 h^\vee/(2\pi)$, where $h^\vee$ is the dual Coxeter number of $G$.