Supercurrents and (Partial) Supersymmetry in Adjoint QCD$_2$ and Its Generalizations

TL;DR

This paper explores supersymmetry in 1+1 dimensional adjoint QCD, constructing a supercurrent, analyzing quantum anomalies, and generalizing to models with mixed massive and massless fermions, revealing conditions for supersymmetry in various sectors.

Contribution

It constructs a gauge-invariant supercurrent in adjoint QCD$_2$, analyzes its anomaly-based conservation, and generalizes the framework to models with both massive and massless fermions, identifying cases with full supersymmetry.

Findings

Supercurrent construction reveals supersymmetry at specific fermion masses.

Quantum anomaly plays a crucial role in supercurrent conservation.

Certain models exhibit both massive and massless sectors as supersymmetric.

Abstract

$1+1$-dimensional $SU(N)$ gauge theory coupled to an adjoint Majorana fermion, also known as adjoint QCD$_2$, has the surprising feature that at fermion mass $\sqrt{\frac{g^2 N}{2 \pi}}$ it exhibits supersymmetry. In this paper, we obtain a deeper insight into how the supersymmetry works by constructing the gauge invariant, Lorentz covariant supercurrent $j_{\mu A}$. Its conservation relies crucially on the presence of a quantum anomaly. We generalize this construction to a class of models where, in addition to an adjoint Majorana fermion of an appropriate mass, the gauge theory is coupled to some collection of massless fermions ($SU(N)$ may be replaced by a more general gauge group). In general, these models have a supersymmetric massive sector and a non-supersymmetric CFT sector [arXiv:2202.04017], but there are cases in which both sectors are supersymmetric. An example of such a gapless, fully supersymmetric model is $SU(N)$ gauge theory coupled to three adjoint Majorana fermions, of which two are massless and the third has mass $\sqrt{\frac{3g^2 N}{2\pi}}$.