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

TL;DR

The paper investigates adjoint QCD$_2$ in 1+1 dimensions and shows that supersymmetry emerges at a specific adjoint fermion mass $m = \sqrt{ \frac{g^2 N}{2\pi} }$ via a gauge-invariant, Lorentz-covariant supercurrent $j_{\mu A}$ whose conservation hinges on quantum anomalies. It extends the construction to models with massless fermions and arbitrary gauge groups, revealing a pattern of a supersymmetric massive sector coupled to a non-supersymmetric or partially supersymmetric CFT sector; in special cases the infrared coset sector is itself supersymmetric, yielding fully supersymmetric gapless theories. Concrete examples include SU$(N)$ with three adjoints (two massless, one massive $m=\sqrt{\frac{3 g^2 N}{2\pi}}$) and a two-adjoint model (one massive, one massless) which is fully SUSY at $m=\sqrt{\frac{g^2 N}{\pi}}$, with vanishing string tension and topological vacua. The authors also analyze U(1) cases via the Schwinger model anomaly and discuss generalizations to arbitrary gauge groups with current algebras at level $k$ and modified central charges, providing a unified framework for partial and full SUSY in adjoint QCD$_2$ and its generalizations. These results illuminate how anomaly structure and coset CFTs control the distribution of supersymmetry across massive and gapless sectors, with potential implications for non-perturbative SUSY in low-dimensional gauge theories and lattice studies.

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 $\sqrt{\frac{g^2 N}{2 π}}$ it exhibits supersymmetry. In this paper, we obtain a deeper insight into how the supersymmetry works by constructing the gauge invariant, Lorentz covariant supercurrent $j_{μA}$. 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 $\sqrt{\frac{3g^2 N}{2π}}$.