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

• URL: http://arxiv.org/abs/2406.17079v1
• Date: Mon, 24 Jun 2024 19:07:42 GMT
• Title: Small Circle Expansion for Adjoint QCD$_2$ with Periodic Boundary Conditions
• Authors: Ross Dempsey, Igor R. Klebanov, Silviu S. Pufu, Benjamin T. Søgaard,
• Abstract summary: Supersymmetry is found at the adjoint mass-squared $g2 hvee/ (2pi)$, where $hvee$ is the dual Coxeter number of $G$. We generalize our results to other gauge groupsG$, for which supersymmetry is found at the adjoint mass-squared $g2 hvee/ (2pi)$, where $hvee$ is the dual Coxeter number of $G$.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $G$.

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