Ground states of Class S $$ \mathcal{S} $$ theory on ADE singularities and dual Chern-Simons theory

Abstract

Abstract In radial quantization, the ground states of a gauge theory on ADE singularities ℝ4 /Γ are characterized by flat connections that are maps from Γ to the gauge group. We study Class S $$ \mathcal{S} $$ theory of type a 1 $$ {\mathfrak{a}}_1 $$ = su 2 $$ \mathfrak{su}(2) $$ on a Riemann surface of genus g > 1, without punctures. The fundamental building block of Class S $$ \mathcal{S} $$ theory is the trifundamental Trinion theory — a low energy limit of two M5 branes compactified on the three-punctured Riemann sphere. We show, through the superconformal index, that the supersymmetric Casimir energy of the trifundamental theory imposes a constraint on the set of allowed flat connections, which agrees with the prediction of a duality relating the ground state Hilbert space of Class S $$ \mathcal{S} $$ on ADE singularities to the Hilbert space of a certain dual Chern-Simons theory whose gauge group is given by the McKay correspondence. The conjecture is shown to hold for Γ = ℤ k , agreeing with the previous results of Benini et al. and Alday et al. A non-abelian generalization of this duality is analyzed by considering the example of the dicyclic group Γ = Dic2, corresponding to Chern-Simons gauge group SO(8).

Keywords

• Chern-Simons Theories
• Duality in Gauge Field Theories
• Supersymmetric Gauge Theory
• Topological Field Theories