Analytic Torsion from Chern-Simons theory via the $(2,0)$-theory on Dicyclic Orbifolds of $S^3$

Abstract

The Witten index of the $(2,0)$-theory compactified on spaces of the form $S^3/\Gamma \times S^2$, with a freely acting group $\Gamma$, and with external string sources implemented via timelike surface operator insertions, is expressed in terms of Ray-Singer torsion of $S^3/\Gamma$ and characters of irreducible representations of $\Gamma$. We compute it explicitly for the Dicyclic groups $\Gamma ={\text{Dic}}_k$. The torsion and characters are generally irrational numbers, but they nicely combine to an integer index. Alternatively, the Witten index can be computed from Chern-Simons theory on $S^2$, and Ray-Singer torsion on $S^3/{\text{Dic}}_k$ is thus computable from Chern-Simons theory. The matching of the Witten index calculated by these dual approaches reveals new details about the partition function of the $(2,0)$-theory with surface operators.

Major revisions including the detailed discussion of Chern-Simons duality in section 6, another example in section 7, 67 pages