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

TL;DR

This paper connects the Witten index of the $(2,0)$-theory on certain orbifolds with Ray-Singer torsion and Chern-Simons theory, providing explicit calculations for Dicyclic groups and revealing new insights into the theory's partition function.

Contribution

It introduces a novel relation between the Witten index, Ray-Singer torsion, and Chern-Simons theory for $(2,0)$-theory on Dicyclic orbifolds, with explicit computations and duality checks.

Findings

Explicit computation of the Witten index for Dicyclic groups

Demonstration of the index's integer value despite irrational components

Matching results from different theoretical approaches

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.