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

The Witten index of the $(2,0)$-theory compactified on spaces of the form $S^3\mathrm{/}\mathrm{\Gamma }\times S^2$, with a freely acting group $\mathrm{\Gamma }$, and with external string sources implemented via timelike surface operator insertions, is expressed in terms of Ray-Singer torsion of $S^3\mathrm{/}\mathrm{\Gamma }$ and characters of irreducible representations of $\mathrm{\Gamma }$. We compute it explicitly for the Dicyclic groups $\mathrm{\Gamma }={\text{Dic}}_k$. 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 $S^3\mathrm{/}{\text{Dic}}_k$ 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.