Approximating $\mathrm{SU}(2)$ Chern-Simons theory by finite group gauge theories

Motivated by some previously known facts from mathematical and physics literature, we explore certain relations between 3-dimensional topological gauge theories with continuous and finite gauge groups, commonly known as Chern-Simons (CS) and Dijkgraaf-Witten (DW) theories, respectively. Specifically, we consider the continuous and finite gauge groups to be the same algebraic group over the complex numbers and a finite field, respectively. In this paper, we focus on the $\mathrm{SU}(2)$ example and consider the relationship on the level of the corresponding partition functions on closed 3-manifolds. Mathematically, these are Witten-Reshetikhin-Turaev and DW invariants. We find that the asymptotics of the DW theory when the number of elements of the finite field is large recovers the leading asymptotics of the CS theory at large level when the 3-manifold contains no hyperbolic components. As a byproduct, we develop efficient techniques to count the number of points over finite fields $\mathbb{F}_q$ of $\mathrm{SL}(2)$ representation varieties of fundamental groups of 3-manifolds, possibly with weights pulled back from a chosen class in $H^3(\mathrm{SL}(2,\mathbb{F}_q),\mathrm{U}(1))$.