Analytic framework for self-dual criticality in $\mathbb{Z}_k$ gauge theory with matter

We study the putative multicritical point in 2+1D $\mathbb{Z}_k$ gauge theory where the Higgs and confinement transitions meet. The presence of an $e$-$m$ duality symmetry at this critical point forces anyons with nontrivial braiding to close their gaps simultaneously, giving rise to a critical theory that mixes strong interactions with mutual statistics. An effective U(1) $\times$ U(1) gauge theory with a mutual Chern-Simons term at level $k$ is proposed to describe the vicinity of the multicritical point for $k \geq 4$. We argue analytically that monopoles are irrelevant in the IR CFT and compute the scaling dimensions of the leading duality-symmetric/anti-symmetric operators. In the large $k$ limit, these scaling dimensions approach $3 - \nu_{\rm XY}^{-1}$ as $1/k^2$, where $\nu_{\rm XY}$ is the correlation length exponent of the 3D XY model.