Mirror Symmetry and Mixed Chern-Simons Levels for Abelian 3d $\mathcal{N} = 2$ theories

TL;DR

This paper explores mirror symmetry in abelian 3d $ abla=2$ theories with mixed Chern-Simons levels, revealing multiple duals and equivalences, including applications to knot theories, using partition functions for derivations.

Contribution

It introduces a framework for understanding mirror duals of abelian 3d $ abla=2$ theories with mixed Chern-Simons levels, including new dualities and effective level analyses.

Findings

Multiple mirror duals with different CS levels identified.

Effective Chern-Simons levels are shown to be equivalent in various theories.

Mirror symmetry for knot-related theories analyzed using partition functions.

Abstract

We study the mirror symmetry of abelian 3d $\mathcal{N}=2$ theories with mixed Chern-Simons levels by turning them into $\mathcal{T}_{A,N}$ theories that are defined as $N$ copies of $U(1)-[1]$ theory coupled together by mixed Chern-Simons levels $k_{ij}$. We find that $\mathcal{T}_{A,N}$ theories have many mirror dual theories with different mixed CS levels and FI parameters. As an example, we analyze $U(1)_k+ N_C \,\mathbf{C} + N_{AC} \, \mathbf{AC}$ theories by transforming these theories into certain $\mathcal{T}_{A,N}$ theories and find many equivalent effective Chern-Simons levels. Finally, we analyze mirror symmetry for theories corresponding to knots. In this work we use sphere partition functions and vortex partition functions to derive dual theories.