# Refined 3d-3d Correspondence

## TL;DR

This paper investigates the correspondence between Seifert 3-manifolds and 3d $
abla=2$ supersymmetric theories, providing a method to compute their partition functions and linking it to refined Chern-Simons theory with complex gauge groups.

## Contribution

It introduces a prescription for calculating squashed three-sphere partition functions of 3d theories from boundary conditions in 4d theories, extending to include links via Wilson-'t Hooft loops, and connects to refined Chern-Simons theory.

## Key findings

- Derived a method for partition function computation from boundary conditions.
- Extended the framework to include links with Wilson-'t Hooft loops.
- Constructed an analytic continuation of the refined Chern-Simons S-matrix.

## Abstract

We explore aspects of the correspondence between Seifert 3-manifolds and 3d $\mathcal{N}=2$ supersymmetric theories with a distinguished abelian flavour symmetry. We give a prescription for computing the squashed three-sphere partition functions of such 3d $\mathcal{N}=2$ theories constructed from boundary conditions and interfaces in a 4d $\mathcal{N}=2^*$ theory, mirroring the construction of Seifert manifold invariants via Dehn surgery. This is extended to include links in the Seifert manifold by the insertion of supersymmetric Wilson-'t Hooft loops in the 4d $\mathcal{N}=2^*$ theory. In the presence of a mass parameter for the distinguished flavour symmetry, we recover aspects of refined Chern-Simons theory with complex gauge group, and in particular construct an analytic continuation of the $S$-matrix of refined Chern-Simons theory.