# Braids, Walls, and Mirrors

## TL;DR

This paper develops a geometric framework connecting 4d N=2 theories, their wall-crossing phenomena, and 3d N=2 theories via M5-brane configurations, revealing new mirror symmetries and dualities.

## Contribution

It introduces a novel geometric approach to derive 3d N=2 theories from 4d theories using M5-branes and three-manifolds, linking wall-crossing to 3d mirror symmetry.

## Key findings

- Wall-crossing in 4d theories corresponds to mirror pairs in 3d.
- The framework applies to theories from M5-branes on Riemann surfaces and three-manifolds.
- Explicit examples include mirror symmetry for N_f=1 SQED and the XYZ model.

## Abstract

We construct 3d, N=2 supersymmetric gauge theories by considering a one-parameter `R-flow' of 4d, N=2 theories, where the central charges vary while preserving their phase order. Each BPS state in 4d leads to a BPS particle in 3d, and thus each chamber of the 4d theory leads to a distinct 3d theory. Pairs of 4d chambers related by wall-crossing, R-flow to mirror pairs of 3d theories. In particular, the 2-3 wall-crossing for the A_2 Argyres-Douglas theory leads to 3d mirror symmetry for N_f=1 SQED and the XYZ model. Although our formalism applies to arbitrary N=2 models, we focus on the case where the parent 4d theory consists of pairs of M5-branes wrapping a Riemann surface, and develop a general framework for describing 3d N=2 theories engineered by wrapping pairs of M5-branes on three-manifolds. Each 4d chamber, which corresponds to a dual 3d description, maps to a particular tetrahedral decomposition of the UV 3d geometry. In the IR the physics is captured by a single recombined M5-brane which is a branched double cover of the original UV three-manifold. The braiding of branch loci and the geometry of branch sheets play a key role in encoding the physics.