# (0,2) Dualities and the 4-Simplex

## TL;DR

This paper explores the connection between 2d $
abla=(0,2)$ duality interfaces and triangulated 4-manifolds, establishing dualities related to Pachner moves and linking 2d dualities with 4-manifold geometry.

## Contribution

It introduces a Lagrangian 2d $
abla=(0,2)$ duality interface for the 3d XYZ and SQED models associated with the 4-simplex, and develops a dictionary relating triangulated 4-manifolds to 2d $
abla=(0,2)$ interfaces.

## Key findings

- Identified IR dualities of interfaces corresponding to Pachner moves.
- Checked dualities using supersymmetric half-indices.
- Connected Pachner moves to abelian $
abla=(0,2)$ trialities.

## Abstract

We propose that a simple, Lagrangian 2d $\mathcal{N}=(0, 2)$ duality interface between the 3d $\mathcal{N}=2$ XYZ model and 3d $\mathcal{N}=2$ SQED can be associated to the simplest triangulated 4-manifold: the 4-simplex. We then begin to flesh out a dictionary between more general triangulated 4-manifolds with boundary and 2d $\mathcal{N}=(0, 2)$ interfaces. In particular, we identify IR dualities of interfaces associated to local changes of 4d triangulation, governed by the (3,3), (2,4), and (4,2) Pachner moves. We check these dualities using supersymmetric half-indices. We also describe how to produce stand-alone 2d theories (as opposed to interfaces) capturing the geometry of 4-simplices and Pachner moves by making additional field-theoretic choices, and find in this context that the Pachner moves recover abelian $\mathcal{N}=(0,2)$ trialities of Gadde-Gukov-Putrov. Our work provides new, explicit tools to explore the interplay between 2d dualities and 4-manifold geometry that has been developed in recent years.