# 3d $\mathcal{N}=2$ theories and plumbing graphs: adding matter, gauging, and new dualities

## TL;DR

This paper extends the framework of 3d $
abla$=2 gauge theories associated with plumbing graphs by incorporating matter fields, analyzing their properties, and exploring dualities through graph transformations, revealing new dualities and connections to knot theory.

## Contribution

It introduces a generalized graph-based encoding of 3d $
abla$=2 theories with matter, enabling the analysis of dualities and operations via graph transformations.

## Key findings

- Graph operations correspond to gauging and integrating out nodes.
- New dualities are derived from gauging global symmetries.
- Superpotentials are represented by triangle graphs with consistency conditions.

## Abstract

Recently, a large class of 3d $\mathcal{N} = 2$ gauge theories with mixed Chern-Simons levels, corresponding to plumbing 3-manifolds, has been identified. In this paper we generalize these theories by including in their content chiral multiples, and analyze their properties. We find that the content of such theories can be encoded in graphs, which generalize plumbing graphs, and various operations in these theories can be represented in terms of transformations of such graphs. The operations in question include gauging global symmetries, integrating out gauge nodes, which for theories without chiral multiplets corresponds to Kirby moves, and ST-transformations that involve chiral multiplets. The dualities such as mirror triality and SQED-XYZ duality can be also represented in terms of graphs, and enable us to find many new dual theories by gauging global symmetries. In particular, we find that gauged SQED-XYZ duality leads to other dualities, which take the same form as operations of linking and unlinking discussed in the context of knots-quivers correspondence. We also find that the superpotential can be encoded in an interesting class of triangle graphs that satisfy certain consistency conditions, we discuss decoupling and Higgsing of chiral multiplets, as well as interpretation of various phenomena in terms of brane webs.