Fusions and Dualities for 3d Theories $T[M_3]$

TL;DR

This paper explores the geometric and topological operations on 3d theories derived from three-manifolds, revealing how dualities and fusions relate to surgeries and manipulations of these manifolds.

Contribution

It introduces new geometric interpretations for operations like handle slides, gaugings, and flips, and connects these to dualities and fusion identities in 3d theories.

Findings

Handle slides and flips correspond to specific dualities.

Fusion identities describe connected sums of matter circles.

Many abelian dualities can be understood through geometric operations.

Abstract

We study 3d theories determined by three-manifolds. Previously, we found that some basic 3d dualities relate to the surgeries of three-manifolds and defined gauge circles and matter circles. In this note, we discuss some operations including handle slides, gaugings and flips of mass parameters, and the corresponding geometric interpretations for these operations. We note that a fusion identity could describe the fusion, or in other words, the connected sum of matter circles. Many abelian dualities could arise from this fusion and inherit geometric interpretations thereof.