Gauge Theories Labelled by Three-Manifolds

TL;DR

This paper establishes a duality between the geometry of triangulated 3-manifolds and 3d N=2 gauge theories, linking topological invariants with field theory properties, and providing a functorial framework for boundary conditions in 4d theories.

Contribution

It introduces a novel dictionary connecting 3-manifold triangulations and 3d N=2 gauge theories, interpreting topological invariants as physical quantities and enabling a functorial approach to boundary conditions.

Findings

Partition functions are invariant under triangulation changes.

Duality between topological invariants and gauge theory quantities.

The construction is functorial with respect to cobordisms.

Abstract

We propose a dictionary between geometry of triangulated 3-manifolds and physics of three-dimensional N=2 gauge theories. Under this duality, standard operations on triangulated 3-manifolds and various invariants thereof (classical as well as quantum) find a natural interpretation in field theory. For example, independence of the SL(2) Chern-Simons partition function on the choice of triangulation translates to a statement that S^3_b partition functions of two mirror 3d N=2 gauge theories are equal. Three-dimensional N=2 field theories associated to 3-manifolds can be thought of as theories that describe boundary conditions and duality walls in four-dimensional N=2 SCFTs, thus making the whole construction functorial with respect to cobordisms and gluing.