From 3D topological quantum field theories to 4D models with defects

TL;DR

This paper introduces a method to extend (2+1)D topological quantum field theories with defects to (3+1)D models using Heegard splittings, demonstrated with BF theory, enabling new insights into higher-dimensional TQFTs.

Contribution

The authors propose a novel approach to lift states and operators from (2+1)D to (3+1)D TQFTs with defects using Heegard splittings, facilitating the study of more complex models.

Findings

Heegard splittings encode 3D topology with line defects into 2D surfaces.

Curvature excitation operators in 3D are derived from ribbon operators in 2D BF theory.

The method paves the way for constructing 3D models based on unitary fusion categories.

Abstract

(2+1) dimensional topological quantum field theories with defect excitations are by now quite well understood, while many questions are still open for (3+1) dimensional TQFTs. Here we propose a strategy to lift states and operators of a (2+1) dimensional TQFT to states and operators of a (3+1) dimensional theory with defects. The main technical tool are Heegard splittings, which allow to encode the topology of a three--dimensional manifold with line defects into a two--dimensional Heegard surface. We apply this idea to the example of BF theory which describes locally flat connections. This shows in particular how the curvature excitation generating surface operators of the (3+1) dimensional theory can be obtained from closed ribbon operators of the (2+1) dimensional BF theory. We hope that this technique allows the construction and study of more general models based on unitary fusion categories.