# Decomposing Quantum Field Theories

## TL;DR

This paper explores the decomposition and gluing of quantum field theories, particularly focusing on 3D supersymmetric theories within 4D theories derived from 6D models, revealing their partition functions as topological quantum field theories.

## Contribution

It introduces a framework for understanding how 3D theories relate to 4D theories via domain walls and interprets their partition functions as topological quantum field theories.

## Key findings

- 3D $S^3$ partition functions can be viewed as topological quantum field theories.
- Decomposition and gluing concepts help explore the space of quantum field theories.
- Connections between supersymmetric theories and topological quantum field theories are established.

## Abstract

Motivated by the question of exploring the "theory space" of quantum field theories, we review the concept of "decomposing" and "gluing" quantum field theories. We explain this in the context of three-dimensional $\mathcal{N}=2$ supersymmetric domain walls inside four-dimensional $\mathcal{N}=2$ theories, where the latter arise from the 6d $(2,0)$ theory compactified on Riemann surfaces. We find that $S^3$ partition functions of the three-dimensional theories can be interpreted as partition functions of a three-dimensional topological quantum field theory.