# A GNS construction of three-dimensional abelian Dijkgraaf-Witten theories

## TL;DR

This paper generalizes the GNS construction to develop a topological quantum field theory framework for three-dimensional abelian Dijkgraaf-Witten theories, connecting invariants of manifolds with algebraic structures.

## Contribution

It introduces a GNS-based universal construction to extend manifold invariants to topological field theories, specifically applying it to abelian Dijkgraaf-Witten theories.

## Key findings

- Reformulation of Dijkgraaf-Witten theories via GNS construction
- Recovery of vector spaces from groupoid cardinalities
- Extension of invariants to topological quantum field theories

## Abstract

We give a detailed account of the so-called "universal construction" that aims to extend invariants of closed manifolds, possibly with additional structure, to topological field theories and show that it amounts to a generalization of the GNS construction. We apply this construction to an invariant defined in terms of the groupoid cardinality of groupoids of bundles to recover Dijkgraaf-Witten theories, including the vector spaces obtained as a linearization of spaces of principal bundles.