Extra structure and the universal construction for the Witten-Reshetikhin-Turaev TQFT

TL;DR

This paper explores the universal construction of the Witten-Reshetikhin-Turaev TQFT, demonstrating that removing additional structures leads to infinite-dimensional vector spaces, challenging the usual finite-dimensional requirement.

Contribution

It shows how the universal construction applied to a simplified cobordism category results in infinite-dimensional vector spaces, providing new insights into TQFT structures.

Findings

Universal construction yields infinite-dimensional spaces for the torus

Removing extra structures affects the dimensionality of the associated vector spaces

Highlights limitations of finite-dimensional assumptions in TQFTs

Abstract

A TQFT is a functor from a cobordism category to the category of vector spaces, satisfying certain properties. An important property is that the vector spaces should be finite dimensional. For the WRT TQFT, the relevant 2+1-cobordism category is built from manifolds which are equipped with an extra structure such as a p_1-structure, or an extended manifold structure. We perform the universal construction of Blanchet, Habegger, Masbaum and Vogel on a cobordism category without this extra structure and show that the resulting quantization functor assigns an infinite dimensional vector space to the torus.