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

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 2+1 -cobordism category is built from manifolds which are equipped with an extra structure such as a p 1 p_1 -structure or an extended manifold structure. We perform the universal construction of Blanchet, Habegger, Masbaum, and Vogel (1992) on a cobordism category without this extra structure and show that the resulting quantization functor assigns an infinite dimensional vector space to the torus.