1-dimensional TQFT's assign to every 1-manifold (disjoint union of circles) a vector space and to every surface a linear map between the vector spaces that correspond to the boundary manifolds. So this is a functor from the cobordism category of 1-manifolds to the category of vector spaces. What are some examples of higher dimensional TQFT's? ie functors from the cobordism category of n-manifolds to the category of vector spaces.

  • $\begingroup$ As already said, two dim TQFT corresponds to Frobenius algebra, because every surface has a "pair of pants" decomposition. The simplest example of $2$ dim TQFT I know : take a triangulated surface, a vector space $V$, an element $a \in V \otimes V \otimes V$ and a contraction $\mu \in V^* \otimes V^*$. Triangulating your surface, you can put a copy of $a$ on every triangle, taking the big tensor product of all $a$ and using edges for contracting. You will end with a scalar which is an invariant of the surface. $a$ and $\mu$ have to verify some conditions (invariance under Pachner moves). $\endgroup$ – user171326 Jun 29 '17 at 17:12

Two dimensional non-extended TQFTs correspond to Frobenius algebras and there's a book about that (only the short version is linked tho).

Extended TQFTs are a technically more daunting, but conceptually easier to define gadget: if you believe this paper (you should; a more user-friendly presentation is contained in this master thesis) the $\infty$-category (in some model: the paper uses Segal spaces -am I right?-) of $n$-dimensional cobordisms happen to be the free symmetric monoidal $(\infty,n)$-category on a single generator.

For what I can understand and remember, the issue with the nonextended version is that without higher category theory you cannot properly address Baez-Dolan conjecture. As the dimension goes up and up, you can not expect to have a nice classification of the manifolds in your cobordism category in terms of "generators and relations" as it happens in dimension two.

Three-dimensional TQFTs are studied in a beautiful, but difficult, book by Turaev, because of their link with knot theory, but I would appear ridicolous if I attempted to explain more.

All this material, however, comes from my bad memory and from a rather unthoughtful googling operation: you can certainly find something that suits better your curiosity if you interrogate it too (Plus, I'm far from being an expert in each of the things I cited.) :)

There are, in fact, several different approached to the subject, and it might be dispersive not to choose one on which you want to concentrate first: just to mention one, https://arxiv.org/abs/0806.1079

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