Tittel: A curious connection between 2-dimensional topological quantum field theories and commutative Frobenius algebras
Veileder: Claudia Scheimbauer
Sammendrag: The cobordism category $nCob^{or}$ consists of manifolds as its objects and oriented cobordisms between these manifolds as its morphisms. It has a monoidal product, namely the disjoint union, and a natural symmetric structure. The same is true for the category of vector spaces $Vect_k$ over a field $k$ equipped with the usual tensor product. Such categories are called symmetric monoidal categories and functors respecting these structures are called symmetric monoidal functors. In particular, a symmetric monoidal functor from $nCob^{or}$ into $Vect_k$ is called an $n$-dimensional topological quantum field theory, or TQFT for short.
There is a special kind of an algebra called a Frobenius algebra and there is a connection between these algebras and topology. This connection allows us to understand the commutative Frobenius algebras by understanding $2Cob^{or}$ and $2$-dimensional TQFTs. This will be the main theme of this text.
The well known classication theorems for manifolds of dimensional one and two allows us to completely understand the $2$-dimensional cobordism category $2Cob^{or}$ which in turn gives us a handful of relations there. We will also discover that the defining relations for a Frobenius algebra, when presented in a graphical way, are of a topological nature. Since the symmetric monoidal functors preserves these relations, we will see that the image of the circle under a $2$-dimensional TQFT is in fact a commutative Frobenius algebra in $Vect_k$.
From there we move on to the main result (corollary 7.2) in this text, which is an equivalence of categories, namely the category of two-dimensionals TQFTs and the category of commutative Frobenius algebras. It other words, we are going to prove that
$$ 2TQFT^{or}_k \simeq cFA_k. $$