Topological quantum field theories

Abstract

[en] Topological quantum field theories (TQFTs) are functors from the category of bordisms to the category of vector spaces that preserve their monoidal structure. Such functors arose in Physics but have proven to be useful in various fields of Mathematics. TQFTs give topological and geometric invariants of manifolds, and thus may help in understanding and classifying them. In this work, however, we perform the reverse process: the completely known classification of 1- and 2-dimensional manifolds will serve as the ground that permits us comprehend TQFTs in these dimensions and determine their underlying structure. In particular, we give a detailed description of 1- and 2-dimensional TQFTs in terms of finite-dimensional vector spaces and commutative Frobenius algebras, respectively. We conclude by trying to elucidate the relation between TQFTs and Physics. We discuss the common structural properties shared by Hilbert spaces and spacetimes, which motivate the connection of quantum theory with general relativity via TQFTs.