Open–closed strings: Two-dimensional extended TQFTs and Frobenius algebras

Abstract

We study a special sort of 2-dimensional extended Topological Quantum Field Theories (TQFTs). These are defined on open–closed cobordisms by which we mean smooth compact oriented 2-manifolds with corners that have a particular global structure in order to model the smooth topology of open and closed string worldsheets. We show that the category of open–closed TQFTs is equivalent to the category of knowledgeable Frobenius algebras. A knowledgeable Frobenius algebra ( A , C , ı , ı * ) consists of a symmetric Frobenius algebra A, a commutative Frobenius algebra C, and an algebra homomorphism ı : C → A with dual ı * : A → C , subject to some conditions. This result is achieved by providing a description of the category of open–closed cobordisms in terms of generators and the well-known Moore–Segal relations. In order to prove the sufficiency of our relations, we provide a normal form for such cobordisms which is characterized by topological invariants. Starting from an arbitrary such cobordism, we construct a sequence of moves (generalized handle slides and handle cancellations) which transforms the given cobordism into the normal form. Using the generators and relations description of the category of open–closed cobordisms, we show that it is equivalent to the symmetric monoidal category freely generated by a knowledgeable Frobenius algebra. Our formalism is then generalized to the context of open–closed cobordisms with labeled free boundary components, i.e. to open–closed string worldsheets with D-brane labels at their free boundaries.