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

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

2005-10-31

arxiv.org/abs/math/0510664v3

Topology Appl. 155 No. 7 (2008) 623-666

Mathematics

Algebraic Topology

47 pages; LaTeX2e with xypic and pstricks macros; corrected typos

Scientific paper

10.1016/j.topol.2007.11.005

We study a special sort of 2-dimensional extended Topological Quantum Field Theories (TQFTs) which we call open-closed 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,i,i^*) consists of a symmetric Frobenius algebra A, a commutative Frobenius algebra C, and an algebra homomorphism i:C->A with dual i^*:A->C, subject to some conditions. This result is achieved by providing a generators and relations description of the category of open-closed cobordisms. 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.

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