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

نویسندگان

  • Aaron D. Lauda
  • Hendryk Pfeiffer
چکیده

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. Mathematics Subject Classification (2000): 57R56, 57M99, 81T40, 58E05, 19D23, 18D35.

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تاریخ انتشار 2005