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Journal of Mathematics
Volume 2013, Article ID 407068, 25 pages
http://dx.doi.org/10.1155/2013/407068
Research Article

Twin TQFTs and Frobenius Algebras

Department of Mathematics, California State University, Fresno, 5245 North Backer Avenue M/S PB 108, Fresno, CA 93740-8001, USA

Received 12 November 2012; Accepted 16 December 2012

Academic Editor: Roberto A. Kraenkel

Copyright © 2013 Carmen Caprau. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We introduce the category of singular 2-dimensional cobordisms and show that it admits a completely algebraic description as the free symmetric monoidal category on a twin Frobenius algebra, by providing a description of this category in terms of generators and relations. A twin Frobenius algebra consists of a commutative Frobenius algebra , a symmetric Frobenius algebra , and an algebra homomorphism with dual , satisfying some extra conditions. We also introduce a generalized 2-dimensional Topological Quantum Field Theory defined on singular 2-dimensional cobordisms and show that it is equivalent to a twin Frobenius algebra in a symmetric monoidal category.