On the structure of open-closed topological field theory in two dimensions

I discuss the general formalism of two-dimensional topological field theori es defined on open-closed oriented Riemann surfaces, starting from an exten sion of Segal's geometric axioms. Exploiting the topological sewing constra ints allows for the identification of the algebraic structure governing suc h systems. I give a careful treatment of bulk-boundary and boundary-bulk co rrespondences, which are responsible for the relation between the closed an d open sectors. The fact that these correspondences need not be injective n or surjective has interesting implications for the problem of classifying ' boundary conditions'. In particular, I give a clear geometric derivation of the (topological) boundary state formalism and point out some of its limit ations. Finally, I formulate the problem of classifying ton-shell) boundary extensions of a given closed topological field theory in purely algebraic terms and discuss reducibility of boundary extensions. (C) 2001 Elsevier Sc ience B.V. All rights reserved.