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

TL;DR

This paper explores the algebraic and geometric structures of two-dimensional open-closed topological field theories, focusing on boundary conditions, state formalism, and classification of boundary extensions.

Contribution

It extends Segal's axioms to open-closed theories, analyzes boundary conditions, and formulates classification problems algebraically.

Findings

Identifies algebraic structures from sewing constraints.

Provides a geometric derivation of boundary state formalism.

Discusses classification and reducibility of boundary extensions.

Abstract

I discuss the general formalism of two-dimensional topological field theories defined on open-closed oriented Riemann surfaces, starting from an extension of Segal's geometric axioms. Exploiting the topological sewing constraints allows for the identification of the algebraic structure governing such systems. I give a careful treatment of bulk-boundary and boundary-bulk correspondences, which are responsible for the relation between the closed and open sectors. The fact that these correspondences need not be injective nor 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 limitations. Finally, I formulate the problem of classifying (on-shell) boundary extensions of a given closed topological field theory in purely algebraic terms and discuss their reducibility.