Cochain-Valued Theory in Topological Field Theory

Abstract

We determine the category of boundary conditions in the case that the closed string algebra is semisimple. We find that sewing constraints - the most primitive form of worldsheet locality - already imply that D-branes are vector bundles on spacetime. This expository paper describes sewing conditions in two-dimensional open and closed topological field theory. We include a description of the G-equivariant case, where G is a finite group. We comment on extensions to cochain - valued theories and various applications.