D-branes and K-theory in 2D topological field theory

TL;DR

This paper explores the mathematical structure of D-branes in 2D topological field theories, showing they correspond to G-twisted vector bundles and providing sewing constraints and Morse theory proofs.

Contribution

It offers a detailed description of sewing conditions, extends to G-equivariant cases, and proves that D-branes are G-twisted vector bundles in semisimple closed string algebras.

Findings

D-branes are G-twisted vector bundles on spacetime.

Sewing constraints imply worldsheet locality.

Morse theory provides uniform proofs of sewing theorems.

Abstract

This expository paper describes sewing conditions in two-dimensional open/closed topological field theory. We include a description of the G-equivariant case, where G is a finite group. We determine the category of boundary conditions in the case that the closed string algebra is semisimple. In this case we find that sewing constraints -- the most primitive form of worldsheet locality -- already imply that D-branes are (G-twisted) vector bundles on spacetime. We comment on extensions to cochain-valued theories and various applications. Finally, we give uniform proofs of all relevant sewing theorems using Morse theory.