Transgression of D-branes

TL;DR

This paper develops a loop space perspective on D-branes in string theory, establishing an equivalence with classical descriptions and connecting to topological quantum field theories through Frobenius algebra structures.

Contribution

It introduces a novel loop space framework for D-branes using bundles of Frobenius algebras and bimodules, linking geometric and algebraic approaches.

Findings

Proves equivalence between classical and loop space perspectives on D-branes.

Connects D-brane structures to Frobenius algebras in topological quantum field theory.

Shows the loop space approach as a smooth family of reflection-positive Frobenius algebras.

Abstract

Closed strings can be seen either as one-dimensional objects in a target space or as points in the free loop space. Correspondingly, a B-field can be seen either as a connection on a gerbe over the target space, or as a connection on a line bundle over the loop space. Transgression establishes an equivalence between these two perspectives. Open strings require D-branes: submanifolds equipped with vector bundles twisted by the gerbe. In this paper we develop a loop space perspective on D-branes. It involves bundles of simple Frobenius algebras over the branes, together with bundles of bimodules over spaces of paths connecting two branes. We prove that the classical and our new perspectives on D-branes are equivalent. Further, we compare our loop space perspective to Moore-Segal/Lauda-Pfeiffer data for open-closed 2-dimensional topological quantum field theories, and exhibit it as a smooth family of reflection-positive, colored knowledgable Frobenius algebras.