2-vector bundles, D-branes and Frobenius Manifolds

Abstract: We show that if $M$ is a Frobenius manifold of dimension $n$ such that $T_{x} M$ is semisimple for every $x \in M$, then there exists a canonical 2-vector bundle $\mathcal{B}$ over $M$ of rank $n$. This 2-vector bundle encodes the information about the maximal category of $D$-branes associated to the open closed topological field theories defined by the Frobenius algebras $T_{x} M$. In particular this construction answers a conjecture of Graeme Segal in~\cite{segal07:_what_is_ellip_objec}. We also explain the relation of the labels of the $D$-branes to Azumaya algebras and twisted vector bundles on the spectral cover $S$ of $M$.