Two-vector bundles and elliptic objects

We (Baas, Dundas, Rognes) define a two-vector bundle over a base space X as a kind of family of two-vector spaces (in the sense of Kapranov and Voevodsky) parametrized over X. The rank 1 case recovers the notion of a Dixmier-Douady gerbe over X. The equivalence classes of two-vector bundles over X form an abelian monoid, whose Grothendieck group completion is the zero-th generalized cohomology group of X represented by the algebraic K-theory of the symmetric bimonoidal category of two-vector spaces. We argue that this K-theory agrees with the algebraic K-theory of the S-algebra (= A-infinity ring spectrum) representing connective topological K-theory, which by explicit computation (Ausoni, Rognes) is a connective, integrally defined form of elliptic cohomology, i.e., has chromatic complexity two. Two-vector bundles are thus geometric objects over X which provide the "cycles" for an elliptic cohomology theory at X. We also wish to indicate how a two-vector bundle over X leads to a virtual (anomaly) vector bundle over the free loop space of X, and an associated action functional for compact oriented surfaces over X.