Abstract: (joint with Ausoni, Baas, Richter and Rognes)

Two vector bundles give rise to a geometrically defined cohomology theory extrapolating past the theory of vector bundles (K-theory) and differential forms (de Rham cohomology), capturing information related to cobordisms of manifolds beyond K-theory and deRham cohomology's reach.

The analytic and differential geometric understanding of two vector bundles is still very much in its infancy. There was a hope that an "integration of determinants through loops" construction would give an integral functor from two vector bundles to quantum field theories. However, the fact that the commutative ring spectrum representing complex K-theory does not support a determinant rules this out.

The first obstruction has a geometric interpretation: the one-dimensional two vector bundle represented by the Dirac monopole over the three sphere splits virtually.