A talk that I gave:
Correspondences of cohesive linear homotopy types and Quantization
Oxford, July 5-8 2014
Geometric quantization (the orbit method) is really the process of forming the push-forward in twisted equivariant K-theory of the prequantum line bundle on phase space. Incidentally, this is also the definition of D-brane charge in string theory. In this talk I first explain how all the ingredients that go into these statements have a characterization in Goodwillie-tangent spaces to those infinity-toposes which, following Lawvere, I call “cohesive”. A prequantized Lagrangian correspondence here is a kind of pure motive of cohesive higher stacks equipped with cocycles in bivariant twisted differential generalized cohomology. I then discuss how transfer through such correspondences by twisted Umkehr maps serves as a path integral quantization for topological field theories and close by indicating some examples.