Schreiber Correspondences of cohesive linear homotopy types and Quantization

A talk that I gave:

• Correspondences of cohesive linear homotopy types and Quantization

  talk at Symmetries and correspondences in number theory, geometry, algebra, physics: intra-disciplinary trends

  Oxford, July 5-8 2014

  (pdf)

based on

• Quantization via Linear homotopy types (arXiv:1402.7041)

Abstract

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.

Related talks

• Motivic quantization of local prequantum field theory, talk at GAP XI – Higher Geometry and Quantum Field Theory , Pittsburgh August 2013

Related projects

• differential cohomology in a cohesive topos

  • Synthetic Quantum Field Theory

    • Quantum gauge field theory in Cohesive homotopy type theory

    • Type-semantics for quantization

    • Higher toposes of laws of motion

  • cohomology

    • Principal ∞-bundles – models and general theory

    • Fivebrane structures

    • Cosimplicial C-infinity rings and the de Rham complex of Euclidean space

  • ∞-Chern-Weil theory

    • L-∞ algebra connections

    • Cech cocycles for differential characteristic classes

    • Twisted differential structures

      • Twisted Differential String and Fivebrane Structures

      • The moduli 3-stack of the C-field

  • ∞-Chern-Simons theory

    • A higher stacky perspective on Chern-Simons theory

    • Higher Chern-Weil Derivation of AKSZ Sigma-Models

    • 7d Chern-Simons theory and the 5-brane

    • Extended higher cup-product Chern-Simons theories

  • ∞-Wess-Zumino-Witten theory

    • The brane bouquet

  • Higher geometric prequantum theory

    • Classical field theory via Cohesive homotopy types

    • L-∞ algebras of local observables from higher prequantum bundles

    • Local prequantum field theory

  • Motivic quantization of local prequantum field theory

    • Geometric quantization of symplectic and Poisson manifolds

    • Cohomological quantization of local prequantum boundary field theory