Schreiber Motivic quantization of local prequantum field theory

A talk that I have once given:

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• Urs Schreiber

  
  

  Motivic quantization of local prequantum field theory

  
  

  at GAP XI – Higher Geometry and Quantum Field Theory , Pittsburgh August 2013

  & at Twists, generalised cohomology and applications, Münster, October 2013

  
  

  notes: web

  video: rec, YT

about quantization of local prequantum field theory in the context of differential cohomology in a cohesive topos, motivated from the archetyical example of extended geometric quantization of 2d Chern-Simons theory.

Based on

• Quantization via Linear homotopy types

  arXiv:1402.7041

with more details in

• Nuiten (2013)

Abstract

The first part of the talk indicates how topological local prequantum field theory is naturally presented by higher correspondences in the slice of a cohesive (∞,1)-topos over an ∞-group of units of some E-∞ ring $E$. These are correspondences of moduli stacks of fields (spaces of field trajectories) where the correspondence space is equipped with a cocycle in bivariant twisted E-cohomology. The second part of the talk indicates how quantization of such prequantum data is given by pull-push in twisted E-cohomology, sending correspondences to morphisms of $E$-∞-modules. This is analogous to how Chow motives yield cocycles in motivic cohomology. The existence condition on the required orientation in generalized cohomology are the quantum anomaly cancellation conditions. Finally we indicate a list of examples of holographic quantization of boundary field theories this way: the Poisson manifold at the boundary of the non-perturbative Poisson sigma-model (2d Chern-Simons theory), the charged particle at the boundary of the open string and the heterotic string at the boundary of the M2-brane.

The first part is joint work with Domenico Fiorenza and Hisham Sati. The second part is joint work with Joost Nuiten.

Details

For details see

• Joost Nuiten, Cohomological quantization of local prequantum boundary field theory, thesis (2013)

• Urs Schreiber, motivic quantization

For background see also

• Synthetic Quantum Field Theory

• Higher Chern-Simons local prequantum field theory

Related talks

• Correspondences of cohesive linear homotopy types and Quantization, talk at Symmetries and correspondences: intra-disciplinary trends, Oxford, July 5-8 2014

Related projects

• differential cohomology in a cohesive topos

  • Synthetic Quantum Field Theory

    • Quantum gauge field theory in Cohesive homotopy type theory

    • Motivic quantization of local prequantum field theory

    • Type-semantics for quantization

  • cohomology

    • Principal ∞-bundles – models and general theory

    • Fivebrane structures

  • ∞-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

    • Obstruction theory for parameterized higher WZW terms

  • Higher geometric prequantum theory

    • Classical field theory via higher differential geometry

    • Local prequantum field theory

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

    • Higher extensions of mapping class groups

  • Motivic quantization of local prequantum field theory

    • Geometric quantization of symplectic and Poisson manifolds

    • Cohomological quantization of local prequantum boundary field theory