Schreiber Higher Prequantum Geometry

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• Urs Schreiber

  Higher Prequantum Geometry

  (arXiv:1601.05956, v2, talk recording)

  chapter in:

  Gabriel Catren, Mathieu Anel (eds.)

  New Spaces for Mathematics and Physics

  Cambridge University Press 2021

  doi:10.1017/9781108854399.008

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This is an expository survey of

• Local prequantum field theory via Synthetic variational calculus

and its formalization in differential cohomology in a cohesive topos.

Related surveys include

• Synthetic prequantum field theory in a cohesive homotopy topos

• Modern Physics formalized in Modal Homotopy Type Theory

Related talk slides include

• Prequantum field theory (pdf)

• Synthetic Quantum Field Theory

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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

    • Modern Physics formalized in Modal Homotopy Type Theory

  • 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

    • Structure Theory for Higher WZW Terms

    • Obstruction theory for parameterized higher WZW terms

    • The WZW term of the M5-brane

    • Lie n-algebras of BPS charges

    • From higher to exceptional geometry

  • Higher geometric prequantum theory

    • Classical field theory via Cohesive homotopy types

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

    • Local prequantum field theory

    • Higher extensions of mapping class groups

    • Higher theta functions and higher CS-WZW holography

  • Motivic quantization of local prequantum field theory

    • Geometric quantization of symplectic and Poisson manifolds

    • Cohomological quantization of local prequantum boundary field theory

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