Schreiber Higher geometric prequantum theory

An article that we have written:

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• Domenico Fiorenza, $\;$ Chris Rogers $\;$and$\;$ Urs Schreiber:

  
  

  Higher geometric prequantum theory

  arXiv:1304.0236v1

  
  

  published version renamed and more expository:

  
  

  Higher $U(1)$-gerbe connections in geometric prequantization

  Reviews in Mathematical Physics 28 06 (2016)

  doi:10.1142/S0129055X16500124

  arXiv:1304.0236v2

  v3: pdf

  
  

  lecture notes:

  
  

  Higher geometric prequantum theory and The brane bouquet

  talk at Bayrischzell workshop 2013

  pdf slides

on the higher geometric refinement of prequantum field theory and geometric prequantization formulated generally in cohesive higher geometry.

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In the companion article L-∞ algebras of local observables from higher prequantum bundles we discuss the L-∞ algebraic aspects in more detail. See also the discussion at higher Atiyah groupoid.

For more background see

• Higher Prequantum Geometry

• Local prequantum field theory

and

• Classical field theory via Cohesive homotopy types

• Motivic quantization of local prequantum field theory.

Applications include

• Obstruction theory for parameterized higher WZW terms.

Abstract

We develop the refinement of geometric prequantum theory to higher geometry (higher stacks), where a prequantization is given by a higher principal connection (a higher gerbe with connection). We show how there is canonically a tower of higher “gauge groupoids” assigned to a higher prequantization and establish the higher and integrated Kostant-Souriau extension of higher Hamiltonian symplectomorphisms fairly generally. In the special case of higher differential geometry over smooth manifolds we find the L-∞ algebra extension of Hamiltonian vector fields which is the higher-Poisson bracket of local observables and show that it is equivalent to the construction in n-plectic geometry proposed by Rogers. Finally we indicate examples of applications of higher prequantum theory in the extended geometric quantization of local quantum field theories and specifically in string geometry.

Errata

• The differential concretification formula given in Remark 2.10.2 (p. 26) of the article is wrong as stated: it works in degree 1 but produces spurious contributions in degree 0.

  This error had kindly been pointed out by Marco Benini and Alexander Schenkel.

  (The error results from assuming that the evident model of the map $\sharp_1 [X, \mathbf{B}G_{conn}] \to \sharp_1 [X, \mathbf{B}G]$ by simplicial presheaves over the site of Cartesian manifolds $U$ — namely in degree 0 by sending horizontal smooth $\mathfrak{g}$-valued forms on $X \times U$ to the point, and in degree 1 by sending their gauge transformations with potentially discontinuous families of component functions to these families of component functions — is a $U$-wise Kan fibration. If it were, then the homotopy fiber product proposed in 2.10.2 would be computed by the 1-categorical fiber product with this map, which does give the claimed result. Alas, the map fails to satisfy the Kan lifting property in lowest degree. When computing the proposed homotopy fiber product properly it comes out right in degree 1, but it picks up spurious contributions in degree 0.)

  A correct formula for the differential concretification process of $[X, \mathbf{B}G_{conn}]$ was published in

  • Section 3.2 (pp. 17) of

    Marco Benini,$\;$ Alexander Schenkel,$\;$ Urs Schreiber:

    The stack of Yang-Mills fields on Lorentzian manifolds

    CMP 359 (2018) 765-820

    arXiv:1704.01378

    doi:10.1007/s00220-018-3120-1

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

  • 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