Higher U(1)-gerbe connections in geometric prequantization

Abstract

We promote geometric prequantization to higher geometry (higher stacks), where a prequantization is given by a higher principal connection (a higher gerbe with connection). We show fairly generally how there is canonically a tower of higher gauge groupoids and Courant groupoids assigned to a higher prequantization, and establish the corresponding Atiyah sequence as an integrated Kostant-Souriau infinity-group extension of higher Hamiltonian symplectomorphisms by higher quantomorphisms. We also exhibit the infinity-group cocycle which classifies this extension and discuss how its restrictions along Hamiltonian infinity-actions yield higher Heisenberg cocycles. In the special case of higher differential geometry over smooth manifolds we find the L-infinity-algebra extension of Hamiltonian vector fields -- which is the higher Poisson bracket of local observables -- and show that it is equivalent to the construction proposed by the second author in n-plectic geometry. Finally we indicate a list of examples of applications of higher prequantization in the extended geometric quantization of local quantum field theories and specifically in string geometry.

Title changed. Exposition revised. 55 pages

References in corpus (5)

• Principal infinity-bundles -- General theory
• Extended higher cup-product Chern-Simons theories
• L-infinity algebras of local observables from higher prequantum bundles
• Principal infinity-bundles - Presentations
• AQFT from n-functorial QFT

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• The Rational Higher Structure of M-theory
• Lie n-algebras of BPS charges
• The stack of Yang-Mills fields on Lorentzian manifolds
• Smooth 2-Group Extensions and Symmetries of Bundle Gerbes
• Fluxes, bundle gerbes and 2-Hilbert spaces
• Global Double Field Theory is Higher Kaluza-Klein Theory
• Principal $\infty$-Bundles and Smooth String Group Models
• Towards an extended/higher correspondence -- Generalised geometry, bundle gerbes and global Double Field Theory
• Smooth generalized symmetries of quantum field theories
• Chern correspondence for higher principal bundles
• Quantum field theoretic representation of Wilson surfaces: I higher coadjoint orbit theory
• Gerbes in Geometry, Field Theory, and Quantisation