Wednesday, April 3, 2013

1304.0236 (Domenico Fiorenza et al.)

Higher geometric prequantum theory    [PDF]

Domenico Fiorenza, Christopher L. Rogers, Urs Schreiber
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 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 extensions 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 prequantum theory in the extended geometric quantization of local quantum field theories and specifically in string geometry.
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