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.
Higher U (1)-gerbe connections in geometric prequantization / Fiorenza, Domenico; Rogers, Christopher L.; Schreiber, Urs. - In: REVIEWS IN MATHEMATICAL PHYSICS. - ISSN 0129-055X. - STAMPA. - 28:6(2016), p. 1650012. [10.1142/S0129055X16500124]
Higher U (1)-gerbe connections in geometric prequantization
FIORENZA, DOMENICO;
2016
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.File | Dimensione | Formato | |
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