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 [Formula: see text]-group extension of higher Hamiltonian symplectomorphisms by higher quantomorphisms. We also exhibit the [Formula: see text]-group cocycle which classifies this extension and discuss how its restrictions along Hamiltonian [Formula: see text]-actions yield higher Heisenberg cocycles. In the special case of higher differential geometry over smooth manifolds, we find the [Formula: see text]-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 [Formula: see text]-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.