Higher U(1)-gerbe connections in geometric prequantization

TL;DR

This paper extends geometric prequantization to higher geometry using higher gerbes, establishing a framework for higher gauge groupoids, Atiyah sequences, and applications in quantum field and string theories.

Contribution

It introduces a higher geometric prequantization framework with associated gauge groupoids, Atiyah sequences, and cocycles, connecting to higher Poisson structures and applications in physics.

Findings

Constructs a tower of higher gauge groupoids and Courant groupoids.

Establishes an Atiyah sequence as an infinity-group extension.

Identifies higher Heisenberg cocycles and applications in quantum field theories.

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.