Shifted geometric quantization

TL;DR

This paper extends geometric quantization to shifted symplectic structures, defining key concepts and demonstrating applications to various geometric objects, including a derived version of the quantization commutes with reduction principle.

Contribution

It introduces the framework of geometric quantization for shifted symplectic stacks, including Lagrangian fibrations and prequantizations, and proves a derived quantization reduction theorem.

Findings

Defined Lagrangian fibrations and prequantizations for shifted symplectic stacks

Applied the theory to symplectic groupoids, Hamiltonian spaces, and moduli spaces

Proved a derived analog of the 'quantization commutes with reduction' principle

Abstract

We introduce geometric quantization in the setting of shifted symplectic structures. We define Lagrangian fibrations and prequantizations of shifted symplectic stacks and their geometric quantization. In addition, we study many examples including symplectic groupoids, Hamiltonian spaces and moduli spaces of flat connections. In the case of Hamiltonian spaces we prove a derived analog of the "quantization commutes with reduction" principle.