Deformation quantisation for $(-2)$-shifted symplectic structures

TL;DR

This paper develops a framework for quantising $(-2)$-shifted symplectic structures on derived algebraic stacks using solutions to a quantum master equation, linking to de Rham cohomology and Borel--Moore homology.

Contribution

It introduces a novel notion of $E_{-1}$ quantisation for $(-2)$-shifted Poisson structures and relates these to cohomological invariants and Borisov--Joyce invariants.

Findings

Quantisations are parametrised by power series in de Rham cohomology.

Quantisations induce classes in Borel--Moore homology for derived schemes.

In many cases, these classes relate closely to Borisov--Joyce invariants.

Abstract

We formulate a notion of $E_{-1}$ quantisation of $(-2)$-shifted Poisson structures on derived algebraic stacks, depending on a flat right connection on the structure sheaf, as solutions of a quantum master equation. We then parametrise $E_{-1}$ quantisations of $(-2)$-shifted symplectic structures by constructing a map to power series in de Rham cohomology. For derived schemes, we show that these quantisations give rise to classes in Borel--Moore homology, and for a large class of examples we show that the classes are closely related to Borisov--Joyce invariants.