Quantisation of derived Poisson structures
J. P. Pridham
TL;DR
The paper addresses deformation quantisation of $0$-shifted Poisson structures on derived Artin/D-moduli spaces by exploiting an anti-involution on the Hochschild complex to obtain essentially unique deformations. Central to the approach is formality between the $E_2$- and $P_2$-algebras realized via even Drinfeld associators and the Grothendieck–Teichmüller group, which yields quasi-isomorphisms between the Hochschild complex and the polyvector fields under perfect cotangent complexes. This leads to existence results for self-dual DQ algebroid quantisations on spaces with LCI singularities, and extends to derived DM/Artin stacks, $ ext{C}^ ext{∞}$, and analytic settings, including global quantisations via étale descent. The framework further provides quantisations of $1$-shifted co-isotropic structures, via curved $A_ ext{∞}$ and BD$_2$-algebra formalisms in Tate categories, and establishes uniqueness results through obstruction-theory analyses in stacky and Tate contexts. Collectively, the results deliver a comprehensive deformation-quantisation theory for derived geometric objects, with explicit functoriality and self-duality properties and broad applicability to LCI and non-smooth cases.
Abstract
We prove that every $0$-shifted Poisson structure on a derived Artin $n$-stack admits a curved $A_{\infty}$ deformation quantisation whenever the stack has perfect cotangent complex; in particular, this applies to LCI schemes, where it gives a DQ algebroid quantisation. Whereas the Kontsevich--Tamarkin approach to quantisation for smooth varieties hinges on invariance of the Hochschild complex under affine transformations, we instead exploit the observation that the Hochschild complex carries an anti-involution, and that such anti-involutive deformations of the complex of polyvectors are essentially unique. We also establish analogous statements for deformation quantisations in $\mathcal{C}^{\infty}$ and analytic settings.