An outline of shifted Poisson structures and deformation quantisation in derived differential geometry
J. P. Pridham
TL;DR
This work adapts shifted Poisson structures and deformation quantisation from derived algebraic geometry to derived differential geometry by working with dg \mathcal{C}^{\infty}-rings and both stacky and derived enhancements (NQ-manifolds, dg manifolds, and their higher analogues). It defines shifted Poisson structures via polyvector fields and Maurer–Cartan formalisms, proving the equivalence with shifted symplectic structures in the nondegenerate case and extending the framework to super and dg gradiations. Quantisation is developed for 0-, -1-, and -2-shifted structures using BD_k operads, brace algebras, and de Rham-type complexes, yielding curved $A_{\infty}$, BV$_{\infty}$, and related deformations, with careful treatment of spin structures and Lagrangian data. The paper then extends these constructions to Lie groupoids, higher Lie groupoids, and derived stacks, providing functoriality results, descent principles, and Morita-invariance statements that ensure the invariants depend only on differentiable stacks. Collectively, the results supply a coherent differential-geometric counterpart to the algebro-geometric theory, enabling controlled quantisations and derived-intersection phenomena in a smooth setting. The developments promise new tools for studying derived moduli in differential geometry, including derived intersections, transgressions, and quantised Fukaya-type categories in a purely differential framework.
Abstract
We explain how to translate several recent results in derived algebraic geometry to derived differential geometry. These concern shifted Poisson structures on NQ-manifolds, Lie groupoids, smooth stacks and derived generalisations, and include existence and classification of various deformation quantisations.