Derived Algebraic Geometry and Deformation Quantization

Abstract

This is a report on recent progress concerning the interactions between derived algebraic geometry and deformation quantization. We present the notion of derived algebraic stacks, of shifted symplectic and Poisson structures, as well as the construction of deformation quantization of shifted Poisson structures. As an application we propose a general construction of the quantization of the moduli space of $G$-bundles on an oriented space of arbitrary dimension.

Contribution to the ICM 2014 (a bit more typos corrected, ref added)

References in corpus (12)

• A 'Darboux Theorem' for shifted symplectic structures on derived Artin stacks, with applications
• Lagrangian structures on mapping stacks and semi-classical TFTs
• Multiple Derived Lagrangian Intersections
• Thom-Sebastiani & Duality for Matrix Factorizations
• Completions and derived de Rham cohomology
• Operations on derived moduli spaces of branes
• Noncommutative Motives I: A Universal Characterization of the Motivic Stable Homotopy Theory of Schemes
• A d-shifted Darboux theorem
• Generalized Donaldson-Thomas theory over fields K $\ne$ C
• Derived Algebraic Geometry
• Invariants topologiques des Espaces non commutatifs
• Noncommutative Motives II: K-Theory and Noncommutative Motives

Cited by in corpus (12)

• Integrating quantum groups over surfaces
• Orientability for gauge theories on Calabi-Yau manifolds
• Birational geometry for d-critical loci and wall-crossing in Calabi-Yau 3-folds
• Deformation quantisation for (-1)-shifted symplectic structures and vanishing cycles
• Categorification of algebraic quantum field theories
• Semiorthogonal decompositions of stable pair moduli spaces via d-critical flips
• Homotopy L-infinity spaces
• Lectures on shifted Poisson geometry
• Vanishing theorems for representation homology and the derived cotangent complex
• Local Operators from the Space of Vacua of Four Dimensional SUSY Gauge Theories
• Classical BV formalism for group actions
• A Physical Origin for Singular Support Conditions in Geometric Langlands Theory