Derived Algebraic Geometry and Deformation Quantization

. 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.

[1]  Vittoria Bussi.  Generalized Donaldson-Thomas theory over fields K $\neq$ C , 2014, 1403.2403.

[2]  Bertrand Toen,et al.  Derived Algebraic Geometry , 2014, 1401.1044.

[3]  A. Blanc.  Invariants topologiques des espaces non-commutatifs. , 2013, 1307.6430.

[4]  B. Toën,et al.  Operations on derived moduli spaces of branes , 2013 .

[5]  D. Calaque.  Lagrangian structures on mapping stacks and semi-classical TFTs , 2013, 1306.3235.

[6]  R. Pandharipande,et al.  ALMOST CLOSED 1-FORMS , 2012, Glasgow Mathematical Journal.

[7]  D. Joyce,et al.  Symmetries and stabilization for sheaves of vanishing cycles , 2012, 1211.3259.

[8]  B. Bhatt.  Completions and derived de Rham cohomology , 2012, 1207.6193.

[9]  B. Toën.  Derived Azumaya algebras and generators for twisted derived categories , 2012, Inventiones mathematicae.

[10]  J. Francis.  The tangent complex and Hochschild cohomology of $\mathcal {E}_n$-rings , 2011, Compositio Mathematica.

[11]  Anatoly Preygel.  Thom-Sebastiani and duality for matrix factorizations, and results on the higher structures of the Hochschild invariants , 2012 .

[12]  B. Toën,et al.  Shifted symplectic structures , 2011, 1111.3209.

[13]  Gonçalo Tabuada.  A Guided Tour Through the Garden of Noncommutative Motives , 2011, 1108.3787.

[14]  J. Lurie.  Moduli Problems for Ring Spectra , 2011 .

[15]  Carlos T. Simpson,et al.  Homotopy Theory of Higher Categories: THE MODEL STRUCTURE , 2010, 1001.4071.

[16]  J.P.Pridham.  Representability of derived stacks , 2010, 1011.2189.

[17]  A. Kapustin.  Topological Field Theory, Higher Categories, and Their Applications , 2010, 1004.2307.

[18]  M. Kontsevich,et al.  Motivic Donaldson-Thomas invariants: Summary of results , 2009, 0910.4315.

[19]  B. Toën.  Higher and derived stacks: a global overview , 2009 .

[20]  B. Keller,et al.  On Hochschild Cohomology and Morita Deformations , 2009 .

[21]  Bernhard Keller,et al.  On differential graded categories , 2006, math/0601185.

[22]  B. Toën,et al.  Moduli of objects in dg-categories , 2005, math/0503269.

[23]  Goncalo Tabuada.  Algèbre homologique Une structure de catégorie de modèles de Quillen sur la catégorie des dg-catégories , 2004, math/0407338.

[24]  G. Vezzosi,et al.  Homotopical Algebraic Geometry II: Geometric Stacks and Applications , 2004, math/0404373.

[25]  Bertrand Toen,et al.  Homotopical algebraic geometry. I. Topos theory. , 2002, math/0207028.

[26]  C. Simpson.  Algebraic aspects of higher nonabelian Hodge theory , 1999, math/9902067.

[27]  C. Simpson.  Algebraic (geometric) $n$-stacks , 1996, alg-geom/9609014.

[28]  M. Kontsevich.  Enumeration of Rational Curves Via Torus Actions , 1994, hep-th/9405035.

[29]  Nicolai Reshetikhin,et al.  Quantum Groups , 1993 .

[30]  V. Turaev.  Skein quantization of Poisson algebras of loops on surfaces , 1991 .

[31]  M. Artin,et al.  Versal deformations and algebraic stacks , 1974 .