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Saturday, 2 January 2016

Our new article: Deformation quantization on jet manifolds

Our new article: G. Sardanashvily, A. Zamyatin, “Deformation quantization on jet manifolds” in arXiv:1512.06047

Abstract. Deformation quantization conventionally is described in terms of multidifferential operators. Jet manifold technique is well-known provide the adequate formulation of theory of differential operators. We extended this formulation to the multidifferential ones, and consider their infinite order jet prolongation. The infinite order jet manifold is endowed with the canonical flat connection that provides the covariant formula of a deformation star-product.

1 Introduction

2 Deformation quantization of Poisson manifolds
  • 2.1 Gerstenhaber’s deformation of algebras
  • 2.1.1 Formal deformation
  • 2.1.2 Deformation of rings
  • 2.2 Star-product
  • 2.3 Kontsevich’s deformation quantization
  • 2.3.1 Differential graded Lie algebras
  • 2.3.2 L^∞-algebras
  • 2.3.3 Formality theorem

3 Deformation quantization on jet manifolds
  • 3.1 Multidifferential operators on C^∞(X)
  • 3.2 Deformations of C^∞(X)
  • 3.3 Jet prolongation of multidifferential operators
  • 3.4 Star-product in a covariant form

4 Appendix
  • 4.1 Fibre bundles
  • 4.2 Differential forms and multivector fields
  • 4.3 First order jet manifolds
  • 4.4 Higher and infinite order jets
  • 4.5 Hochschild cohomology
  • 4.6 Chevalley–Eilenberg cohomology of Lie algebras

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