Introduction to deformation quantization

Yoshiaki Maeda, Hiroshige Kajiura (note: Ryo Takamura)

The University of Tokyo, Lectures in Mathematical Sciences 20 (May 2002),

I.S.B.N. 1-000-01609-9

1. Introduction 1.1 From commutative geometry to noncommutative geometry 1.2 Proposal of describing geometry in terms of noncommutative rings 1.3 Outline of this lecture

2. Poisson algebra 2.1 Poisson algebra 2.2 Examples of Poisson algebras

3. Deformation quantization 3.1 Deformation quantization 3.2 Moyal product 3.3 Remarks on deformation quantization

4. Hochschild cohomology 4.1 Fundamental properties of hochschild cohomologies 4.2 Hochschild cohomology of multi-differential operators

5. Equivalence of deformation quantizations 5.1 Degree two de-Rham cohomology and the equivalence 5.2 Auto-equivalence

6. Deligne’s relative classes 6.1 Deligne’s relative class and Cech cohomology 6.2 Injectivity of {Deformation quantizations} → {Deligne’s relative classes} 6.3 Surjectivity of {Deformation quantizations} → {Deligne’s relative classes}

7. Weyl manifolds 7.1 Formal Weyl manifolds 7.2 Symmetric product 7.3 ν-differential 7.4 Weyl coordinates 7.5 Weyl functions on each local patch 7.6 Weyl manifolds

8. Fedosov connection 8.1 Symplectic connections 8.2 Lift of the connections to those on the Weyl algebra bundle 8.3 Filtration in a Weyl algebra 8.4 Flat connection and recursion 8.5 Horizontal section of the flat connection 8.6 A simple example 8.7 An extension of the flat connection and a relation to others

9. Cohomology equations 9.1 Cohomology equations 9.2 Various results on the cohomology equations

10. Kontsevich’s construction 10.1 Differential graded Lie algebras 10.2 L_{∞}-algebras 10.3 L_{∞}-algebras and their Maurer-Cartan equations 10.4 Compactifications and the existence of the L_{∞}-map in the case of M = R^{n} 10.5 Kontsevich’s deformation quantization formula and its applications

10.5.1 Simple examples of the deformation quantization formula

11. Path integral expression of the deformation quantization formula

11.1 Poisson-sigma model 11.1.1 Kontsevich’s graphs and the Poisson-sigma model, 11.1.2 The symplectic manifold case, 11.1.3 Treatment of the Poisson-sigma model as a gauge theory and the necessity of

the BV-formalism

11.2 The Batalin-Vilkovisky (BV) formalism 11.2.1 Determination of the space of fields, 11.2.2 The BV bracket and the master equation, 11.2.3 Quantum field theory in terms of the path integral and gauge fixing, 11.2.4 Quantum BV master equation, 11.2.5 Notations

11.3 Quantization of the Poisson-sigma model in terms of the BV-formalism 11.3.1 Determination of the BV-action, 11.3.2 Superfield formalism, 11.3.3 Gauge fixing, 11.3.4 Feynman rules and the perturbative expansion, 11.3.5 The perturbative expansion and the deformation quantization formula, 11.3.6 Renormalization

11.4 A relation to the L_{∞}-algebra

12. A Noncommutative two-dimensional sphere 12.1 A noncommutative three sphere 12.2 A noncommutative two sphere 12.3 Noncommutative Kähler manifolds

13. Convergence problem of deformation quantization 13.1 Deformtation quantization of Fréchet Poisson algebras 13.2 Convergence of the deformtation quantization of Fréchet Poisson algebras 13.3 Product formula for e^{itxy}