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Formality Theory pp 61–80Cite as

Kontsevich’s Formula and Globalization

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Abstract

In this chapter, we give a sketchy exposition of the Kontsevich formula, which allows us to define locally a star product for any Poisson manifold and we introduce some further developments of the Kontsevich theory; in particular, we introduce briefly the globalization of star products by Cattaneo-Felder-Tomassini and Dolgushev. There are many interesting problems that come up after Kontsevich’s theory: in the last section of this chapter we aim to introduce some of these questions.

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Notes

  1. 1.

    \(\fancyscript{H}_n\) is a non-compact smooth \(2n\)-dimensional manifold and we introduce an orientation on \(\fancyscript{H}_n\) using its complex structure.

  2. 2.

    The definition of Lie algebra cohomology is recalled in Appendix A.

References

  1. I. Batalin, G. Vilkovisky, Gauge algebra and quantization. Phys. Lett. B 102(1), 27–31 (1981)

    Article  ADS  MathSciNet  Google Scholar 

  2. I. Batalin, G. Vilkovisky, Quantization of gauge theories with linearly dependent generators. Phys. Rev. D 28, 2567–2582 (1983)

    Google Scholar 

  3. S. Bates, A. Weinstein, Lectures on the geometry of quantization, Berkeley Mathematics Lecture Notes. vol. 8, AMS, Providence, (1997)

    Google Scholar 

  4. P. Bieliavsky, V. Gayral, Deformation Quantization for Actions of Kählerian Lie Groups. Mem. Am. Math. Soc. (to appear) (2011)

    Google Scholar 

  5. F. Bonechi, A. Cattaneo, M. Zabzine, Geometric quantization and non-perturbative Poisson sigma model. Adv. Theor. Math. Phys. 10(5), 683–712 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  6. M. Bordemann, The deformation quantization of certain super-Poisson brackets and BRST cohomology. ed. by G. Dito, D. Sternheimer, in Conference Moshé Flato (1999)

    Google Scholar 

  7. M. Bordemann, M. Brischle, C. Emmrich, S. Waldmann, Subalgebras with Converging Star Products in Deformation Quantization: An Algebraic Construction for \(\mathbb{C}P^n\). J. Math. Phys. 37(12), 6311–6323 (1996)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  8. S. Bosch, Lectures of Formal and Rigid Geometry, Lecture Notes in Mathematics (Springer, Berlin, 2014)

    Book  Google Scholar 

  9. A. Cattaneo, G. Felder, A path integral approach to the Kontsevich quantization formula. Comm. Math. Phys. 212, 591–611 (2000)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  10. A.S. Cattaneo, G. Felder, L. Tomassini, From local to global deformation quantization of Poisson manifolds. Duke Math. J., 115, 329–352 (2002)

    Google Scholar 

  11. A.S. Cattaneo, G. Felder, On the globalization of Kontsevich’s star product and the perturbative Poisson sigma model. Prog. Theor. Phys. Suppl. 144, 38–53 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  12. A.S. Cattaneo, G. Felder, L. Tomassini, Deformation Quantization, IRMA Lectures in Mathematics and Theoretical Physics. Fedosov connections on jet bundles and deformation quantization, (2002)

    Google Scholar 

  13. A.S. Cattaneo, Formality and star products. Poisson Geometry, Deformation Quantization and Group Representations, vol. 323 in London Mathematical Society Lecture Note series, (Cambridge University Press, 2005), pp. 79–144

    Google Scholar 

  14. M. Crainic, Prequantization and Lie brackets. J. Symplectic. Geom., 2(4), (2004)

    Google Scholar 

  15. G. Dito, The necessity of wheels in universal quantization formulas. arXiv:1308.4386

  16. V.A. Dolgushev, Covariant and equivariant formality theorems. Adv. Math. 191(1), 147–177 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  17. V.A. Dolgushev, A proof of Tsygan’s formality conjecture for an arbitrary smooth manifold. Ph.D thesis, MIT (2005)

    Google Scholar 

  18. P. Etingov, MIT Lecture notes. http://ocw.mit.edu/courses/mathematics/18-238-geometry-and-quantum-field-theory-fall-2002/lecture-notes/

  19. B.V. Fedosov, A simple geometrical construction of deformation quantization. J. Differ. Geom. 40, 213–238 (1994)

    MATH  MathSciNet  Google Scholar 

  20. I.M. Gelfand, Some problems of differential geometry and the calculation of the cohomology of Lie algebras of vector fields. Sov. Math. Dokl. 12, 1367–1370 (1971)

    Google Scholar 

  21. E. Hawkins, The correspondence between geometric quantization and formal deformation quantization. arXiv:math/9811049 (1998)

  22. E. Hawkins, Geometric quantization of vector bundles and the correspondence with deformation quantization. Comm. Math. Phys. 215(2), 409–432 (2000)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  23. E. Hawkins, A groupoid approach to quantization. J. Symplectic Geom. 6(1), 61–125 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  24. V. Hinich, Tamarkin’s proof of Kontsevich’s formality theorem. Forum Math. 15, 591–614 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  25. B. Keller, Deformation quantization after Kontsevich and Tamarkin. In: Déformation, quantification, théorie de Lie, vol. 20 (Panoramas et Synthèses, Société mathématique de France, 2005), pp. 19–62

    Google Scholar 

  26. A.A. Kirillov, Geometric quantization. In Dynamical Systems IV, Encyclopaedia of Mathematical Sciences. (Springer, Berlin, 1990)

    Google Scholar 

  27. I. Kolar, P.W. Michor, J. Slovak, Natural operations in differential geometry (Springer, 2010)

    Google Scholar 

  28. M. Kontsevich, Deformation quantization of Poisson manifolds. Lett. Math. Phys. 66, 157–216 (2003)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  29. B. Kostant, Quantization and unitary representation. In Lectures in Modern Analysis and Applications III, vol. 170 of Lecture Notes in Mathematics (Springer, Berlin, 1970), pp. 87–208

    Google Scholar 

  30. B. Kostant, On the definition of quantization. In: Géométrie Symplectique et Physique Mathématique, vol. 237 of Colloques Intern. CNRS, (1975)

    Google Scholar 

  31. C. Laurent-Gengoux, P. Xu, Quantization of pre-quasi-symplectic groupoids and their hamiltonian spaces. In: ed. by J.E. Marsden, T.S. Ratiu, The Breadth of Symplectic and Poisson Geometry: Festschrift in Honor of Alan Weinstein, vol. 232 of Progress in Mathematics. Birkhäuser, (2005)

    Google Scholar 

  32. E. Lerman, Geometric quantization: a crash course. Contemp. Math. 583, 147–174 (2012)

    Article  MathSciNet  Google Scholar 

  33. K.C.H. Mackenzie, Lie Groupoids and Lie Algebroids in Differential Geometry, Number 124 in London Mathamatical Society Lecture notes series (Cambridge University Press, Cambridge, 1987)

    Book  MATH  Google Scholar 

  34. K.C.H. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, London Math. Soc. Lecture notes series (Cambridge University Press, Cambridge, 2005)

    Book  MATH  Google Scholar 

  35. C.M. Marle, Lie, Symplectic and Poisson Groupoids and Their Lie Algebroids, Encyclopedia of Mathematical Physics (Elsevier, Amsterdam, 2006)

    Google Scholar 

  36. E. Miranda, From action-angle coordinates to geometric quantization and back, In: Finite Dimensional Integrable Systems in Geometry and Mathematical, 2011, (2011)

    Google Scholar 

  37. H. Omori, Y. Maeda, N. Miyazaki, A. Yoshida, Deformation quantization of Fréchet-Poisson algebras of Heisenberg type. Contemp. Math. 288, 391–395 (2001)

    Article  Google Scholar 

  38. M.E. Peskin, D.V. Schroeder, An Introduction to Quantum Field Theory (Westview Press, Boulder, 1995)

    Google Scholar 

  39. M. Polyak, Quantization of linear Poisson structures and degrees of maps. Lett. Math. Phys. 66(1–2), 15–35 (2003)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  40. M. Rieffel, Deformation quantization and operator algebra. Proceedings of Symposia in Pure Mathematics, vol. 51, (1990)

    Google Scholar 

  41. M. Rieffel, Quantization and \(C^*\)-algebras. Contemp. Math., 167, (1994)

    Google Scholar 

  42. W.G. Ritter, Geometric quantization. arXiv:math-ph/0208008

  43. L.H. Ryder, Quantum Field Theory (Cambridge University Press, Cambridge, 1985)

    MATH  Google Scholar 

  44. D.J. Saunders, The Geometry of Jet Bundles, Number 142 in London Math. Soc. Lecture notes series (Cambridge University Press, Cambridge, 1989)

    Google Scholar 

  45. B. Shoikhet, Vanishing of the Kontsevich Integrals of the Wheels. Lett. Math. Phys. 56, 141–149 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  46. D.E. Tamarkin, Another proof of M. Kontsevich formality theorem. arXiv:math/9803025, (1998)

  47. D.E. Tamarkin, Formality of chain operad of little discs. Lett. Math. Phys. 66(1–2), 65–72 (2003)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  48. A. Weinstein, Symplectic groupoids and Poisson manifolds. Bull. Amer. Math. Soc. 16(1), 101–104 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  49. A. Weinstein, Symplectic groupoids, geometric quantization, and irrational rotation algebras, In Symplectic Geometry, Groupoids and Integrable Systems, vol. 20 of Mathematical Sciences Research Institute Publications (Springer, Berlin, 1991)

    Google Scholar 

  50. A. Weinstein, Tangential deformation quantization and polarized symplectic groupoids, In: Deformation Theory and Symplectic Geometry, vol. 20 of Mathematical Physics Studies (Springer, Berlin, 1996)

    Google Scholar 

  51. A. Weinstein, P. Xu, Extensions of symplectic groupoids and quantization. J. Reine Angew. Math. 417, 159–189 (1991)

    MATH  MathSciNet  Google Scholar 

  52. T. Willwacher, Counterexample to the quantizability of modules. Lett. Math. Phys. 81(3), 265–280 (2007)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  53. T. Willwacher, The obstruction to the existence of a loopless star product. arXiv:1309.7921, (2013)

  54. N.M.J. Woodhouse, Geometric Quantization, 2nd edn. (Oxford University Press, Oxford, 1997)

    MATH  Google Scholar 

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  1. Department of Mathematics, University of Würzburg, Würzburg, Germany

    Chiara Esposito

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Esposito, C. (2015). Kontsevich’s Formula and Globalization. In: Formality Theory. SpringerBriefs in Mathematical Physics, vol 2. Springer, Cham. https://doi.org/10.1007/978-3-319-09290-4_4

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