Formality theorems: from associators to a global formulation

Gilles Halbout[1]

  • [1] Institut de Recherche Mathématique Avancée Université Louis Pasteur 7, rue René Descartes 67084 Strasbourg Cedex FRANCE

Annales mathématiques Blaise Pascal (2006)

  • Volume: 13, Issue: 2, page 313-348
  • ISSN: 1259-1734

Abstract

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Let M be a differential manifold. Let Φ be a Drinfeld associator. In this paper we explain how to construct a global formality morphism starting from Φ . More precisely, following Tamarkin’s proof, we construct a Lie homomorphism “up to homotopy" between the Lie algebra of Hochschild cochains on C ∞ ( M ) and its cohomology ( Γ ( M , Λ T M ) , [ - , - ] S ). This paper is an extended version of a course given 8 - 12 March 2004 on Tamarkin’s works. The reader will find explicit examples, recollections on G ∞ -structures, explanation of the Etingof-Kazhdan quantization-dequantization theorem, of Tamarkin’s cohomological obstruction and of globalization process needed to get the formality theorem. Finally, we prove here that Tamarkin’s formality maps can be globalized.

How to cite

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Halbout, Gilles. "Formality theorems: from associators to a global formulation." Annales mathématiques Blaise Pascal 13.2 (2006): 313-348. <http://eudml.org/doc/10533>.

@article{Halbout2006,
abstract = {Let $M$ be a differential manifold. Let $\Phi $ be a Drinfeld associator. In this paper we explain how to construct a global formality morphism starting from $\Phi $. More precisely, following Tamarkin’s proof, we construct a Lie homomorphism “up to homotopy" between the Lie algebra of Hochschild cochains on $C^\{\infty \}(M)$ and its cohomology $(\Gamma (M,\Lambda TM), ~[-,-]_S$). This paper is an extended version of a course given 8 - 12 March 2004 on Tamarkin’s works. The reader will find explicit examples, recollections on $G_\infty $-structures, explanation of the Etingof-Kazhdan quantization-dequantization theorem, of Tamarkin’s cohomological obstruction and of globalization process needed to get the formality theorem. Finally, we prove here that Tamarkin’s formality maps can be globalized.},
affiliation = {Institut de Recherche Mathématique Avancée Université Louis Pasteur 7, rue René Descartes 67084 Strasbourg Cedex FRANCE},
author = {Halbout, Gilles},
journal = {Annales mathématiques Blaise Pascal},
keywords = {formality theorem; deformation quantization; dequantization},
language = {eng},
month = {7},
number = {2},
pages = {313-348},
publisher = {Annales mathématiques Blaise Pascal},
title = {Formality theorems: from associators to a global formulation},
url = {http://eudml.org/doc/10533},
volume = {13},
year = {2006},
}

TY - JOUR
AU - Halbout, Gilles
TI - Formality theorems: from associators to a global formulation
JO - Annales mathématiques Blaise Pascal
DA - 2006/7//
PB - Annales mathématiques Blaise Pascal
VL - 13
IS - 2
SP - 313
EP - 348
AB - Let $M$ be a differential manifold. Let $\Phi $ be a Drinfeld associator. In this paper we explain how to construct a global formality morphism starting from $\Phi $. More precisely, following Tamarkin’s proof, we construct a Lie homomorphism “up to homotopy" between the Lie algebra of Hochschild cochains on $C^{\infty }(M)$ and its cohomology $(\Gamma (M,\Lambda TM), ~[-,-]_S$). This paper is an extended version of a course given 8 - 12 March 2004 on Tamarkin’s works. The reader will find explicit examples, recollections on $G_\infty $-structures, explanation of the Etingof-Kazhdan quantization-dequantization theorem, of Tamarkin’s cohomological obstruction and of globalization process needed to get the formality theorem. Finally, we prove here that Tamarkin’s formality maps can be globalized.
LA - eng
KW - formality theorem; deformation quantization; dequantization
UR - http://eudml.org/doc/10533
ER -

References

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