Formality theorems: from associators to a global formulation

Halbout, Gilles

Annales mathématiques Blaise Pascal, Tome 13 (2006) no. 2, pp. 313-348.

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 $C^{\infty }\left(M\right)$ 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.

[1] Bauesi, J. H. The double bar and cobar constructions, Compos. Math, Volume 43 (1981), pp. 331-341 | Numdam | MR | Zbl

[2] Dolgushev, V. Covariant and equivariant formality theorems, Adv. Math., Volume 191 (2005), pp. 147-177 | DOI | MR | Zbl

[3] Drinfeld, V. G. Quasi-Hopf algebras, Leningrad Math. J., Volume 1 (1990), pp. 1419-1457 | MR

[4] Drinfeld, V. G. Quantum groups, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), Amer. Math. Soc., Providence, RI, 1993, pp. 798-820 | MR

[5] Enriquez, B. A cohomological construction of quantization functors of Lie bialgebras, Adv. Math., Volume 197 (2005), pp. 430-479 | DOI | MR | Zbl

[6] Etingof, P.; Kazhdan, D. Quantization of Lie bialgebras. I, Selecta Math. (N.S.), Volume 2 (1996), pp. 1-41 | DOI | MR | Zbl

[7] Etingof, P.; Kazhdan, D. Quantization of Lie bialgebras. II, III, Selecta Math. (N.S.), Volume 4 (1998), p. 213-231, 233-269 | DOI | MR | Zbl

[8] Fedosov, B. A simple geometrical construction of deformation quantization, J. Diff. Geom., Volume 40 (1994), pp. 213-238 | MR | Zbl

[9] Gerstenhaber, M.; Voronov, A. Homotopy G-algebras and moduli space operad, Internat. Math. Res. Notices, Volume 3 (1995), pp. 141-153 | DOI | MR | Zbl

[10] Ginot, G. Homologie et modèle minimal des algèbres de Gerstenhaber, Ann. Math. Blaise Pascal, Volume 11 (2004), pp. 95-127 | DOI | Numdam | MR | Zbl

[11] Ginot, G.; Halbout, G. A formality theorem for Poisson manifold, Lett. Math. Phys., Volume 66 (2003), pp. 37-64 | DOI | MR | Zbl

[12] Ginzburg, V.; Kapranov, M. Koszul duality for operads, Duke Math. J., Volume 76 (1994), pp. 203-272 | DOI | MR | Zbl

[13] Halbout, G. Formule d’homotopie entre les complexes de Hochschild et de de Rham, Compositio Math., Volume 126 (2001), pp. 123-145 | DOI | MR | Zbl

[14] Hinich, V. Tamarkin’s proof of Kontsevich’s formality theorem, Forum Math., Volume 15 (2003), pp. 591-614 | DOI | MR | Zbl

[15] Hochschild, G.; Kostant, B.; Rosenberg, A. Differential forms on regular affine algebras, Transactions AMS, Volume 102 (1962), pp. 383-408 | DOI | MR | Zbl

[16] Kassel, C. Homologie cyclique, caractère de Chern et lemme de perturbation, J. Reine Angew. Math., Volume 408 (1990), pp. 159-180 | DOI | MR | Zbl

[17] Khalkhali, M. Operations on cyclic homology, the X complex, and a conjecture of Deligne, Comm. Math. Phys., Volume 202 (1999), pp. 309-323 | DOI | MR | Zbl

[18] Kontsevich, M. Formality conjecture. Deformation theory and symplectic geometry, Math. Phys. Stud., Volume 20 (1996), pp. 139-156 | MR | Zbl

[19] Kontsevich, M. Deformation quantization of Poisson manifolds, I, Lett. Math. Phys., Volume 66 (2003), pp. 157-216 | DOI | MR | Zbl

[20] Kontsevich, M.; Soibelman, Y. Deformations of algebras over operads and the Deligne conjecture (2000), pp. 255-307 | MR | Zbl

[21] Lecomte, P. B. A.; Wilde, M. De A homotopy formula for the Hochschild cohomology, Compositio Math., Volume 96 (1995), pp. 99-109 | Numdam | MR | Zbl

[22] Tamarkin, D. Another proof of M. Kontsevich’s formality theorem (1998) (math.QA/9803025)

[23] Voronov, A.; Publ., Kluwer Acad. Homotopy Gerstenhaber algebras, Conférence Moshé Flato 1999, Vol. II (Dijon), Math. Phys. Stud., 22, 2000, pp. 307-331 | MR | Zbl