Formality theorems: from associators to a global formulation
Annales mathématiques Blaise Pascal, Volume 13 (2006) no. 2, p. 313-348

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,ΛTM),[-,-] 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.

@article{AMBP_2006__13_2_313_0,
     author = {Halbout, Gilles},
     title = {Formality theorems: from associators to a global formulation},
     journal = {Annales math\'ematiques Blaise Pascal},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {13},
     number = {2},
     year = {2006},
     pages = {313-348},
     doi = {10.5802/ambp.220},
     mrnumber = {2275450},
     zbl = {1112.53067},
     language = {en},
     url = {http://www.numdam.org/item/AMBP_2006__13_2_313_0}
}
Halbout, Gilles. Formality theorems: from associators to a global formulation. Annales mathématiques Blaise Pascal, Volume 13 (2006) no. 2, pp. 313-348. doi : 10.5802/ambp.220. http://www.numdam.org/item/AMBP_2006__13_2_313_0/

[1] Bauesi, J. H. The double bar and cobar constructions, Compos. Math, Tome 43 (1981), pp. 331-341 | Numdam | MR 632433 | Zbl 0478.57027

[2] Dolgushev, V. Covariant and equivariant formality theorems, Adv. Math., Tome 191 (2005), pp. 147-177 | Article | MR 2102846 | Zbl 02134411

[3] Drinfeld, V. G. Quasi-Hopf algebras, Leningrad Math. J., Tome 1 (1990), pp. 1419-1457 | MR 1047964

[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 934283

[5] Enriquez, B. A cohomological construction of quantization functors of Lie bialgebras, Adv. Math., Tome 197 (2005), pp. 430-479 | Article | MR 2173841 | Zbl 02231207

[6] Etingof, P.; Kazhdan, D. Quantization of Lie bialgebras. I, Selecta Math. (N.S.), Tome 2 (1996), pp. 1-41 | Article | MR 1403351 | Zbl 0863.17008

[7] Etingof, P.; Kazhdan, D. Quantization of Lie bialgebras. II, III, Selecta Math. (N.S.), Tome 4 (1998), p. 213-231, 233-269 | Article | MR 1669953 | Zbl 0915.17009

[8] Fedosov, B. A simple geometrical construction of deformation quantization, J. Diff. Geom., Tome 40 (1994), pp. 213-238 | MR 1293654 | Zbl 0812.53034

[9] Gerstenhaber, M.; Voronov, A. Homotopy G-algebras and moduli space operad, Internat. Math. Res. Notices, Tome 3 (1995), pp. 141-153 | Article | MR 1321701 | Zbl 0827.18004

[10] Ginot, G. Homologie et modèle minimal des algèbres de Gerstenhaber, Ann. Math. Blaise Pascal, Tome 11 (2004), pp. 95-127 | Article | Numdam | MR 2077240 | Zbl 02207860

[11] Ginot, G.; Halbout, G. A formality theorem for Poisson manifold, Lett. Math. Phys., Tome 66 (2003), pp. 37-64 | Article | MR 2064591 | Zbl 1066.53145

[12] Ginzburg, V.; Kapranov, M. Koszul duality for operads, Duke Math. J., Tome 76 (1994), pp. 203-272 | Article | MR 1301191 | Zbl 0855.18006

[13] Halbout, G. Formule d’homotopie entre les complexes de Hochschild et de de Rham, Compositio Math., Tome 126 (2001), pp. 123-145 | Article | MR 1827641 | Zbl 1007.16008

[14] Hinich, V. Tamarkin’s proof of Kontsevich’s formality theorem, Forum Math., Tome 15 (2003), pp. 591-614 | Article | MR 1978336 | Zbl 01916218

[15] Hochschild, G.; Kostant, B.; Rosenberg, A. Differential forms on regular affine algebras, Transactions AMS, Tome 102 (1962), pp. 383-408 | Article | MR 142598 | Zbl 0102.27701

[16] Kassel, C. Homologie cyclique, caractère de Chern et lemme de perturbation, J. Reine Angew. Math., Tome 408 (1990), pp. 159-180 | Article | MR 1058987 | Zbl 0691.18002

[17] Khalkhali, M. Operations on cyclic homology, the X complex, and a conjecture of Deligne, Comm. Math. Phys., Tome 202 (1999), pp. 309-323 | Article | MR 1689975 | Zbl 0952.16008

[18] Kontsevich, M. Formality conjecture. Deformation theory and symplectic geometry, Math. Phys. Stud., Tome 20 (1996), pp. 139-156 | MR 1480721 | Zbl 1149.53325

[19] Kontsevich, M. Deformation quantization of Poisson manifolds, I, Lett. Math. Phys., Tome 66 (2003), pp. 157-216 | Article | MR 2062626 | Zbl 1058.53065

[20] Kontsevich, M.; Soibelman, Y. Deformations of algebras over operads and the Deligne conjecture (2000), pp. 255-307 | MR 1805894 | Zbl 0972.18005

[21] Lecomte, P. B. A.; Wilde, M. De A homotopy formula for the Hochschild cohomology, Compositio Math., Tome 96 (1995), pp. 99-109 | Numdam | MR 1323727 | Zbl 0842.16006

[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 1805923 | Zbl 0974.16005