Formality theorem for differential graded manifolds

Liao, Hsuan-Yi; Stiénon, Mathieu; Xu, Ping

doi:10.1016/j.crma.2017.11.017

Homological algebra/Differential geometry

Formality theorem for differential graded manifolds
[Théorème de formalité pour les variétés différentielles graduées]

Présenté par : the Editorial Board

Liao, Hsuan-Yi ¹ ; Stiénon, Mathieu ¹ ; Xu, Ping ¹

¹ Department of Mathematics, Pennsylvania State University, USA

Comptes Rendus. Mathématique, Tome 356 (2018) no. 1, pp. 27-43.

Résumé
Abstract

Nous prouvons un théorème de formalité pour les variétés lisses différentielles graduées. Plus précisément, nous prouvons qu'il existe, pour toute variété différentielle graduée $(M, Q)$ , un quasi-isomorphisme $L_{\infty}$ de l'algèbre de Lie différentielle graduée $({}_{\oplus}T_{poly}^{•} (M), [Q, -], [-, -])$ dans l'algèbre de Lie différentielle graduée $({}_{\oplus}D_{poly}^{•} (M), 〚 m + Q, - 〛, 〚 -, - 〛)$ , dont le premier coefficient de Taylor (1) est égal à la composée $hkr \circ {({td}_{(M, Q)}^{\nabla})}^{\frac{1}{2}} : {}_{\oplus}T_{poly}^{•} (M) \to {}_{\oplus}D_{poly}^{•} (M)$ de l'action (par contraction) de ${({td}_{(M, Q)}^{\nabla})}^{\frac{1}{2}} \in \prod_{k \geq 0} {(Ω^{k} (M))}^{k}$ sur ${}_{\oplus}T_{poly}^{•} (M)$ avec l'application de Hochschild–Kostant–Rosenberg et (2) respecte les structures d'algèbres associatives en cohomologie. Comme application, nous prouvons la conjecture de Kontsevich–Shoikhet : il existe un théorème de type Kontsevich–Duflo valable pour toute variété différentielle graduée de dimension finie.

We establish a formality theorem for smooth dg manifolds. More precisely, we prove that, for any finite-dimensional dg manifold $(M, Q)$ , there exists an $L_{\infty}$ quasi-isomorphism of dglas from $({}_{\oplus}T_{poly}^{•} (M), [Q, -], [-, -])$ to $({}_{\oplus}D_{poly}^{•} (M), 〚 m + Q, - 〛, 〚 -, - 〛)$ whose first Taylor coefficient (1) is equal to the composition $hkr \circ {({td}_{(M, Q)}^{\nabla})}^{\frac{1}{2}} : {}_{\oplus}T_{poly}^{•} (M) \to {}_{\oplus}D_{poly}^{•} (M)$ of the action of ${({td}_{(M, Q)}^{\nabla})}^{\frac{1}{2}} \in \prod_{k \geq 0} {(Ω^{k} (M))}^{k}$ on ${}_{\oplus}T_{poly}^{•} (M)$ (by contraction) with the Hochschild–Kostant–Rosenberg map and (2) preserves the associative algebra structures on the level of cohomology. As an application, we prove the Kontsevich–Shoikhet conjecture: a Kontsevich–Duflo-type theorem holds for all finite-dimensional smooth dg manifolds.

Reçu le : 2017-11-02
Accepté le : 2017-11-23
Publié le : 2017-12-06

DOI : 10.1016/j.crma.2017.11.017

Affiliations des auteurs :

Liao, Hsuan-Yi ¹ ; Stiénon, Mathieu ¹ ; Xu, Ping ¹

¹ Department of Mathematics, Pennsylvania State University, USA

@article{CRMATH_2018__356_1_27_0,
     author = {Liao, Hsuan-Yi and Sti\'enon, Mathieu and Xu, Ping},
     title = {Formality theorem for differential graded manifolds},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {27--43},
     publisher = {Elsevier},
     volume = {356},
     number = {1},
     year = {2018},
     doi = {10.1016/j.crma.2017.11.017},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2017.11.017/}
}

TY  - JOUR
AU  - Liao, Hsuan-Yi
AU  - Stiénon, Mathieu
AU  - Xu, Ping
TI  - Formality theorem for differential graded manifolds
JO  - Comptes Rendus. Mathématique
PY  - 2018
SP  - 27
EP  - 43
VL  - 356
IS  - 1
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2017.11.017/
DO  - 10.1016/j.crma.2017.11.017
LA  - en
ID  - CRMATH_2018__356_1_27_0
ER  -

%0 Journal Article
%A Liao, Hsuan-Yi
%A Stiénon, Mathieu
%A Xu, Ping
%T Formality theorem for differential graded manifolds
%J Comptes Rendus. Mathématique
%D 2018
%P 27-43
%V 356
%N 1
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2017.11.017/
%R 10.1016/j.crma.2017.11.017
%G en
%F CRMATH_2018__356_1_27_0

Liao, Hsuan-Yi; Stiénon, Mathieu; Xu, Ping. Formality theorem for differential graded manifolds. Comptes Rendus. Mathématique, Tome 356 (2018) no. 1, pp. 27-43. doi : 10.1016/j.crma.2017.11.017. http://www.numdam.org/articles/10.1016/j.crma.2017.11.017/

Bibliographie
Cité par

[1] Bandiera, R. Descent of Deligne–Getzler ∞-groupoids, 2017 | arXiv

[2] Bandiera, R.; Chen, Z.; Stiénon, M.; Xu, P. Shifted derived Poisson manifolds associated with Lie pairs, 2017 | arXiv

[3] Bayen, F.; Flato, M.; Fronsdal, C.; Lichnerowicz, A.; Sternheimer, D. Deformation theory and quantization. I. Deformations of symplectic structures, Ann. Phys., Volume 111 (1978) no. 1, pp. 61-110 MR 0496157 (58 #14737a)

[4] Calaque, D.; Dolgushev, V.; Halbout, G. Formality theorems for Hochschild chains in the Lie algebroid setting, J. Reine Angew. Math., Volume 612 (2007), pp. 81-127 (MR 2364075)

[5] Calaque, D.; Rossi, C.A. Lectures on Duflo Isomorphisms in Lie Algebra and Complex Geometry, EMS Ser. Lect. Math., European Mathematical Society (EMS), Zürich, 2011 (MR 2816610)

[6] Calaque, D.; Van den Bergh, M. Hochschild cohomology and Atiyah classes, Adv. Math., Volume 224 (2010) no. 5, pp. 1839-1889 (MR 2646112)

[7] Cattaneo, A.S.; Felder, G. Relative formality theorem and quantisation of coisotropic submanifolds, Adv. Math., Volume 208 (2007) no. 2, pp. 521-548 (MR 2304327)

[8] Cattaneo, A.S.; Felder, G.; Tomassini, L. From local to global deformation quantization of Poisson manifolds, Duke Math. J., Volume 115 (2002) no. 2, pp. 329-352 (MR 1944574)

[9] Cattaneo, A.S.; Fiorenza, D.; Longoni, R. On the Hochschild–Kostant–Rosenberg map for graded manifolds, Int. Math. Res. Not. (2005) no. 62, pp. 3899-3918 MR 2202177 (2007e:58004)

[10] Chen, Z.; Xiang, M.; Xu, P. Atiyah and Todd classes arising from integrable distributions, 2017 | arXiv

[11] Dolgushev, V. Covariant and equivariant formality theorems, Adv. Math., Volume 191 (2005) no. 1, pp. 147-177 MR 2102846 (2006c:53101)

[12] Dolgushev, V.; Tamarkin, D.; Tsygan, B. Formality theorems for Hochschild complexes and their applications, Lett. Math. Phys., Volume 90 (2009) no. 1–3, pp. 103-136 (MR 2565036)

[13] Dolgushev, V.; Tamarkin, D.; Tsygan, B. Formality of the homotopy calculus algebra of Hochschild (co)chains, 2008 | arXiv

[14] Duflo, M. Caractères des groupes et des algèbres de Lie résolubles, Ann. Sci. Éc. Norm. Supér. (4), Volume 3 (1970), pp. 23-74 (MR 0269777)

[15] Emmrich, C.; Weinstein, A. (Prog. Math.), Volume vol. 123, Birkhäuser Boston, Boston, MA (1994), pp. 217-239 (MR 1327535)

[16] Fedosov, B.V. A simple geometrical construction of deformation quantization, J. Differ. Geom., Volume 40 (1994) no. 2, pp. 213-238 MR 1293654 (95h:58062)

[17] Fiorenza, D.; Manetti, M. $L_{\infty}$ structures on mapping cones, Algebra Number Theory, Volume 1 (2007) no. 3, pp. 301-330 (MR 2361936)

[18] Gerstenhaber, M.; Schack, S.D. Algebraic cohomology and deformation theory, Il Ciocco, 1986 (NATO Adv. Stud. Inst. Ser., Ser. C, Math. Phys. Sci.), Volume vol. 247, Kluwer Acad. Publ., Dordrecht (1988), pp. 11-264 (MR 981619)

[19] Kontsevich, M. Deformation quantization of Poisson manifolds, Lett. Math. Phys., Volume 66 (2003) no. 3, pp. 157-216 MR 2062626 (2005i:53122)

[20] H.-Y. Liao, M. Stiénon, Formal exponential map for graded manifolds, Int. Math. Res. Not., rnx130, . | DOI

[21] Liao, H.-Y.; Stiénon, M.; Xu, P. Formality and Kontsevich–Duflo-type theorems for Lie pairs, 2016 | arXiv

[22] Manchon, D.; Torossian, C. Cohomologie tangente et cup-produit pour la quantification de Kontsevich, Ann. Math. Blaise Pascal, Volume 10 (2003) no. 1, pp. 75-106 (MR 1990011)

[23] Manchon, D.; Torossian, C. Erratum: “Tangent cohomology and cup-product for the Kontsevich quantization”, Ann. Math. Blaise Pascal, Volume 10 (2003) no. 1, pp. 75-106 MR1990011 (in French) Ann. Math. Blaise Pascal, 11, 1, 2004, pp. 129-130 MR 2077241

[24] M. Manetti, Lie methods in deformation theory, work in progress.

[25] Manin, Y.I. Gauge Field Theory and Complex Geometry, Grundlehren Math. Wiss., Fundamental Principles of Mathematical Sciences, vol. 289, Springer-Verlag, Berlin, 1997 (Translated from the 1984 Russian original by N. Koblitz and J.R. King, with an appendix by Sergei Merkulov. MR 1632008)

[26] Mehta, R.A. Q-algebroids and their cohomology, J. Symplectic Geom., Volume 7 (2009) no. 3, pp. 263-293 (MR 2534186)

[27] Mehta, R.A. Supergroupoids, double structures, and equivariant cohomology, 2006 | arXiv

[28] Mehta, R.A.; Stiénon, M.; Xu, P. The Atiyah class of a dg-vector bundle, C. R. Math. Acad. Sci. Paris, Volume 353 (2015) no. 4, pp. 357-362 (MR 3319134)

[29] Pevzner, M.; Torossian, C. Isomorphisme de Duflo et la cohomologie tangentielle, J. Geom. Phys., Volume 51 (2004) no. 4, pp. 487-506 (MR 2085348)

[30] Shoikhet, B. On the Duflo formula for $L_{\infty}$ -algebras and Q-manifolds, 1998 | arXiv

[31] Tamarkin, D.E. Operadic Proof of M. Kontsevich's Formality Theorem, ProQuest LLC, Ann Arbor, MI, 1999 Thesis (Ph.D.)–The Pennsylvania State University MR 2699544

[32] Willwacher, T. The homotopy braces formality morphism, Duke Math. J., Volume 165 (2016) no. 10, pp. 1815-1964 (MR 3522653)

[33] Xu, P. Quantum groupoids, Commun. Math. Phys., Volume 216 (2001) no. 3, pp. 539-581 MR 1815717 (2002f:17033)

Cité par Sources :

^☆ Research partially supported by NSF grants DMS-1406668 and DMS-1707545.