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Letters in Mathematical Physics

, Volume 90, Issue 1–3, pp 103–136 | Cite as

Formality Theorems for Hochschild Complexes and Their Applications

  • Vasiliy Dolgushev
  • Dmitry Tamarkin
  • Boris Tsygan
Open Access
Article

Abstract

We give a popular introduction to formality theorems for Hochschild complexes and their applications. We review some of the recent results and prove that the truncated Hochschild cochain complex of a polynomial algebra is non-formal.

Mathematics Subject Classification (2000)

19D55 53D55 

Keywords

Hochschild complexes deformation quantization 

Notes

Acknowledgements

We would like to thank J. Stasheff for discussions and for his useful comments on the first version of our manuscript. D.T. and B.T. are supported by NSF grants. The work of V.D. is partially supported by the Grant for Support of Scientific Schools NSh-3036.2008.2.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. 1.
    Alekseev A., Meinrenken E.: On the Kashiwara–Vergne conjecture. Invent. Math. 164, 615–634 (2006)MATHCrossRefMathSciNetADSGoogle Scholar
  2. 2.
    Andler M., Dvorsky A., Sahi S.: Deformation quantization and invariant distributions. C. R. Acad. Sci. Paris Sér. I Math. 330(2), 115–120 (2000)MATHMathSciNetADSGoogle Scholar
  3. 3.
    Andler M., Dvorsky A., Sahi S.: Kontsevich quantization and invariant distributions on Lie groups. Ann. Sci. Ecole Norm. Sup. (4) 35(3), 371–390 (2002)MATHMathSciNetGoogle Scholar
  4. 4.
    Andler M., Sahi S., Torossian C.: Convolution of invariant distributions: proof of the Kashiwara–Vergne conjecture. Lett. Math. Phys. 69, 177–203 (2004)MATHCrossRefMathSciNetADSGoogle Scholar
  5. 5.
    Arnal D., Manchon D., Masmoudi M.: Choix des signes pour la formalité de M. Kontsevich. Pac. J. Math. 203(1), 23–66 (2002)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Babenko I.K., Taimanov I.A.: Massey products in symplectic manifolds. Sb. Math. 191(7–8), 1107–1146 (2000)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Bayen F., Flato M., Fronsdal C., Lichnerowicz A., Sternheimer D.: Deformation theory and quantization. I. Deformations of symplectic structures. Ann. Phys. (N.Y.) 111, 61–110 (1978)MATHCrossRefMathSciNetADSGoogle Scholar
  8. 8.
    Bayen F., Flato M., Fronsdal C., Lichnerowicz A., Sternheimer D.: Deformation theory and quantization, II. Physical applications. Ann. Phys. (N.Y.) 111, 111–151 (1978)MATHCrossRefMathSciNetADSGoogle Scholar
  9. 9.
    Belov-Kanel A., Kontsevich M.: Automorphisms of the Weyl algebra. Lett. Math. Phys. 74(2), 181–199 (2005) arXiv:math/0512169MATHCrossRefMathSciNetADSGoogle Scholar
  10. 10.
    Berezin F.A.: Quantization. Izv. Akad. Nauk. 38, 1116–1175 (1974)MathSciNetGoogle Scholar
  11. 11.
    Berezin F.A.: General concept of quantization. Commun. Math. Phys. 40, 153–174 (1975)CrossRefMathSciNetADSGoogle Scholar
  12. 12.
    Berger C., Fresse B.: Combinatorial operad actions on cochains. Math. Proc. Camb. Philos. Soc. 137(1), 135–174 (2004)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Berger C., Moerdijk I.: Axiomatic homotopy theory for operads. Comment. Math. Helv. 78(4), 805–831 (2003) arXiv:math/0206094MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Boardmann, J.M., Vogt, R.M.: Homotopy Invariant Algebraic Structures on Topological Spaces. Lecture Notes in Mathematics, vol. 347. Springer, Berlin (1973)Google Scholar
  15. 15.
    Borisov D.V.: G -structure on the deformation complex of a morphism. J. Pure Appl. Algebra 210(3), 751–770 (2007)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Brylinski J.-L.: A differential complex for Poisson manifolds. J. Differ. Geom. 28(1), 93–114 (1988)MATHMathSciNetGoogle Scholar
  17. 17.
    Brylinski J.-L., Zuckerman G.: The outer derivation of a complex Poisson manifold. J. Reine Angew. Math. 506, 181–189 (1999)MATHMathSciNetGoogle Scholar
  18. 18.
    Calaque, D., Halbout, G.: Weak quantization of Poisson structures. arXiv:0707.1978Google Scholar
  19. 19.
    Calaque, D., Van den Bergh, M.: Hochschild cohomology and Atiyah classes. arXiv:0708.2725Google Scholar
  20. 20.
    Calaque, D., Van den Bergh, M.: Global formality at the G -level. arXiv:0710.4510Google Scholar
  21. 21.
    Calaque D., Rossi C.A.: Shoikhet’s conjecture and Duflo isomorphism on (Co)invariants. SIGMA 4, 060 (2008) arXiv:0805.2409MathSciNetGoogle Scholar
  22. 22.
    Căldăraru A.: The Mukai pairing. II. The Hochschild–Kostant–Rosenberg isomorphism. Adv. Math. 194(1), 34–66 (2005)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Cattaneo, A.S.: Deformation quantization and reduction. In: Poisson Geometry in Mathematics and Physics. Contemporary Mathematics, vol. 450, pp. 79–101. American Mathematical Society, Providence (2008)Google Scholar
  24. 24.
    Cattaneo A.S., Felder G.: A path integral approach to the Kontsevich quantization formula. Commun. Math. Phys. 212(3), 591–611 (2000) arXiv:math/9902090MATHCrossRefMathSciNetADSGoogle Scholar
  25. 25.
    Cattaneo A.S., Felder G., Tomassini L.: From local to global deformation quantization of Poisson manifolds. Duke Math. J. 115(2), 329–352 (2002) math.QA/0012228MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Cattaneo A.S., Felder G.: Relative formality theorem and quantisation of coisotropic submanifolds. Adv. Math. 208(2), 521–548 (2007)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Cattaneo, A.S., Felder, G.: Effective Batalin–Vilkovisky theories, equivariant configuration spaces and cyclic chains. Progr. Math. (2009, in press). arXiv:0802.1706Google Scholar
  28. 28.
    Cattaneo A.S., Torossian C.: Quantification pour les paires symetriques et diagrammes de Kontsevich. Ann. Sci. Ec. Norm. Sup. 41, 789–854 (2008) arXiv:math/0609693MATHMathSciNetGoogle Scholar
  29. 29.
    Cuntz, J., Skandalis, G., Tsygan, B.: Cyclic homology in non-commutative geometry. In: Encyclopaedia of Mathematical Sciences, vol. 121. Operator Algebras and Non-commutative Geometry, II. Springer, Berlin (2004)Google Scholar
  30. 30.
    Daletski Yu., Gelfand I., Tsygan B.: On a variant of noncommutative geometry. Sov. Math. Dokl. 40(2), 422–426 (1990)Google Scholar
  31. 31.
    Deligne P., Griffiths P., Morgan J., Sullivan D.: Real homotopy theory of Kähler manifolds. Invent. Math. 29(3), 245–274 (1975)MATHCrossRefMathSciNetADSGoogle Scholar
  32. 32.
    Dolgushev V.A.: Covariant and Equivariant Formality Theorems. Adv. Math. 191(1), 147–177 (2005) arXiv:math/0307212MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Dolgushev V.A.: A Formality Theorem for Hochschild Chains. Adv. Math. 200(1), 51–101 (2006) math.QA/0402248MATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Dolgushev, V.A.: A Proof of Tsygan’s formality conjecture for an arbitrary smooth manifold. PhD thesis, MIT. math.QA/0504420Google Scholar
  35. 35.
    Dolgushev V.A.: The Van den Bergh duality and the modular symmetry of a Poisson variety. Selecta Math. (N.S.) 14(2), 199–228 (2009) arXiv:math/0612288MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Dolgushev, V.A.: Erratum to: “A Proof of Tsygan’s Formality Conjecture for an Arbitrary Smooth Manifold.” arXiv:math/0703113Google Scholar
  37. 37.
    Dolgushev V.A., Rubtsov V.N.: An algebraic index theorem for Poisson manifolds. J. Reine Angew. Math. (Crelles J.) 2009(633), 77–113 (2009)MathSciNetGoogle Scholar
  38. 38.
    Dolgushev V., Tamarkin D., Tsygan B.: The homotopy Gerstenhaber algebra of Hochschild cochains of a regular algebra is formal. J. Noncommut. Geom. 1(1), 1–25 (2007) arXiv:math/0605141MathSciNetCrossRefGoogle Scholar
  39. 39.
    Dolgushev, V.A., Tamarkin, D.E., Tsygan, B.L.: Formality of the homotopy calculus algebra of Hochschild (co)chains. arXiv:0807.5117Google Scholar
  40. 40.
    Drinfeld V.G.: Quasi-Hopf algebras. Leningr. Math. J. 1(6), 1419–1457 (1990)MathSciNetGoogle Scholar
  41. 41.
    Etingof, P., Ginzburg, V.: Noncommutative del Pezzo surfaces and Calabi–Yau algebras. arXiv:0709.3593Google Scholar
  42. 42.
    Gelfand I.M., Fuchs D.V.: Cohomology of the algebra of formal vector fields. Izv. Akad. Nauk., Math. Ser. (In Russian) 34, 322–337 (1970)Google Scholar
  43. 43.
    Gelfand I.M., Kazhdan D.A.: Some problems of differential geometry and the calculation of cohomologies of Lie algebras of vector fields. Sov. Math. Dokl. 12(5), 1367–1370 (1971)Google Scholar
  44. 44.
    Gerstenhaber M.: The cohomology structure of an associative ring. Ann. Math. 78, 267–288 (1963)CrossRefMathSciNetGoogle Scholar
  45. 45.
    Getzler E.: A Darboux theorem for Hamiltonian operators in the formal calculus of variations. Duke Math. J. 111(3), 535–560 (2002)MATHCrossRefMathSciNetGoogle Scholar
  46. 46.
    Getzler E.: Lie theory for nilpotent L-infinity algebras. Ann. Math. 170(1), 271–301 (2009) arXiv:math/0404003MATHCrossRefMathSciNetGoogle Scholar
  47. 47.
    Getzler, E., Jones, J.D.S.: Operads, homotopy algebra and iterated integrals for double loop spaces. hep-th/9403055Google Scholar
  48. 48.
    Ginzburg, V.: Calabi–Yau algebras. math.AG/0612139Google Scholar
  49. 49.
    Ginzburg V., Kapranov M.: Koszul duality for operads. Duke Math. J. 76(1), 203–272 (1944)CrossRefMathSciNetGoogle Scholar
  50. 50.
    Goldman W., Millson J.: The deformation theory of representation of fundamental groups in compact Kähler manifolds. Publ. Math. I.H.E.S. 67, 43–96 (1988)MATHMathSciNetGoogle Scholar
  51. 51.
    Halperin S., Stasheff J.: Obstructions to homotopy equivalences. Adv. Math. 32(3), 233–279 (1979)MATHCrossRefMathSciNetGoogle Scholar
  52. 52.
    Henriques A.: Integrating L -algebras. Compos. Math. 144(4), 1017–1045 (2008)MATHCrossRefMathSciNetGoogle Scholar
  53. 53.
    Hinich V.: Homological algebra of homotopy algebras. Comm. Algebra 25(10), 3291–3323 (1997)MATHCrossRefMathSciNetGoogle Scholar
  54. 54.
    Hinich V.: Tamarkin’s proof of Kontsevich formality theorem. Forum Math. 15(4), 591–614 (2003) math.QA/0003052MATHCrossRefMathSciNetGoogle Scholar
  55. 55.
    Hinich V., Schechtman V.: Homotopy Lie algebras. I.M. Gelfand Seminar. Adv. Sov. Math. 16(2), 1–28 (1993)MathSciNetGoogle Scholar
  56. 56.
    Ikeda N.: Two-dimensional gravity and nonlinear gauge theory. Ann. Phys. 235, 435–464 (1994)MATHCrossRefADSGoogle Scholar
  57. 57.
    Kashiwara M., Vergne M.: The Campbell-Hausdorff formula and invariant hyperfunctions. Invent. Math. 47, 249–272 (1978)MATHCrossRefMathSciNetADSGoogle Scholar
  58. 58.
    Kontsevich M.: Deformation quantization of Poisson manifolds. Lett. Math. Phys. 66, 157–216 (2003) q-alg/9709040MATHCrossRefMathSciNetADSGoogle Scholar
  59. 59.
    Kontsevich M. et al.: Formality conjecture. In: Sternheimer, D. (eds) Deformation Theory and Symplectic Geometry., pp. 139–156. Kluwer, Dordrecht (1997)Google Scholar
  60. 60.
    Kontsevich M.: Deformation quantization of algebraic varieties. Moshé Flato memorial conference 2000, Part III (Dijon). Lett. Math. Phys. 56(3), 271–294 (2001)MATHCrossRefMathSciNetGoogle Scholar
  61. 61.
    Kontsevich M.: Operads and motives in deformation quantization. Lett. Math. Phys. 48, 35–72 (1999)MATHCrossRefMathSciNetGoogle Scholar
  62. 62.
    Kontsevich, M., Soibelman, Y.: Deformations of algebras over operads and the Deligne conjecture. In: Proceedings of the Moshé Flato Conference. Mathematical Physics Studies, vol. 21, pp. 255–307, Kluwer, Dordrecht (2000)Google Scholar
  63. 63.
    Kontsevich, M., Soibelman, Y.: Notes on A-infinity algebras, A-infinity categories and non-commutative geometry. I. math.RA/0606241Google Scholar
  64. 64.
    Koszul, J.L.: Crochet de Schouten–Nijenhuis et cohomologie, Astérisque (1985). Numéro Hors Série, pp. 257–271Google Scholar
  65. 65.
    Lada T., Stasheff J.: Introduction to SH Lie algebras for physicists. Int. J. Theor. Phys. 32(7), 1087–1103 (1993)MATHCrossRefMathSciNetGoogle Scholar
  66. 66.
    Lambrechts, P., Volic, I.: Formality of the little N-disks operad. arXiv:0808.0457Google Scholar
  67. 67.
    Lichnerowicz A.: Les variétés de Poisson et leurs algèbres de Lie associées. J. Differ. Geom. 12(2), 253–300 (1977)MATHMathSciNetGoogle Scholar
  68. 68.
    Loday J.-L.: Cyclic Homology, Grundlehren der mathematischen Wissenschaften, vol. 301. Springer, Berlin (1992)Google Scholar
  69. 69.
    Lyakhovich S.L., Sharapov A.A.: BRST theory without Hamiltonian and Lagrangian. J. High Energy Phys. 3(011), 22 (2005)Google Scholar
  70. 70.
    Manchon D., Torossian C.: Cohomologie tangente et cup-produit pour la quantification de Kontsevich. Ann. Math. Blaise Pascal 10(1), 75–106 (2003)MATHMathSciNetGoogle Scholar
  71. 71.
    Markl M.: Models for operads. Comm. Algebra 24, 1471–1500 (1996)MATHCrossRefMathSciNetGoogle Scholar
  72. 72.
    May J.P.: Infinite loop space theory. Bull. Am. Math. Soc. 83(4), 456–494 (1977)MATHCrossRefMathSciNetGoogle Scholar
  73. 73.
    May J.P.: Matrix Massey products. J. Algebra 12, 533–568 (1969)MATHCrossRefMathSciNetGoogle Scholar
  74. 74.
    McClure, J.E., Smith, J.H.: A solution of Deligne’s Hochschild cohomology conjecture. In: Recent Progress in Homotopy Theory (Baltimore, MD, 2000). Contemporary Mathematics, vol. 293, pp. 153–193. American Mathematical Society, Providence (2002). math.QA/9910126Google Scholar
  75. 75.
    Quillen D.: Rational homotopy theory. Ann. Math. 90(2), 205–295 (1969)CrossRefMathSciNetGoogle Scholar
  76. 76.
    Schaller P., Strobl T.: Poisson structure induced (topological) field theories. Modern Phys. Lett. A 9(33), 3129–3136 (1994)MATHCrossRefMathSciNetADSGoogle Scholar
  77. 77.
    Schlessinger, M., Stasheff, J.: Deformation theory and rational homotopy type, University of North Carolina preprint (1979)Google Scholar
  78. 78.
    Schlessinger M., Stasheff J.: The Lie algebra structure of tangent cohomology and deformation theory. J. Pure Appl. Algebra 38(2–3), 313–322 (1985)MATHCrossRefMathSciNetGoogle Scholar
  79. 79.
    Shoikhet B.: A proof of the Tsygan formality conjecture for chains. Adv. Math. 179(1), 7–37 (2003) math.QA/0010321MATHCrossRefMathSciNetGoogle Scholar
  80. 80.
    Stasheff J.: Homological reduction of constrained Poisson algebras. J. Differ. Geom. 45, 221–240 (1997)MATHMathSciNetGoogle Scholar
  81. 81.
    Swan R.G.: Hochschild cohomology of quasiprojective schemes. J. Pure Appl. Algebra 110(1), 57–80 (1996)MATHCrossRefMathSciNetGoogle Scholar
  82. 82.
    Tamarkin, D.: Another proof of M. Kontsevich formality theorem. math.QA/9803025Google Scholar
  83. 83.
    Tamarkin D.: Formality of chain operad of little discs. Lett. Math. Phys. 66(1–2), 65–72 (2003) math.QA/9809164MATHCrossRefMathSciNetADSGoogle Scholar
  84. 84.
    Tamarkin D.: What do DG categories form?. Compos. Math. 143(5), 1335–1358 (2007) math.CT/0606553MATHCrossRefMathSciNetGoogle Scholar
  85. 85.
    Tamarkin, D., Tsygan, B.: Cyclic formality and index theorems, Talk given at the Moshé Flato Conference (2000). Lett. Math. Phys. 56(2), 85–97 (2001)Google Scholar
  86. 86.
    Torossian C.: Sur la conjecture combinatoire de Kashiwara–Vergne. J. Lie Theory 12(2), 597–616 (2002)MATHMathSciNetGoogle Scholar
  87. 87.
    Tsygan, B.: Formality conjectures for chains. In: Differential Topology, Infinite-dimensional Lie Algebras, and Applications. American Mathematical Society Translations, Series 2, vol. 194, pp. 261–274. American Mathematical Society, Providence (1999)Google Scholar
  88. 88.
    Van den Bergh, M.: A relation between Hochschild homology and cohomology for Gorenstein rings. Proc. Am. Math. Soc. 126(5), 1345–1348 (1998); Erratum to “A Relation between Hochschild Homology and Cohomology for Gorenstein Rings.” Proc. Am. Math. Soc. 130(9), 2809–2810 (2002)Google Scholar
  89. 89.
    Van den Bergh M.: On global deformation quantization in the algebraic case. J. Algebra 315(1), 326–395 (2007)MATHCrossRefMathSciNetGoogle Scholar
  90. 90.
    Vey J.: Déformation du crochet de Poisson sur une variété symplectique. Comment. Math. Helv. 50, 421–454 (1975)MATHCrossRefMathSciNetGoogle Scholar
  91. 91.
    Voronov, A.: Quantizing Poisson manifolds. In: Perspectives on Quantization (South Hadley, MA, 1996). Contemporary Mathematics, vol. 214, pp. 189–195. American Mathematical Society, Providence (1998)Google Scholar
  92. 92.
    Voronov, A.A.: Homotopy Gerstenhaber algebras. In: Proceedings of the Moshé Flato Conference. Mathematical Physics Studies, vol. 22, pp. 307–331. Kluwer, Dordrecht (2000)Google Scholar
  93. 93.
    Weinstein A.: The modular automorphism group of a Poisson manifold. J. Geom. Phys. 23(3–4), 379–394 (1997)MATHCrossRefMathSciNetADSGoogle Scholar
  94. 94.
    Willwacher, T.: Formality of cyclic chains. arXiv:0804.3887Google Scholar
  95. 95.
    Yekutieli A.: The continuous Hochschild cochain complex of a scheme. Can. J. Math. 54(6), 1319–1337 (2002)MATHMathSciNetGoogle Scholar
  96. 96.
    Yekutieli A.: Deformation quantization in algebraic geometry. Adv. Math. 198(1), 383–432 (2005) math.AG/0310399MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2009

Authors and Affiliations

  • Vasiliy Dolgushev
    • 1
  • Dmitry Tamarkin
    • 2
  • Boris Tsygan
    • 2
  1. 1.Department of MathematicsUniversity of California at RiversideRiversideUSA
  2. 2.Mathematics DepartmentNorthwestern UniversityEvanstonUSA

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