Noncommutative Calculus and the Gauss–Manin Connection

  • V. A. Dolgushev
  • D. E. Tamarkin
  • B. L. TsyganEmail author
Part of the Progress in Mathematics book series (PM, volume 287)


To Murray Gerstenhaber on his 80th and to Jim Stasheff on his 70th birthday

After an overview of noncommutative differential calculus, we construct parts of it explicitly and explain why this construction agrees with a fuller version obtained from the theory of operads.

AMS 2010 Subject Codes: 19D55, 18G55

Key words

Hochschild homology Cyclic homology Homotopy algebras Connections 


  1. 1.
    Barannikov, S.: Quantum periods, I. Semi-infinite variations of Hodge structures, Int. Math. Res. Not. 23, 1243–1264 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Baranovsky, V.: A universal enveloping of an L algebra (2007), arxiv:0706.1396Google Scholar
  3. 3.
    Bressler, P., Nest, R., Tsygan, B.: Riemann-Roch theorems via deformation quantization, I, II. Adv. Math. 167(1), 1–25, 26–73 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bressler, P., Gorokhovsky, A., Nest, R., Tsygan, B.: Algebraic index theorem for symplectic deformations of gerbes (to appear)Google Scholar
  5. 5.
    Căldăraru, A.: The Mukai pairing, I: the Hochschild structure, Preprint, arXiv:math/0308079Google Scholar
  6. 6.
    Căldăraru, A.: The Mukai pairing, II: the Hochschild-Kostant-Rosenberg isomorphism. Adv. Math. 194(1), 34–66 (2005), arXiv:math/0308080MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Căldăraru, A., Willerton, S.: The Mukai pairing, I: a categorical approach. N. Y. J. Math. 16, 61–98 (2010), arXiv:0707.2052MathSciNetzbMATHGoogle Scholar
  8. 8.
    Cartan, H., Eilenberg, S.: Homological Algebra. Princeton University Press, NJ (1956)zbMATHGoogle Scholar
  9. 9.
    Costello, K.: Topological conformal field theories and Calabi-Yau categories, Adv. Math. 210(1), 165–214 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Daletski, Yu., Gelfand, I., Tsygan, B.: On a variant of noncommutative geometry. Soviet Math. Dokl. 40(2), 422–426 (1990)MathSciNetGoogle Scholar
  11. 11.
    Dolgushev, V.A.: A Proof of Tsygan’s Formality Conjecture for an Arbitrary Smooth Manifold, PhD thesis. MIT, Cambridge, MA, math.QA/0504420Google Scholar
  12. 12.
    Dolgushev, V.A., 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/0605141MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Dolgushev, V.A., Tamarkin, D., Tsygan, B.: Formality of the homotopy calculus algebra of Hochschild (co)chains (2008), arXiv:0807.5117Google Scholar
  14. 14.
    Dolgushev, V.A., Tamarkin, D., Tsygan, B.: Formality theorems for Hochschild complexes and their applications. Lett. Math. Phys. 90(1–3), 103–136 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Dorfman, I.Ya., Gelfand, I.M.: Hamiltonian operators and algebraic structures related to them. Funct. Anal. Appl. 13, 13–30 (1979)zbMATHGoogle Scholar
  16. 16.
    Gerstenhaber, M.: The cohomology structure of an associative ring. Ann. Math. (2) 78, 267–288 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Getzler, E.: Cartan homotopy formulas and the Gauss-Manin connection in cyclic homology. Quantum deformations of algebras and their representations (Ramat-Gan, 1991/1992; Rehovot, 1991/1992), 65–78, Israel Math. Conf. Proc., 7, Bar-Ilan Univ., Ramat-Gan (1993)Google Scholar
  18. 18.
    Getzler, E., Jones, J.D.S.: A algebras and the cyclic bar complex. Illinois J. Math. 34, 256–283 (1990)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Getzler, E., Jones, J.D.S.: Operads, homotopy algebra and iterated integrals for double loop spaces. Preprint hep-th9403055Google Scholar
  20. 20.
    Ginzburg, V., Schedler, T.: Free products, cyclic homology, and the Gauss-Manin connection, arXiv:0803.3655Google Scholar
  21. 21.
    Hinich, V.: Tamarkin’s proof of Kontsevich formality theorem. Forum Math. 15(4), 591–614 (2003), math.QA/0003052MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Hood, C.E., Jones, J.D.S.: Some algebraic properties of cyclic homology groups. K-theory 1, 361–384 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kadeishvili, T.V.: On the homology theory of fibered spaces. Uspekhi Mat. Nauk 35(3), 183–188 (1980)MathSciNetGoogle Scholar
  24. 24.
    Katzarkov, L., Kontsevich, M., Pantev, T.: Hodge theoretic aspects of mirror symmetry, From Hodge theory to integrability and TQFT tt* geometry. Proc. Sympos. Pure Math. AMS 78, Providence, RI, (2008), arXiv:0806.0107Google Scholar
  25. 25.
    Keller, B.: A algebras, modules and functor categories. Trends in representation theory of algebras and related topics. Contemp. Math. AMS 406, 67–93 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Kontsevich, M.: Operads and motives in deformation quantization. Lett. Math. Phys. 48, 35–72 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Kontsevich, M., Soibelman, Y.: Deformations of algebras over operads and the Deligne conjecture, Conférence Moshe Flato 1999, vol. 1 (Dijon), pp. 255–307. Math. Phys. Stud., 21. Kluwer, Dordrecht (2000)Google Scholar
  28. 28.
    Kontsevich, M., Soibelman, Y.: Notes on A-infinity algebras, A-infinity categories and non-commutative geometry. I, math.RA/0606241Google Scholar
  29. 29.
    Lada, T., Stasheff, J.: Introduction to SH Lie algebras for physicists. Int. J. Theor. Phys. 32(7), 1087–1103 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Loday, J.-L.: Cyclic homology. Gründlehren der mathematischen Wissenschaften, 301. Springer, Berlin (1992)Google Scholar
  31. 31.
    Markarian, N.: The Atiyah class, Hochschild cohomology and the Riemann-Roch theorem, J. Lond. Math. Soc. (2) 79(1), 129–143 (2009), arXiv:math/0610553MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Merkulov, S.: An L algebra of an unobstructed deformation functor. Int. Math. Res. Not. 3, 147–164 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Merkulov, S.: Exotic automorphisms of the Schouten algebra of polyvector fields, arXiv:0809.2385Google Scholar
  34. 34.
    Merkulov, S., Vallette, B.: Deformation theory of representations of prop(erad)s (2007), arXiv:0707.0889Google Scholar
  35. 35.
    Nest, R., Tsygan, B.: On the cohomology ring of an algebra. In: Advances in geometry. Progress in Mathematics, vol. 172, pp. 337–370. Birkhäuser, Boston, MA, (1999)Google Scholar
  36. 36.
    Ramadoss, A.: The Mukai pairing and integral transforms in Hochschild homology, arXiv:0805.1760Google Scholar
  37. 37.
    Shoikhet, B.: A proof of the Tsygan formality conjecture for chains. Adv. Math. 179(1), 7–37 (2003), math.QA/0010321MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Stasheff, J.: Homotopy associativity of H-spaces, I and II, Trans. AMS 108, 275–312 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Tamarkin, D.: Another proof of M. Kontsevich formality theorem, math.QA/ 9803025Google Scholar
  40. 40.
    Tamarkin, D.: Formality of chain operad of little discs. Lett. Math. Phys. 66(1–2), 65–72 (2003), math.QA/9809164MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Tamarkin, D.: What do DG categories form? Compos. Math. 143(5), 1335–1358 (2007); math.CT/0606553MathSciNetzbMATHGoogle Scholar
  42. 42.
    Tamarkin, D., Tsygan, B.: Cyclic formality and index theorems. Talk given at the Moshé Flato Conference. Lett. Math. Phys. 56(2), 85–97 (2001)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Tamarkin, D., Tsygan, B.: The ring of differential operators on forms in noncommutative calculus. Graph patterns in mathematics and theoretical physics. Proc. Symp. Pure Math., vol. 73, pp. 105–131. AMS, RI (2005)Google Scholar
  44. 44.
    Toën, B., Vezzosi, G.: A note on Chern character, loop spaces and derived algebraic geometry. Proc. of the 2007 Abel Symposium, arXiv:0804.1274Google Scholar
  45. 45.
    Tsygan, B.: On the Gauss-Manin connection in cyclic homology. Methods Funkt. Anal. Topol. 13(1), 83–94 (2007)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • V. A. Dolgushev
    • 1
    • 2
  • D. E. Tamarkin
    • 1
    • 2
  • B. L. Tsygan
    • 1
    • 2
    Email author
  1. 1.Department of MathematicsUniversity of California at RiversideRiversideUSA
  2. 2.Mathematics DepartmentNorthwestern UniversityEvanstonUSA

Personalised recommendations