Selecta Mathematica

, Volume 19, Issue 1, pp 1–47 | Cite as

The minimal model for the Batalin–Vilkovisky operad

  • Gabriel C. Drummond-Cole
  • Bruno ValletteEmail author


The purpose of this paper is to explain and to generalize, in a homotopical way, the result of Barannikov–Kontsevich and Manin, which states that the underlying homology groups of some Batalin–Vilkovisky algebras carry a Frobenius manifold structure. To this extent, we first make the minimal model for the operad encoding BV-algebras explicit. Then, we prove a homotopy transfer theorem for the associated notion of homotopy BV-algebra. The final result provides an extension of the action of the homology of the Deligne–Mumford–Knudsen moduli space of genus 0 curves on the homology of some BV-algebras to an action via higher homotopical operations organized by the cohomology of the open moduli space of genus zero curves. Applications in Poisson geometry and Lie algebra cohomology and to the Mirror Symmetry conjecture are given.


Operad Batalin–Vilkovisky algebra Moduli spaces of curves Homotopy algebra Frobenius manifold 

Mathematics Subject Classification (1991)

Primary 18D50 Secondary 18G55 53D45 55P48 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Apostolov V., Gualtieri M.: Generalized Kähler manifolds, commuting complex structures, and split tangent bundles. Commun. Math. Phys. 271(2), 561–575 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Barannikov S.: Non-commutative periods and mirror symmetry in higher dimensions. Commun. Math. Phys 228(2), 281–325 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bousfield A.K., Curtis E.B., Kan D.M., Quillen D.G., Rector D.L., Schlesinger J.W.: The mod-p lower central series and the Adams spectral sequence. Topology 5, 331–342 (1966)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Beilinson, A., Drinfeld, V.: Chiral Algebras, American Mathematical Society Colloquium Publications, vol. 51. American Mathematical Society, Providence, RI (2004)Google Scholar
  5. 5.
    Bellier, O.: Koszul duality of operads over a Hopf algebra (work in progress) (2011)Google Scholar
  6. 6.
    Berglund, A.: Homological perturbation theory for algebras over operads. arXiv:0909.3485 (2009)Google Scholar
  7. 7.
    Barannikov S., Kontsevich M.: Frobenius manifolds and formality of Lie algebras of polyvector fields. Int. Math. Res. Not. 4, 201–215 (1998)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Berger C., Moerdijk I.: Axiomatic homotopy theory for operads. Comment. Math. Helv. 78(4), 805–831 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Brylinski J.-L.: A differential complex for Poisson manifolds. J. Differ. Geom. 28(1), 93–114 (1988)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Batalin I.A., Vilkovisky G.A.: Gauge algebra and quantization. Phys. Lett. B 102(1), 27–31 (1981)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Cavalcanti, G.R.: New aspects of the ddc-lemma. arXiv:math/0501406 (2005)Google Scholar
  12. 12.
    Cavalcanti G.R.: Formality in generalized Kähler geometry. Topol. Appl. 154(6), 1119–1125 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Carr M.P., Devadoss S.L.: Coxeter complexes and graph-associahedra. Topol. Appl. 153(12), 2155–2168 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Costello, K., Gwilliam, O.: Factorization algebras in perturbative quantum field theory. Available on the home pages of the authors (2011)Google Scholar
  15. 15.
    Cao, H.-D., Zhou, J.: DGBV algebras and mirror symmetry. In: First International Congress of Chinese Mathematicians (Beijing, 1998), AMS/IP Studies in Advances Mathematics, vol. 20. American Mathematical Soceity, Providence, RI, pp. 279–289 (2001)Google Scholar
  16. 16.
    Drummond-Cole, G.: Homotopically trivializing the circle in the framed little disks. arXiv:1112.1129 (2011)Google Scholar
  17. 17.
    Drummond-Cole, G.: Formal formality of the hypercommutative algebras of low dimensional Calabi–Yau varieties. arXiv:1201.6111 (2012)Google Scholar
  18. 18.
    De Concini C., Procesi C.: Wonderful models of subspace arrangements. Selecta Math. N.S. 1(3), 459–494 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Drummond-Cole G., Vallette B.: ∞-operads, BV , and hypercommutative. Oberwolfach Rep 28, 1566–1569 (2009)Google Scholar
  20. 20.
    Devadoss S., Forcey S.: Marked tubes and the graph multiplihedron. Algebr. Geom. Topol. 8(4), 2081–2108 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Deligne P., Griffiths P., Morgan J., Sullivan D.: Real homotopy theory of Kähler manifolds. Invent. Math. 29(3), 245–274 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Dotsenko, V., Khoroshkin, A.: Free resolutions via Gröbner bases. ArXiv e-prints (2009)Google Scholar
  23. 23.
    Dotsenko V., Khoroshkin A.: Gröbner bases for operads. Duke Math. J. 153(2), 363–396 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Galvez-Carrillo, I., Tonks, A., Vallette, B.: Homotopy Batalin–Vilkovisky algebras, J. Noncommut. Geom. arXiv:0907.2246 (2011, to appear)Google Scholar
  25. 25.
    Getzler E.: Batalin–Vilkovisky algebras and two-dimensional topological field theories. Commun. Math. Phys. 159(2), 265–285 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Getzler E.: Two-dimensional topological gravity and equivariant cohomology. Commun. Math. Phys. 163(3), 473–489 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Getzler, E.: Operads and Moduli Spaces of Genus 0 Riemann Surfaces. The Moduli Space of Curves (Texel Island, 1994). Progress in Mathematics, vol. 129, pp. 199–230. Birkhäuser Boston, Boston, MA (1995)Google Scholar
  28. 28.
    Givental, A.B.: Gromov–Witten invariants and quantization of quadratic Hamiltonians, Mosc. Math. J. 1(4), 551–568, 645. Dedicated to the memory of I.G. Petrovskii on the occasion of his 100th anniversary (2001)Google Scholar
  29. 29.
    Givental A.B.: Semisimple Frobenius structures at higher genus. Internat. Math. Res. Not. 23, 1265–1286 (2001)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Getzler, E., Jones, J.D.S.: Operads, homotopy algebra and iterated integrals for double loop spaces. hep-th/9403055 (1994)Google Scholar
  31. 31.
    Granåker, J.: Strong homotopy properads. Int. Math. Res. Not. IMRN 14 (2007)Google Scholar
  32. 32.
    Ginzburg V., Schedler T.: Differential operators and BV structures in noncommutative geometry. Selecta Math. N.S. 16(4), 673–730 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Gualtieri, M.: Generalized complex geometry. arXiv:math/0401221 (2004)Google Scholar
  34. 34.
    Hinich V.: Homological algebra of homotopy algebras. Commun. Algebra 25(10), 3291–3323 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Hirschhorn, P.S.: Model Categories and Their Localizations. Mathematical Surveys and Monographs, vol. 99. American Mathematical Society, Providence, RI (2003)Google Scholar
  36. 36.
    Hitchin N.: Generalized Calabi–Yau manifolds. Q. J. Math. 54(3), 281–308 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Hoefel, E., Livernet, M.: OCHA and Leibniz pairs, towards a Koszul duality. arXiv:math/1104.3607 (2011)Google Scholar
  38. 38.
    Hirsh, J., Millès, J.: Curved Koszul duality theory, (2010, to appear Math. Ann.)Google Scholar
  39. 39.
    Hoffbeck E.: A Poincaré–Birkhoff–Witt criterion for Koszul operads. Manuscripta Math. 131(1–2), 87–110 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Kontsevich M., Manin Yu.: Gromov–Witten classes, quantum cohomology, and enumerative geometry. Comm. Math. Phys. 164(3), 525–562 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Khoroshkin, A., Markarian, N., Shadrin, S.: On quasi-isomorphism of hycom and bv/δ (2011, in preparation)Google Scholar
  42. 42.
    Kontsevich, M.: Homological algebra of mirror symmetry. In: Proceedings of the International Congress of Mathematicians, vols. 1, 2, pp. 120–139 (Zürich, 1994) Birkhäuser, Basel (1995)Google Scholar
  43. 43.
    Kontsevich M.: Deformation quantization of Poisson manifolds. Lett. Math. Phys. 66(3), 157–216 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Koszul, J.-L.: Crochet de Schouten–Nijenhuis et cohomologie, Astérisque (1985) Numero Hors Serie, The Mathematical Heritage of Élie Cartan, Lyon, pp. 257–271 (1984)Google Scholar
  45. 45.
    Kodaira K., Spencer D.C.: On deformations of complex analytic structures. I, II. Ann. Math. 2(67), 328–466 (1958)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Kodaira K., Spencer D.C.: On deformations of complex analytic structures. III. Stability theorems for complex structures. Ann. Math. 71, 43–76 (1960)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Li, Y.: On deformations of generalized complex structures: the generalized Calabi–Yau case. arXiv:hep-th/0508030 (2005)Google Scholar
  48. 48.
    Lichnerowicz A.: Les variétés de Poisson et leurs algèbres de Lie associées. J. Differ. Geom. 12(2), 253–300 (1977)MathSciNetzbMATHGoogle Scholar
  49. 49.
    Loday J.-L.: Realization of the Stasheff polytope. Arch. Math. 83(3), 267–278 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Losev A., Shadrin S.: From Zwiebach invariants to Getzler relation. Commun. Math. Phys. 271(3), 649–679 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Loday, J.-L., Vallette, B.: Algebraic operads, Grundlehren Math. Wiss. 346, Springer, Heidelberg, (2012)Google Scholar
  52. 52.
    Manin, Y.I.: Frobenius Manifolds, Quantum Cohomology, and Moduli Spaces, American Mathematical Society Colloquium Publications, vol. 47. American Mathematical Society, Providence, RI (1999)Google Scholar
  53. 53.
    Markl M.: Models for operads. Commun. Algebra 24(4), 1471–1500 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    Markarian, N.: hycom = bv/δ A blog post available through the URL (2009)
  55. 55.
    Massey, W.S.: Some higher order cohomology operations. In: Symposium internacional de topologí a algebraica. International Symposium on Algebraic Topology. Universidad Nacional Autónoma de México and UNESCO, Mexico City, pp. 145–154 (1958)Google Scholar
  56. 56.
    Mathieu O.: Harmonic cohomology classes of symplectic manifolds. Comment. Math. Helv. 70(1), 1–9 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    Merkulov S.A.: Formality of canonical symplectic complexes and Frobenius manifolds. Int. Math. Res. Not. 14, 727–733 (1998)MathSciNetCrossRefGoogle Scholar
  58. 58.
    Mac Lane, S.: Homology, Classics in Mathematics. Reprint of the 1975 edition. Springer, Berlin (1995)Google Scholar
  59. 59.
    Markl, M., Shnider, S., Stasheff, J.: Operads in Algebra, Topology and Physics. Mathematical Surveys and Monographs, vol. 96. American Mathematical Society, Providence, RI (2002)Google Scholar
  60. 60.
    Merkulov S., Vallette B.: Deformation theory of representations of prop(erad)s. I. J. Reine Angew. Math. 634, 51–106 (2009)MathSciNetzbMATHGoogle Scholar
  61. 61.
    Merkulov S., Vallette B.: Deformation theory of representations of prop(erad)s. II. J. Reine Angew. Math. 636, 123–174 (2009)MathSciNetzbMATHGoogle Scholar
  62. 62.
    Park J.-S.: Semi-classical quantum field theories and Frobenius manifolds. Lett. Math. Phys. 81(1), 41–59 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  63. 63.
    Priddy S.B.: Koszul resolutions. Trans. Am. Math. Soc. 152, 39–60 (1970)MathSciNetzbMATHCrossRefGoogle Scholar
  64. 64.
    Peters, C.A.M., Steenbrink, J.H.M.: Mixed Hodge structures. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 52, Springer, Berlin (2008)Google Scholar
  65. 65.
    Quillen D.: Rational homotopy theory. Ann. Math. 90, 205–295 (1969)MathSciNetzbMATHCrossRefGoogle Scholar
  66. 66.
    Retakh V.S.: Lie–Massey brackets and n-homotopically multiplicative maps of differential graded Lie algebras. J. Pure Appl. Algebra 89(1–2), 217–229 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  67. 67.
    Rezk, C.W.: Spaces of algebra structures and cohomology of operads. Ph.D. thesis, MIT (1996)Google Scholar
  68. 68.
    Sullivan D.: Infinitesimal computations in topology. Inst. Hautes Études Sci. Publ. Math. 47, 269–331 (1978)Google Scholar
  69. 69.
    Terilla J.: Smoothness theorem for differential BV algebras. J. Topol. 1(3), 693–702 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  70. 70.
    Van der Laan, P.: Operads up to homotopy and deformations of operad maps. arXiv:math.QA/0208041 (2002)Google Scholar
  71. 71.
    Van der Laan, P.: Coloured Koszul duality and strongly homotopy operads. arXiv:math.QA/0312147 (2003)Google Scholar
  72. 72.
    Wang J.S.P.: On the cohomology of the mod-2 Steenrod algebra and the non-existence of elements of Hopf invariant one. Ill. J. Math 11, 480–490 (1967)zbMATHGoogle Scholar
  73. 73.
    Xu P.: Gerstenhaber algebras and BV-algebras in Poisson geometry. Commun. Math. Phys. 200(3), 545–560 (1999)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Mathematics DepartmentNorthwestern UniversityEvanstonUSA
  2. 2.Laboratoire J.A. DieudonnéUniversité de Nice Sophia-AntipolisNice Cedex 02France

Personalised recommendations