Letters in Mathematical Physics

, Volume 89, Issue 1, pp 33–49 | Cite as

Abstract Hodge Decomposition and Minimal Models for Cyclic Algebras

  • Joseph Chuang
  • Andrey LazarevEmail author


We show that an algebra over a cyclic operad supplied with an additional linear algebra datum called Hodge decomposition admits a minimal model whose structure maps are given in terms of summation over trees. This minimal model is unique up to homotopy.

Mathematics Subject Classification (2000)

18D50 57T30 81T18 16E45 


cyclic operad cobar-construction Hodge decomposition minimal model a-infinity algebra 


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  1. 1.
    Ballard, M.: Sheaves on local Calabi–Yau varieties. arXiv:0801.3499Google Scholar
  2. 2.
    Barnes D., Lambe L.: A fixed point approach to homological perturbation theory. Proc. Am. Math. Soc. 112(3), 881–892 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bondal A., Van den Bergh M.: Generators and representability of functors in commutative and noncommutative geometry. Moscow Math. J. 3, 1–36 (2003)zbMATHADSMathSciNetGoogle Scholar
  4. 4.
    Chuang, J., Lazarev, A.: Dual Feynman transform for modular operads. Commun. Number Theory Phys. 1, 605–649 (2007). arXiv:0704.2561Google Scholar
  5. 5.
    Chuang, J., Lazarev, A.: Feynman diagrams and minimal models for operadic algebras. arXiv:0802.3507v1Google Scholar
  6. 6.
    Costello, K.: Topological conformal field theories and Calabi-Yau categories. Adv. Math. 210(1), 165–214 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Getzler, E., Kapranov, M.M.: Cyclic operads and cyclic homology. Geometry, topology, and physics. In: Conf. Proc. Lecture Notes Geom. Topology, vol. IV, pp. 167–201. Int. Press, Cambridge (1995)Google Scholar
  8. 8.
    Getzler E., Kapranov M.M.: Modular operads. Compos. Math. 110(1), 65–126 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Griffiths P., Harris J.: Principles of algebraic geometry. Pure and applied mathematics. Wiley, New York (1978)Google Scholar
  10. 10.
    Hamilton A., Lazarev A.: Characteristic classes of A-infinity algebras. J. Homotopy Relat. Struct. 3(1), 65–111 (2008)MathSciNetGoogle Scholar
  11. 11.
    Hinich V., Vaintrob A.: Cyclic operads and algebra of chord diagrams. Selecta Math. (N.S.) 8(2), 237–282 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Kadeishvilic, T.V.: The algebraic structure in the homology of an A(∞)-algebra. Soobshch. Akad. Nauk Gruzin. SSR 108 (1982)(2), 249–252 (1983) (Russian)Google Scholar
  13. 13.
    Kajiura, H.: Noncommutative homotopy algebras associated with open strings. Rev. Math. Phys. 19, 1–99 (2007) arXiv:math.QA/0306332Google Scholar
  14. 14.
    Kontsevich, M.: Feynman diagrams and low-dimensional topology. In: First European Congress of Mathematics (Paris, 1992), vol. 2, pp. 97–121. Progr. Math., 120. Birkhäuser, Basel (1994)Google Scholar
  15. 15.
    Kontsevich, M., Soibelman, Y.: Deformations of algebras over operads and the Deligne conjecture. In: Conférence Moshé Flato 1999, vol. I (Dijon), pp. 255–307. Math. Phys. Stud., 21. Kluwer, Dordrecht (2000)Google Scholar
  16. 16.
    Kontsevich, M., Soibelman, Y.: Notes on A -algebras, A -categories and non- commutative geometry. I (2006). arXiv:math/0606241Google Scholar
  17. 17.
    Lazarev A.: The Stasheff model of a simply-connected manifold and the string bracket. Proc. Am. Math. Soc. 136, 735–745 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Lazaroiu C. Generating the superpotential on a D-brane category: I. arXiv:hep-th/0610120Google Scholar
  19. 19.
    Markl, M.: Transferring A (strongly homotopy associative) structures. In: Proceedings of the 25th Winter School Geometry and Physics. Palermo: Universita degli Studi di Palermo, 2006. Supplemento ai Rendiconti del Circolo Matematico, pp. 139–151 (2006)Google Scholar
  20. 20.
    Markl, M., Shnider, S., Stasheff, J.: Operads in algebra, topology and physics. In: Mathematical Surveys and Monographs, vol. 96. American Mathematical Society, Providence (2002)Google Scholar
  21. 21.
    Merkulov, S.: Strongly homotopy algebras of a Kähler manifold. Int. Math. Res. Not. (3), 153–164 (1999)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Morita, S.: Geometry of differential forms. In: Translations of Mathematical Monographs, vol. 201. AMS (2001)Google Scholar
  23. 23.
    Polishchuk, A.: Homological mirror symmetry with higher products. In: Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds (Cambridge, MA, 1999). AMS/IP Stud. Adv. Math., vol. 23, pp. 247–259. American Mathematical Society, Providence (2001)Google Scholar

Copyright information

© Springer 2009

Authors and Affiliations

  1. 1.Centre for Mathematical ScienceCity UniversityLondonUK
  2. 2.Department of MathematicsUniversity of LeicesterLeicesterUK

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