Communications in Mathematical Physics

, Volume 159, Issue 2, pp 265–285 | Cite as

Batalin-Vilkovisky algebras and two-dimensional topological field theories

  • E. Getzler


By a Batalin-Vilkovisky algebra, we mean a graded commutative algebraA, together with an operator Δ:AA⊙+1 such that Δ2 = 0, and [Δ,a]−Δa is a graded derivation ofA for allaA. In this article, we show that there is a natural structure of a Batalin-Vilkovisky algebra on the cohomology of a topological conformal field theory in two dimensions. We make use of a technique from algebraic topology: the theory of operads.


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  1. 1.
    Arnold, V.I.: The cohomology ring of the colored braid group. Mat. Zametki5, 227–231 (1969)Google Scholar
  2. 2.
    Beilinson, A., Ginzburg, V.: Infinitesimal structures of moduli space ofG-bundles. Duke Math. J.66, 63–74 (1992)Google Scholar
  3. 3.
    Birman, J.S.: Braids, links and mapping class groups. Ann. Math. Studies, no. 82, Princeton, NJ: Princeton U. Press 1974Google Scholar
  4. 4.
    Boardman, J.M., Vogt, R.M.: Homotopy invariant algebraic structures on topological spaces. Lecture Notes in Mathematics, no.347, Berlin, Heidelberg, New York: Springer 1973Google Scholar
  5. 5.
    Cohen, F.R.: The homology ofC n+1-spaces,n≧0. In: The homology of iterated loop spaces, Lecture Notes in Mathematics, no.533, Berlin, Heidelberg, New York: Springer 1976, pp. 207–351Google Scholar
  6. 6.
    Dijkgraaf, R., Verlinde, H., Verlinde, E.: Topological strings ind<1. Nucl. Phys.B352, 59–86 (1991)CrossRefGoogle Scholar
  7. 7.
    Fadell, E., Neuwirth, L.: Configuration spaces. Math. Scand.10, 111–118 (1962)Google Scholar
  8. 8.
    Fulton, W., MacPherson, R.: A compactification of configuration spaces. To appear, Ann. Math.Google Scholar
  9. 9.
    Gerstenhaber, M.: The cohomology structure of an associative ring. Ann. Math. (2)78, 59–103 (1963)Google Scholar
  10. 10.
    Getzler, E., Jones, J.D.S.: Operads and homotopy algebras. Preprint, 1993Google Scholar
  11. 11.
    Ginzburg, V.A., Kapranov, M.M.: Koszul duality for operads. Preprint 1993Google Scholar
  12. 12.
    Hochschild, G., Kostant, B., Rosenberg, A.: Differential forms on regular affine algebras. Trans. Am. Math. Soc.102, 383–408 (1962)Google Scholar
  13. 13.
    Hořava, P.: Spacetime diffeomorphisms and topologicalW -symmetry in two dimensional topological string theory. Preprint EFI-92-70, hep-th/9202020, Enrico Fermi Institute, 1993Google Scholar
  14. 14.
    Joyal, A., Street, R.: The geometry of tensor calculus, I. Adv. Math.88, 55–112 (1991)CrossRefGoogle Scholar
  15. 15.
    Lerche, W., Vafa, C., Warner, N.P.: Chiral rings inN=2 superconformal field theories. Nucl. Phys.B324, 427–474 (1989)CrossRefGoogle Scholar
  16. 16.
    Lian, B.H., Zuckerman, G.J.: New perspectives on the BRST-algebraic structure of string theory. Preprint hep-th/9211072Google Scholar
  17. 17.
    May, J.P.: The geometry of iterated loop spaces. Lecture Notes in Mathematics, no.271, Berlin, Heidelberg, New York: Springer 1972Google Scholar
  18. 18.
    Schwarz, A., Penkava, M.: On some algebraic structures arising in string theory. Preprint UCD-92-03, University of California, Davis, hep-th/912071Google Scholar
  19. 19.
    Priddy, S.B.: Koszul resolutions. Trans. Am. Math. Soc.152, 39–60 (1970)Google Scholar
  20. 20.
    Schwarz, A.: Geometry of Batalin-Vilkovisky quantization. Preprint hep-th/9205088, University of California, Davis 1992Google Scholar
  21. 21.
    Segal, G.: The definition of conformal field theory. Unpublished manuscriptGoogle Scholar
  22. 22.
    Witten, E.: The anti-bracket formalism. Mod. Phys. Lett.A5, 487–494 (1990)CrossRefGoogle Scholar
  23. 23.
    Witten, E.: On the structure of the topological phase of two-dimensional gravity. Nucl. Phys.B340, 281–332 (1990)CrossRefGoogle Scholar
  24. 24.
    Zwiebach, B.: Closed string field theory: quantum action and the B-V master equation. Nucl. Phys.B390, 33–152 (1993)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • E. Getzler
    • 1
  1. 1.Department of MathematicsMITCambridgeUSA

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