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Homology of Moduli of Curves and Commutative Homotopy Algebras

  • Conference paper
The Gelfand Mathematical Seminars, 1993–1995

Abstract

There have been a number of mathematical results recently identifying algebras over certain operads [4, 11, 12, 15, 17, 18, 19, 20, 26, 28]. See [1, 29] for expository surveys of the basics of operad theory. Before citing any of these results, let us mention some trivial classical examples. Let A denote one of the three words: “commutative,” “associative” and “Lie.” In each of these cases, consider the corresponding operad O(n) = (words in the free A algebra on n generators having exactly one occurrence of each generator) k , n ≥ 1, () k meaning the linear span over the ground field k. When A is commutative, associative or Lie, we will denote the corresponding operad O by C, As and Lie, respectively. Note that C(n)=(x 1…x n )k = k and As(n) = (x σ(1)…x σ(n) | σ ε S n)k = k[S n ]. The main feature of these three operads is that they describe algebras of the corresponding types. More precisely, algebras over an operad O, O = C, As or Lie, (or simply, O-algebras) are exactly A algebras.

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References

  1. Adams, J.F., Infinite loop spaces. Princeton, NJ: Princeton University Press, 1978

    MATH  Google Scholar 

  2. Arnold, V.I., The cohomology ring of the colored braid group, Mat. Zametki 5(1969), 227–231 (English translation: Math. Notes 5 (1969), 138–140

    MathSciNet  Google Scholar 

  3. Axelrod, S., Singer, I.M., Chern-Simons perturbation theory II., J. Diff. Geom. 39, 173–213 (1994), hep-th/9304087

    MathSciNet  MATH  Google Scholar 

  4. Beilinson, A., Ginzburg, V., Infinitesimal structure of moduli spaces of G-bundles, Internat. Math. Research Notices 4 (1992), 63–74

    Article  MathSciNet  Google Scholar 

  5. Cohen, F.R., Artin’s braid groups, classical homotopy theory and sundry other curiosities, Contemp. Math. 78 (1988), 167–206

    Google Scholar 

  6. Deligne, P., Resumé des premiers exposés de A. Grothendieck., In: SGA 7. Lecture Notes in Math., 288, Springer-Verlag, Berlin, Heidelberg, New York, 1972, 1–24.

    Google Scholar 

  7. Deligne, P., Théorie de Hodge, II, Inst. Hautes Etudes Sci. Publ.Math. 40 (1971), 5–57

    Article  MathSciNet  MATH  Google Scholar 

  8. Drinfel’d, V. G., Letter to V. Schechtman, September 18, 1988.

    Google Scholar 

  9. Dubrovin, B., Geometry of 2d topological field theories, preprint SISSA-89/94/FM, July 1994, hep-th /9407018

    Google Scholar 

  10. Fulton, W., Maherson, R., A compactification of configuration spaces, Ann. Math. 139 (1994), 183–225

    Article  MATH  Google Scholar 

  11. Getzler, E., Batalin-Vilkovisky algebras and two-dimensional topological field theories, Commun. Math. Phys. 159 (1994), 265–285, hepth/9212043

    Article  MathSciNet  MATH  Google Scholar 

  12. Getzler, E., Two-dimensional topological gravity and equivariant cohomology,Commun. Math. Phys. 163 (1994), 473–489, hep-th/9305013

    Article  MathSciNet  MATH  Google Scholar 

  13. Getzler, E., Cartan homotopy formulas and the Gauss-Manin connection in cyclic homology, Israel Math. Conf. Proc. 7 (1993), 65–78.

    MathSciNet  Google Scholar 

  14. Getzler, E., Operads and moduli spaces of genus 0 Riemann surfaces, In: The moduli space of curves. Progress in Math., 129, Birkhäuser, Boston, Basel, Berlin, 1995, 199–230

    Chapter  Google Scholar 

  15. Getzler, E., Jones, J.D.S., Operads, homotopy algebra and iterated integrals for double loop spaces. Preprint. Department of Mathematics, MIT, March 1994, hep-th/9403055

    Google Scholar 

  16. Getzler, E., Kapranov, M. M., Modular operads, Preprint, August 1994, dg-ga/9408003

    Google Scholar 

  17. Ginzburg, V., Kapranov, M. M., Koszul duality for operads, Duke Math. J. 76 (1994), 203–272.

    Article  MathSciNet  MATH  Google Scholar 

  18. Hinich, V., Schechtman, V., Homotopy Lie algebras,Adv. Studies Sov. Math., 16 (1993), 1–18.

    MathSciNet  Google Scholar 

  19. Huang, Y.-Z., Operadic formulation of topological vertex algebras and Gerstenhaber or Batalin-Vilkovisky algebras, Commun. Math. Phys. 164 (1994), 105–144, hep-th/9306021

    Article  MATH  Google Scholar 

  20. Huang, Y.-Z., Lepowsky, J., Vertex operator algebras and operads, The Gelfand Mathematics Seminars, 1990–1992, Birkhäuser, Boston, 1993, pp. 145–161, hep-th/9301009.

    Google Scholar 

  21. Kadeishvili, T., A(∞)-algebra structure in cohomology and the rational homotopy type, Forschungsschwerpunkt Geometrie, Universität Heidelberg, Mathematisches Institut 37 (1988) 1–64

    Google Scholar 

  22. Kadeishvili, T., The category of differential coalgebras and the category of;4(∞)-algebras, Proc. Tbilisi Math. Inst. 77 (1985) 50–70

    MathSciNet  MATH  Google Scholar 

  23. Kapranov, M. M., Chow quotients of Grassmannians I, In: The Gelfand Mathematics Seminars, 1990–1992, Birkhäuser, Boston, 1993, pp. 29–110

    Google Scholar 

  24. Keel, S., Intersection theory of moduli space of stable n-pointed curves of genus zero, Trans. Amer. Math. Soc. 330 (1992), 545–574

    Article  MathSciNet  MATH  Google Scholar 

  25. Kimura, T., Operads of Moduli Spaces and Algebraic Structures in Conformal Field Theory, Preprint, University of North Carolina. November 1994.

    Google Scholar 

  26. Kimura, T., Stasheff, J., and Voronov, A. A., On operad structures of moduli spaces and string theory, Commun. Math. Phys. 171 (1995), 1–25, hep-th/9307114

    Article  MathSciNet  MATH  Google Scholar 

  27. Knudsen, F. F., The projectivity of moduli spaces of stable curves, II: the stacks M g,n . Math. Scand. 52 (1983), 161–199

    MathSciNet  MATH  Google Scholar 

  28. Kontsevich, M., Manin, Yu.I., Gromov-Witten classes, quantum cohomology, and enumerative geometry, Commun. Math. Phys. 164 (1994), 525–562, hep-th/9402147

    Article  MathSciNet  MATH  Google Scholar 

  29. Kriz, I., May, J. P., Operads, algebras, and modules, I: definitions and examples. Preprint. University of Chicago 1993

    Google Scholar 

  30. Lang, S., Algebra, Second Edition, Addison-Wesley, 1994.

    Google Scholar 

  31. Lian, B. H., Zuckerman, G. J., New perspectives on the BRST-algebraic structure of string theory, Commun. Math. Phys. 154 (1993), 613–646 hep-th/9211072

    Article  MathSciNet  MATH  Google Scholar 

  32. May, J. P., The geometry of iterated loop spaces, Lecture Notes in Math., 271, Springer-Verlag, Berlin, Heidelberg, New York, 1972

    Google Scholar 

  33. May, J. P., Infinite loop space theory, Bull. Amer. Math. Soc. 83 (1977), 456–494

    Article  MathSciNet  MATH  Google Scholar 

  34. Michaelis, W., Lie coalgebras, Adv. in Math. 38 (1980), 1–54

    Article  MathSciNet  MATH  Google Scholar 

  35. Moore, J.C., Some properties of the loop homology of commutative coalgebras, In: The Steenrod Algebra and its Applications. Lecture Notes in Math., 168 (1970), 232–245

    Google Scholar 

  36. Quillen, D., Rational homotopy theory, Annals of Math. (2) 90 (1969) 205–295.

    Article  MathSciNet  MATH  Google Scholar 

  37. Ree, R., Lie elements and the algebra associated with shuffles, Ann. of Math. 68 (1958), 210–219

    Article  MathSciNet  MATH  Google Scholar 

  38. Ruan, Y., Tian, G., A mathematical theory of quantum cohomology, Math. Res. Lett. 1 (1994), 269–278

    MathSciNet  MATH  Google Scholar 

  39. Schechtman, V.V., Varchenko, A.N., Arrangements of hyperplanes and Lie algebra homology, Invent. Math. 106 (1991), 139–194

    Article  MathSciNet  MATH  Google Scholar 

  40. Schlessinger, M., Stasheff, J.D., The Lie algebra structure of tangent cohomology and deformation theory, J. of Pure and Appl. Alg. 38 (1985), 313–322

    Article  MathSciNet  MATH  Google Scholar 

  41. Segal, G., The definition of conformal field theory. Preprint. Oxford

    Google Scholar 

  42. Smirnov, V. A., On the cochain complex of topological spaces, Math. USSR Sbornik 43 (1992), 133–144.

    Article  Google Scholar 

  43. Stasheff, J.D., On the homotopy associativity of H-spaces, I, Trans. Amer. Math. Soc. 108 (1963), 275–292

    Article  MathSciNet  MATH  Google Scholar 

  44. Stasheff, J.D., On the homotopy associativity of H-spaces, II, Trans. Amer. Math. Soc. 108 (1963), 293–312

    MathSciNet  Google Scholar 

  45. Stasheff, J.D., H-spaces from a homotopy point of view, Lecture Notes in Math., 161 (1970), Springer-Verlag, Berlin, Heidelberg, New York

    Google Scholar 

  46. Stasheff, J.D., Higher homotopy algebras: string field theory and Drinfel’d’s quasi-Hopf algebras, The Proceedings of the XXth International Conference on Differential Geometric Methods in Theoretical Physics,Baruch College, CUNY, June 1991, World Scientific (1992) 408–425.

    Google Scholar 

  47. Stasheff, J.D., Closed string field theory, strong homotopy Lie algebras and the operad actions of moduli space, In: Perspectives on Mathematics and Physics, R.C. Penner and S.T. Yau, eds., International Press, 1994, 265–288.

    Google Scholar 

  48. Witten, E., On the structure of the topological phase of two dimensional gravity, Nucl. Phys. B 340 (1990), 281–332

    Article  MathSciNet  Google Scholar 

  49. Witten, E., Two dimensional gravity and intersection theory on moduli space, Surv. in Diff. Geom. 1 (1991), 243–310.

    MathSciNet  Google Scholar 

  50. Witten, E., Zwiebach, B., Algebraic structures and differential geometry in two-dimensional string theory, Nucl. Phys. B 377 (1992), 55–112.

    Article  MathSciNet  Google Scholar 

  51. Zwiebach, B., Closed string field theory: Quantum action and the Batalin-Vilkovisky master equation, Nucl. Phys. B 390 (1993), 33–152

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Boston University, 02215, Boston, MA, USA

    Takashi Kimura

  2. Department of Mathematics, University of North Carolina, 27599-3250, Chapel Hill, NC, USA

    Jim Stasheff

  3. Department of Mathematics, University of Pennsylvania, 19104-6395, Philadelphia, PA, USA

    Alexander Voronov

Authors
  1. Takashi Kimura
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  2. Jim Stasheff
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  3. Alexander Voronov
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Editors and Affiliations

  1. Department of Mathematics, Rutgers University, 08903, New Brunswick, NJ, USA

    Israel M. Gelfand  & James Lepowsky  & 

  2. Department of Mathematics, Columbia University, 10027, New York, NY, USA

    Mikhail M. Smirnov

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© 1996 Birkhäuser Boston

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Kimura, T., Stasheff, J., Voronov, A. (1996). Homology of Moduli of Curves and Commutative Homotopy Algebras. In: Gelfand, I.M., Lepowsky, J., Smirnov, M.M. (eds) The Gelfand Mathematical Seminars, 1993–1995. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-4082-2_9

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  • DOI: https://doi.org/10.1007/978-1-4612-4082-2_9

  • Publisher Name: Birkhäuser Boston

  • Print ISBN: 978-1-4612-8643-1

  • Online ISBN: 978-1-4612-4082-2

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