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Communications in Mathematical Physics

, Volume 164, Issue 1, pp 105–144 | Cite as

Operadic formulation of topological vertex algebras and Gerstenhaber or Batalin-Vilkovisky algebras

  • Yi-Zhi Huang
Article

Abstract

We give the operadic formulation of (weak, strong) topological vertex algebras, which are variants of topological vertex operator algebras studied recently by Lian and Zuckerman. As an application, we obtain a conceptual and geometric construction of the Batalin-Vilkovisky algebraic structure (or the Gerstenhaber algebra structure) on the cohomology of a topological vertex algebra (or of a weak topological vertex algebra) by combining this operadic formulation with a theorem of Getzler (or of Cohen) which formulates Batalin-Vilkovisky algebras (or Gerstenhaber algebras) in terms of the homology of the framed little disk operad (or of the little disk operad).

Keywords

Neural Network Statistical Physic Complex System Nonlinear Dynamics Quantum Computing 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [BoV] 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
  2. [B] Borcherds, R.E.: Vertex algebras, Kac-Moody algebras, and the monster. Proc. Natl. Acad. Sci. USA83, 3068–3071 (1986)Google Scholar
  3. [C] 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
  4. [DL] Dong, C., Lepowsky, J.: Generalized vertex algebras and relative vertex operators. Research monograph, to be published, 1993Google Scholar
  5. [FFR] Feingold, A.J., Frenkel, I.B., Ries, J.F.X.: Spinor construction of vertex operator algebras, triality andE 8(1). Contemp. Math., Vol.121, Providence, RI: Am. Math. Soc., 1991Google Scholar
  6. [FHL] Frenkel, I.B., Huang, Y.-Z., Lepowsky, J.: On axiomatic approaches to vertex operator algebras and modules. Preprint, 1989; Memoirs Am. Math. Soc.104, 1993Google Scholar
  7. [FLM1] Frenkel, I.B., Lepowsky, J., Neurman, A.: A natural representation of the Fischer-Griess monster with the modular functionJ as character. Proc. Natl. Acad. Sci. USA81, 3256–3260 (1984)Google Scholar
  8. [FLM2] Frenkel, I.B., Lepowsky, J., Meurman, A.: Vertex operator algebras and the monster. Pure and Appl. Math.134, Boston: Academic Press 1988Google Scholar
  9. [FZ] Frenkel, I.B., Zhu, Y.: Vertex operator algebras associated to representations of affine and Virasoro algebras. Duke Math.66, 123–156 (1992)CrossRefGoogle Scholar
  10. [Ger] Gerstenhaber, M.: The cohomology structure of an associative ring. Ann. Math. (2)78, 267–288 (1963)Google Scholar
  11. [Get] Getzler, E.: Batalin-Vilkovisky algebras and two-dimensional topological field theory. Commun. Math. Phys., to appearGoogle Scholar
  12. [H1] Huang, Y.-Z.: On the geometric interpretation of vertex operator algebras. Ph.D. thesis, Rutgers University, 1990; Operads and the geometric interpretation of vertex operator algebras, I. Preprint, to appearGoogle Scholar
  13. [H2] Huang, Y.-Z.: Geometric interpretation of vertex operator algebras. Proc. Natl. Acad. Sci. USA88, 9964–9968 (1991)Google Scholar
  14. [H3] Huang, Y.-Z.: Vertex operator algebras and conformal field theory. Intl. J. Mod. Phys.A7, 2109–2151 (1992)CrossRefGoogle Scholar
  15. [H4] Huang, Y.-Z.: Operads and the geometric interpretation of vertex operator algebras. II. In preparationGoogle Scholar
  16. [HL1] Huang, Y.-Z., Lepowsky, J.: Vertex operator algebras and operads. The Gelfand Mathematical Seminars, 1990–1992. Corwin, L., Gelfand, I., Lepowsky, J., (eds.), Boston: Birkhäuser, 1993, pp. 145–161Google Scholar
  17. [HL2] Huang, Y.-Z., Lepowsky, J.: Operadic formulation of the notion of vertex operator algebra. Preprint, to appear, 1993Google Scholar
  18. [HS] Hinich, V., Schecchtman, V.: Homotopy Lie algebras. Preprint, 1992Google Scholar
  19. [L] Lian, B.H.: On the classification of simple vertex operator algebras. Preprint Commun. Math. Phys., to appearGoogle Scholar
  20. [LZ] Lian, B.H., Zuckerman, G.J.: New perspectives on theBRST-algebraic structure of string theory. Preprint, hep-th/9211072, 1992Google Scholar
  21. [Mas] Massey, W.S.: A basic course in algebraic topology. Graduate Texts in Math., Vol. 127. Berlin, Heidelberg, New York: Springer 1991Google Scholar
  22. [May] May, J.P.: The geometry of iterated loop spaces. Lecture Notes in Mathematics, No. 271. Berlin, Heidelberg, New York: Springer 1972Google Scholar
  23. [MS] Moore, G., Seiberg, N.: Classical and quantum conformal field theory, Comm. Math. Phys.123, 177–254 (1989)CrossRefGoogle Scholar
  24. [PS] Penkava, M., Schwarz, A.: On some algebraic structures arising in string theory. Preprint, hep-the/9212072, 1992Google Scholar
  25. [S1] Stasheff, J.D.: Homotopy associativity ofH-spaces. I. Trans. Am. Math. Soc.108, 215–292 (1963); Homotopy associativity ofH-spaces. II. Trans. Am. Math. Soc.108, 293–312 (1963)Google Scholar
  26. [S2] Stasheff, J.D.: Closed string field theory, strong homotopy Lie algebras and the operad actions of moduli spaces. Preprint, hep-th/9304061, 1993Google Scholar
  27. [T] Tsukada, H.: Vertex operator superalgebras. Comm. Alg.18, 2259–2274 (1990)Google Scholar
  28. [Z] Zwiebach, B.: Closed string field theory: quantum action and theB-V master equation. Nucl. Phys.B390, 33–152 (1993)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • Yi-Zhi Huang
    • 1
  1. 1.Department of MathematicsUniversity of PennsylvaniaPhiladelphiaUSA

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