## Batalin-Vilkovisky algebras and two-dimensional topological field theories.(English)Zbl 0807.17026

In G. Segal’s approach the two-dimensional topological field theory can be described as a category with sets of morphisms identified with certain moduli spaces. It is shown that in these settings the cohomology of topological field theory in two dimensions is endowed with the structure of a graded commutative algebra $$A$$ with the additional operator $$\Delta$$ of degree one such that $$\Delta^ 2=0$$ and $$[\Delta,a]- \Delta(a)$$ is a graded derivation of $$A$$ for arbitrary $$a\in A$$ (Batalin-Vilkovisky algebra). The construction is given in purely topological terms using the language of operads.

### MSC:

 17B69 Vertex operators; vertex operator algebras and related structures 81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics 81T70 Quantization in field theory; cohomological methods 81R99 Groups and algebras in quantum theory
Full Text:

### References:

 [1] Arnold, V.I.: The cohomology ring of the colored braid group. Mat. Zametki5, 227–231 (1969) [2] Beilinson, A., Ginzburg, V.: Infinitesimal structures of moduli space ofG-bundles. Duke Math. J.66, 63–74 (1992) · Zbl 0763.32011 [3] Birman, J.S.: Braids, links and mapping class groups. Ann. Math. Studies, no. 82, Princeton, NJ: Princeton U. Press 1974 [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 1973 · Zbl 0285.55012 [5] Cohen, F.R.: The homology ofC n+1-spaces,n. In: The homology of iterated loop spaces, Lecture Notes in Mathematics, no.533, Berlin, Heidelberg, New York: Springer 1976, pp. 207–351 [6] Dijkgraaf, R., Verlinde, H., Verlinde, E.: Topological strings ind<1. Nucl. Phys.B352, 59–86 (1991) · Zbl 0783.58088 [7] Fadell, E., Neuwirth, L.: Configuration spaces. Math. Scand.10, 111–118 (1962) · Zbl 0136.44104 [8] Fulton, W., MacPherson, R.: A compactification of configuration spaces. To appear, Ann. Math. · Zbl 0820.14037 [9] Gerstenhaber, M.: The cohomology structure of an associative ring. Ann. Math. (2)78, 59–103 (1963) · Zbl 0131.27302 [10] Getzler, E., Jones, J.D.S.: Operads and homotopy algebras. Preprint, 1993 · Zbl 0795.46052 [11] Ginzburg, V.A., Kapranov, M.M.: Koszul duality for operads. Preprint 1993 · Zbl 0855.18006 [12] Hochschild, G., Kostant, B., Rosenberg, A.: Differential forms on regular affine algebras. Trans. Am. Math. Soc.102, 383–408 (1962) · Zbl 0102.27701 [13] Hořava, P.: Spacetime diffeomorphisms and topologicalW symmetry in two dimensional topological string theory. Preprint EFI-92-70, hep-th/9202020, Enrico Fermi Institute, 1993 [14] Joyal, A., Street, R.: The geometry of tensor calculus, I. Adv. Math.88, 55–112 (1991) · Zbl 0738.18005 [15] Lerche, W., Vafa, C., Warner, N.P.: Chiral rings inN=2 superconformal field theories. Nucl. Phys.B324, 427–474 (1989) [16] Lian, B.H., Zuckerman, G.J.: New perspectives on the BRST-algebraic structure of string theory. Preprint hep-th/9211072 · Zbl 0780.17029 [17] May, J.P.: The geometry of iterated loop spaces. Lecture Notes in Mathematics, no.271, Berlin, Heidelberg, New York: Springer 1972 · Zbl 0244.55009 [18] Schwarz, A., Penkava, M.: On some algebraic structures arising in string theory. Preprint UCD-92-03, University of California, Davis, hep-th/912071 [19] Priddy, S.B.: Koszul resolutions. Trans. Am. Math. Soc.152, 39–60 (1970) · Zbl 0261.18016 [20] Schwarz, A.: Geometry of Batalin-Vilkovisky quantization. Preprint hep-th/9205088, University of California, Davis 1992 · Zbl 0786.58017 [21] Segal, G.: The definition of conformal field theory. Unpublished manuscript [22] Witten, E.: The anti-bracket formalism. Mod. Phys. Lett.A5, 487–494 (1990) · Zbl 1020.81931 [23] Witten, E.: On the structure of the topological phase of two-dimensional gravity. Nucl. Phys.B340, 281–332 (1990) [24] Zwiebach, B.: Closed string field theory: quantum action and the B-V master equation. Nucl. Phys.B390, 33–152 (1993)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.