×

New perspectives on the BRST-algebraic structure of string theory. (English) Zbl 0780.17029

The authors give a new interpretation for the action of the symmetry charges on the BRST cohomology in terms of what they call the Gerstenhaber bracket. This interpretation gives rise to a new class of examples of what mathematicians call a Gerstenhaber algebra. Applying their theory to the \(c=1\) model, they give a precise conceptual description of the BRST-Gerstenhaber algebra of this model. They also show that their constructions generalize to any topological conformal field theory.
Reviewer: V.Abramov (Tartu)

MSC:

17B70 Graded Lie (super)algebras
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
81T70 Quantization in field theory; cohomological methods
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Bouwknegt, P., McCarthy, J., Pilch, K.: Commun. Math. Phys.145, 541 (1992) · Zbl 0765.53059
[2] Belavin, A., Polyakov, A.M., Zamolodchikov, A.A.: Infinite conformal symmetry in two dimensional quantum field theory. Nucl. Phys. B241, 333 (1984) · Zbl 0661.17013
[3] Borcherds, R.E.: Vertex operator algebras, Kac-Moody algebras and the Monster. Proc. Natl. Acad. Sci. USA83, 3068 (1986) · Zbl 0613.17012
[4] Drinfel’d, V.G.: Quantum groups. ICM Proceedings, Berkeley, California, USA 798 (1986)
[5] Dijkgraaf, R., Verlinde, H., Verlinde, E.: Notes on topological string theory and 2d quantum gravity. Preprint PUPT-1217, IASSNS-HEP-90/80 · Zbl 0932.81028
[6] Eguchi, T., Yang, S.K.:N=2 superconformal models as topological field theories. Tokyo-preprint UT-564 · Zbl 1020.81833
[7] Frenkel, I.B., Garland, H., Zuckerman, G.J.: Semi-infinite cohomology and string theory. Proc. Natl. Acad. Sci. USA83, 8442 (1986) · Zbl 0607.17007
[8] Frenkel, E.: Determinant formulas for the free field representations of the Virasoro and Kac-Moody algebras. Phys. Lett.285B, 71 (1992)
[9] Frenkel, I.B., Lepowsky, J., Meurman, A.: Vertex operator algebras and the Monster. New York: Academic Press 1988 · Zbl 0674.17001
[10] Frenkel, I.B., Huang, Y., Lepowsky, J.: On axiomatic approaches to vertex operator algebras and modules. Yale-Rutgers preprint. Dept. of Math., 1989
[11] Gerstenhaber, M.: The cohomology structure of an associative ring. Ann. Math.78, No. 2, 267 (1962) · Zbl 0131.27302
[12] Gerstenhaber, M.: On the deformation of rings and algebras. Ann. Math.79, No. 1, 59 (1964) · Zbl 0123.03101
[13] Gerstenhaber, M., Giaquinto, A., Schack, S.: Quantum symmetry. Preprint · Zbl 0762.17013
[14] Kanno, H., Sarmadi, M.H.: BRST cohomology ring in 2d gravity coupled to minimal models. Preprint hepth9207078 · Zbl 0985.81691
[15] Kutasov, D., Martinec, E., Seiberg, N.: Phys. Lett. B276, 437 (1992)
[16] Lada, T., Stasheff, J.: Introduction to sh Lie algebras for physicists. Preprint UNC-MATH-92/2 · Zbl 0824.17024
[17] Lian, B.H.: Semi-infinite homology and 2d quantum gravity. Dissertation, Yale Univ., May 1991
[18] Lian, B.H.: On the classification of simple vertex operator algebras. Univ. of Toronto preprint, February 1992
[19] Lian, B.H., Zuckerman, G.J.: New selection rules and physical states in 2d gravity; conformal gauge. Phys. Lett. B,254, No. 3, 4, 417 (1991) · Zbl 1176.17015
[20] Lian, B.H., Zuckerman, G.J.: 2d gravity withc=1 matter. Phys. Lett. B266, 21 (1991) · Zbl 1176.81116
[21] Lian, B.H., Zuckerman, G.J.: Semi-infinite homology and 2D gravity (I). Commun. Math. Phys.145, 561 (1992) · Zbl 0849.17028
[22] Moore, G., Seiberg, N.: Classical and quantum conformal field theory. Commun. Math. Phys.123, 177–254 (1989) · Zbl 0694.53074
[23] Mukherji, S., Mukhi, S., Sen, A.: Phys. Lett.266B, 337 (1991)
[24] Schouten, J.A.: Über Differentialkomitanten zweier kontravarianter Größen. Nederl. Acad. Wetensch. Proc. Ser. A43, 449–452 (1940) · JFM 66.0817.04
[25] Schlesinger, M., Stasheff, J.: The Lie algebra structure of tangent cohomology and deformation theory. J. Pure Appl. Alg.38, 313 (1985) · Zbl 0576.17008
[26] Stasheff, J.: Differential graded Lie algebras, quasi-Hopf algebras and higher homotopy algebras. UNC Math. Preprint, 1991 · Zbl 0758.17010
[27] Stasheff, J.: Homotopy associativity ofH-spaces. AMS Trans.108, 275 (1963) · Zbl 0114.39402
[28] Witten, E.: Ground ring of the two dimensional string theory. Nucl. Phys. B373, 187 (1992) · Zbl 0815.53090
[29] Witten, E.: The anti-bracket formalism. preprint IASSNS-HEP-90/9
[30] Witten, E.: On background independent open-string field theory. Preprint FERMILAB-PUBIASSNS-HEP-92/53
[31] Witten, E., Zwiebach, B.: Algebraic structures and differential geometry in 2d string theory. Nucl. Phys. B377, 55 (1992)
[32] Wu, Y., Zhu, C.: The complete structure of the cohomology ring and associated symmetries ind=2 string theory. Preprint · Zbl 1043.81722
[33] Zhu, Y.: Vertex operator algebras, elliptic functions and modular forms. Yale Thesis, Dept. of Math., 1990
[34] Zuckerman, G.J.: New applications of Gerstenhaber algebras. (In progress)
[35] Zwiebach, B.: Closed string field theory: Quantum action and the B-V master equation. Preprint IASSNS-HEP-92/41
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.