## 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

### Keywords:

string theory; BRST cohomology; Gerstenhaber algebra
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### 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
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