×

zbMATH — the first resource for mathematics

Determinant bundles and Virasoro algebras. (English) Zbl 0665.17010
This paper studies the interplay between the following objects: (1) The Virasoro algebra and its highest weight representations. (2). The moduli space of curves. (3) Determinant bundles. (4) Conformal field theory.
For example, the Virasoro algebra acts by infinitesimal deformations on the moduli space and on the determinant bundle. Representations of the Virasoro algebra give rise to D-modules on the moduli space which can be interpreted as differential equations for correlation functions in conformal field theory.
Reviewer: A.N.Pressley

MSC:
17B65 Infinite-dimensional Lie (super)algebras
14H10 Families, moduli of curves (algebraic)
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Alessandrini, V., Amati, D.: Properties of dual multiloop amplitudes. Nuovo Cim.4A, N4, 793-844 (1971) · doi:10.1007/BF02731520
[2] Beilinson, A.A.: Residues and adéles. Funct. Anal. Appl.14, N1, 44-45 (1980) (in Russian)
[3] Beilinson, A., Bernstein, J.: Localisation de 700-1-modules. C.R. Acad. Sci. (Paris) Ser. I, t.292, 15-18 (1981) · Zbl 0476.14019
[4] Beilinson, A.A., Manin, Yu.I., Schechtman, V.V.: Sheaves of Virasoro and Neveu-Schwartz algebras. Lecture Notes in Mathematics, Vol. 1289, pp. 52-66. Berlin, Heidelberg, New York: Springer 1987
[5] Belavin, A.A., Knizhnik, V.G.: Algebraic geometry and the geometry of quantum strings. Phys. Lett. B168, 201-206 (1986) · Zbl 0693.58043 · doi:10.1016/0370-2693(86)90963-9
[6] Belavin, A.A., Polyakov, A.M., Zamolodchikov, A.B.: Infinite conformal symmetries in two-dimensional quantum field theory. Nucl. Phys. B241, 333-380 (1984) · Zbl 0661.17013 · doi:10.1016/0550-3213(84)90052-X
[7] Bismut, J., Gillet, H., Soule, C.: Analytic torsion and holomorphic determination bundles, I, II, III. Commun. Math. Phys.115, 49-78, 79-126, 301-351 (1988) · Zbl 0651.32017 · doi:10.1007/BF01238853
[8] Bershadsky, M., Radul, A.: Conformal field theories with additionalZ N symmetry. Int. J. Mod. Phys. A2, N1, 165-178 (1987) · Zbl 1165.81373 · doi:10.1142/S0217751X87000053
[9] Bost, J.B.: Fibrés determinants, determinants regularisés et mesures sur les espaces de modules des courbes complexes. Sem. Bourbaki, exp.676 (Fevr. 1987)
[10] Date, E., Jimbo, M., Kashiwara, M., Miwa, T.: Transformation groups for solution equations. In: Proc. of RIMS Symposium. Jimbo, M., Miwa, T. (eds.), pp. 34-120. Singapore: World Scientific 1983 · Zbl 0571.35098
[11] Deligne, P.: Le determinant de la cohomologie. In: Current trends in arithmetical algebraic geometry. Contemp. Math.67, 93-177 (1987)
[12] Feigin, B.L., Fuchs, D.B.: Representations of the Virasoro algebra. In: Seminar on supermanifolds 5. Leites, D. (ed.), Reports of Dept. Math. Stockholm Univ., N25 (1986)
[13] Knizhnik, V.G.: Analytic fields on Riemann surfaces. II. Commun. Math. Phys.112, 567-590 (1987) · Zbl 0656.58044 · doi:10.1007/BF01225373
[14] Knudsen, F.F., Mumford, D.: The projectivity of the moduli space of stable curves. I. Preliminaries on ?det? and ?div?. Math. Scand.39, 19-55 (1976) · Zbl 0343.14008
[15] Kontsevich, M.L.: Virasoro algebra and Teichmüller spaces. Funct. Anal. Appl.21, N2, 78-79 (1987) (in Russian) · Zbl 0647.58012 · doi:10.1007/BF01078034
[16] Manin, Yu.I.: Critical dimensions of string theories and dualizing sheaf of the moduli space of (super) curves. Funct. Anal. Appl.20, N3, 88-89 (1986) (in Russian) · Zbl 0639.14015
[17] Mumford, D.: Stability of projective varieties. Enseign. Math.23, 39-110 (1977) · Zbl 0363.14003
[18] Quillen, D.: Determinants of Cauchy-Riemann operators over a Riemann surface. Funct. Anal. Appl.19, No. 1, 31-34 (1985) (in Russian) · Zbl 0603.32016 · doi:10.1007/BF01086022
[19] Segal, G., Wilson, G.: Loop groups and equations of KdV type. Publ. Math. IHES 1-64, 63 (1985) · Zbl 0592.35112
[20] Tate, J.: Residues of differentials on curves. Ann. Sci. E.N.S., 4 Ser., t.1, 149-159 (1968) · Zbl 0159.22702
[21] Tjurin, A.N.: On periods of quadratic differentials. Russ. Math. Surv.33, N6, 149-195 (1978) · Zbl 0402.16019 · doi:10.1070/RM1978v033n03ABEH002472
[22] Tsuchiya, A., Kanie, Y.: Fock space representations of the Virasoro algebra. Publ. RIMS22, N2, 259-327 (1986) · Zbl 0604.17008
[23] Zamolodchikov, Al.B.: Conformal scalar field on the hyperelliptic curve and critical Ashkin-Feller multipoint correlation functions. Nucl. Phys. B285, 481-503 (1987) · doi:10.1016/0550-3213(87)90350-6
[24] Toledo, D., Tong, Y.-L.: Duality and intersection theory in complex manifolds. I. Math. Ann.237, 41-77 (1978) · Zbl 0391.32008 · doi:10.1007/BF01351557
[25] Alvarez-Gaumé, L., Gomez, C., Reina, C.: New methods in string theory, preprint CERN (1987)
[26] Arbarello, E., De Concini, C., Kac, V., Procesi, C.: In preparation
[27] Eguchi, T., Ooguri, H.: Conformal and current algebras on a general Riemann surface. Nucl. Phys. B282, 308 (1987) · doi:10.1016/0550-3213(87)90686-9
[28] Kawamoto, N., Namikawa, Yu., Tsuchiya, A., Yamada, Y.: Geometric realisation of conformal field theory on Riemann surfaces. Preprint, Nagoya university (1987) · Zbl 0648.35080
[29] Witten, E.: Quantum field theory, Grassmannians, and algebraic curves. Commun. Math. Phys.113, 529-600 (1988) · Zbl 0636.22012 · doi:10.1007/BF01223238
[30] Gabber, O.: The integrability of characteristic variety. Am. J. Math.103, N3, 445 (1981) · Zbl 0492.16002 · doi:10.2307/2374101
[31] Schechtman, V.V.: Riemann-Roch theorem and Atiyah-Hirzebruch spectral sequence. Usp. Mat. Nauk (=Russ. Math. Surv.),35, N6, 179-180 (1980) (in Russian) · Zbl 0469.14004
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.