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Virasoro-type algebras, Riemann surfaces and strings in Minkowski space. (English. Russian original) Zbl 0659.17012
Funct. Anal. Appl. 21, No. 4, 294-307 (1987); translation from Funkts. Anal. Prilozh. 21, No. 4, 47-61 (1987).
Let \(\Gamma\) denote a compact Riemann surface of genus \(g\) with two distinguished points \(P_+\), \(P_-\). The generalized Heisenberg algebra connected with \(\Gamma\) is defined by the authors as an algebra with generators \(\alpha_ n\) and a central element t, satisfying the relations \([\alpha_ n,\alpha_ m]=\gamma_{nm}\cdot t\), \([\alpha_ n,t]=0\), where \(\alpha_{mn}=(1/2\pi i)\oint A_ m dA_ n\). The \(A_ n\)’s are suitable meromorphic functions on \(\Gamma\), holomorphic away from \(P_+\), \(P_-\) and characterized by their behavior in the neighbourhoods of these points. An analog of the Virasoro algebra connected with \(\Gamma\) is defined as an algebra with generators \(E_ n\) and \(t\) and the relations \[ [E_ n,t]=0,\quad [E_ n,E_ m]=\sum c^ k_{nm} E_{n+m-k}+\chi_{nm} t. \] The \(c^ k_{nm}\) are the structural constants of the algebra \({\mathcal L}^{\Gamma}\) of meromorphic vector fields on \(\Gamma\) with respect to a suitable basis and \(\chi_{nm}\) is a 2-cocycle on \({\mathcal L}^{\Gamma}\).
With this structure the authors give an algebro-geometric model for strings, the ideas of an earlier paper [Funct. Anal. Appl. 21, 126-142 (1987); translation from Funkts. Anal. Prilozh. 21, No.2, 46-63 (1987; Zbl 0634.17010)] are modified.
Reviewer: G. Czichowski

MSC:
17B68 Virasoro and related algebras
14H55 Riemann surfaces; Weierstrass points; gap sequences
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
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[1] I. M. Krichever and S. P. Novikov, ”Virasoro type algebras, Riemann surfaces and structures of the theory of solitons,” Funkts. Anal. Prilozhen.,21, No. 2, 46-63 (1987). · Zbl 0634.17010
[2] I. M. Gel’fand and D. B. Fuks, ”Cohomology of the Lie algebra of vector fields on a circle,” Funkts. Anal. Prilozhen.,2, No. 4, 92-93 (1968). · Zbl 0176.11501
[3] F. Schottky, ”Uber eine specielle Funktion, welche bei einer bestimmten linearen Transformation ihres Arguments unverandert bleibt,” J. Reine Angew. Math.,101, 227-272 (1887). · JFM 19.0424.02
[4] M. Schiffer and N. Hawley, ”Half-order differentials on Riemann surfaces,” Acta Math.,115, 199-236 (1966). · Zbl 0136.06701
[5] J. Schwartz, ”Superstring theory,” Phys. Rep.,89, 223-322 (1982). · Zbl 0578.22027
[6] C. Thorn, ”A detailed study of the physical state conditions in covariantly quantized string theories,” Preprint Inst. Theor. Phys., University of California (1986).
[7] É. I. Zverovich, ”Boundary problems of the theory of analytic functions,” Usp. Mat. Nauk,26, No. 1, 113-181 (1971). · Zbl 0217.10201
[8] B. L. Feigin and D. B. Fuks, ”Skew-symmetric invariant differential operators on the line and Verma modules over the Virasoro algebra,” Funkts. Anal. Prilozhen.,16, No. 2, 47-63 (1982). · Zbl 0493.46061
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