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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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##### References:
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