Virasoro-type algebras, Riemann surfaces and strings in Minkowski space
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.
scientific article; zbMATH DE number 4077503
scientific article