Algebras of Virasoro type, energy-momentum tensor, and decomposition operators on Riemann surfaces
This is the third in a series of papers by the same authors [see ibid. 21, No.2, 46-63 (1987; Zbl 0634.17010)] and 21, No.4, 47-61 (1987; Zbl 0659.17012)] developing a program of the operator quantization of multiloop diagrams in the bosonic string theory. The approach departs from a twice pointed non-singular Riemannian surface \(\Gamma\) as an algebro-geometric model of a bosonic string; the fixed points \(P_{\pm}\) correspond to the conformal compactification of the string world sheet at \(t\to \pm \infty\) in the Minkowski space. The so-called `almost graded' central extensions of certain tensor algebras on \(\Gamma\) play a crucial role in the operator theory of interacting strings; they are analogues of the Virasoro and Heisenberg algebras. The{\S} 1 contains a reminder of the basic ideas in a `more appropriate for the sequel' form. Operator realization of a bosonic string in the Fock space \({\mathcal H}^{\pm}\) of Dirac fermions on \(\Gamma\) is discussed in the {\S}2. In the case of genus \(g>0\) the energy-impulse tensor proves to be ill- defined, and the {\S}3 is devoted to the introduction of its proper substitution, the energy-impulse `pseudotensor' on \(\Gamma\), which is defined invariantly and depends on the triple \(\Gamma\), \(P_+\), \(P_-\) only. The concluding {\S}4 sketches a program of extending the results presented beyond the bosonic sector of the closed string, via the BRST techniques.
scientific article; zbMATH DE number 4120365
scientific article