an:04120365
Zbl 0684.17012
Krichever, I. M.; Novikov, S. P.
Algebras of Virasoro type, energy-momentum tensor, and decomposition operators on Riemann surfaces
EN
Funct. Anal. Appl. 23, No. 1, 19-33 (1989); translation from Funkts. Anal. Prilozh. 23, No. 1, 24-40 (1989).
00180322
1989
j
17B65 81Q30 81S10 81T60 30F99 17B67
normal ordering; almost-graded algebras; operator quantization; bosonic string; Riemannian surface; Fock space; Dirac fermions
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.
V.Pestov
Zbl 0634.17010; Zbl 0659.17012