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Algebras of Virasoro type, energy-momentum tensor, and decomposition operators on Riemann surfaces. (English. Russian original) Zbl 0684.17012
Funct. Anal. Appl. 23, No. 1, 19-33 (1989); translation from Funkts. Anal. Prilozh. 23, No. 1, 24-40 (1989).
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§ 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 §2.
In the case of genus \(g>0\) the energy-impulse tensor proves to be ill- defined, and the §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 §4 sketches a program of extending the results presented beyond the bosonic sector of the closed string, via the BRST techniques.
Reviewer: V.Pestov

17B65 Infinite-dimensional Lie (super)algebras
81Q30 Feynman integrals and graphs; applications of algebraic topology and algebraic geometry
81S10 Geometry and quantization, symplectic methods
81T60 Supersymmetric field theories in quantum mechanics
30F99 Riemann surfaces
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
Full Text: DOI
[1] I. M. Krichever and S. P. Novikov, ”Virasoro type algebras, Riemann surfaces, and structures of soliton theory,” Funkts. Anal. Prilozhen.,21, No. 2, 46-63 (1987). · Zbl 0634.17010
[2] I. M. Krichever and S. P. Novikov, ”Virasoro type algebras, Riemann surfaces, and strings in Minkowsky space,” Funkts. Anal. Prilozhen.,21, No. 4, 47-61 (1987). · Zbl 0659.17012
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[9] B. A. Dubrovin, ”Theta-functions and nonlinear equations,” Usp. Mat. Nauk,36, No. 2, 11-80 (1981). · Zbl 0478.58038
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