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Algebras of Virasoro type, energy-momentum tensor, and decomposition operators on Riemann surfaces. (English) Zbl 0684.17012

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 ${P}_{±}$ correspond to the conformal compactification of the string world sheet at $t\to ±\infty$ 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 ${ℋ}^{±}$ 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

##### MSC:
 17B65 Infinite-dimensional Lie (super)algebras 81Q30 Feynman integrals and graphs; applications of algebraic topology and algebraic geometry 81S10 Geometric quantization, symplectic methods (quantum theory) 81T60 Supersymmetric field theories 30F99 Riemann surfaces (one complex variable) 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
##### References:
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