Virasoro-type algebras, Riemann surfaces and strings in Minkowsky space

dk with two simple poles at the points P±, with residues ±1 and purely imaginary periods with respect to all contours on . The real part of the corresponding integral (z) = Rek(z) is single-valued on and represents “time.” The level lines (z) = const represent the positions of the string at the present time. To the collection m = m+ + m of strings corresponds a Riemann surface with two collections of points P+,i,P ,j, i = 1,...,m+, j = 1,...,m with a dierential dk, with real residues c+i,c j at all points P±, c+ > 0, c < 0, and purely imaginary periods on . In exactly the same way the function (z) = Rek(z) is single-valued and plays the role of “time”. As ! ±1 the contours T = const split into free strings. The connected components concentrated near the points P± play the role of asymptotically free “in” and “out” strings. In [1] a rich collection of algebraic objects connected with this picture for m = 1 was constructed, which for genus g = 0 reduce to the theory of the Virasoro algebra and its representations. In the present paper we demonstrate that these algebraic forms arise in the process of quantization of strings on such algebrogeometric models, “diagrams”. To the asymptotic “in” and “out” states correspond the ordinary Fock spaces of free strings—small contours T = const near the points P±. The global algebrogeometric objects on a surface with distinguished points P±. permit one in principle to trace the whole course of the interaction. The algebrogeometric objects in the theory of Polyakov, Belavin, Knizhnik, etc., of strings in Euclidean space-time, as is known, lead to problems on the space of moduli of Riemann surfaces. The algebraic forms we introduce do not appear in this theory. But to the minds of the authors the approach we develop relates the algebrogeometric theory of strings with traditional ideas of operator quantum theory of strings in Minkowsky space and lets us use the mathematical techniques of the method of finite-zone integration in the theory of solitons.