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Geometric realization of conformal field theory on Riemann surfaces. (English) Zbl 0648.35080
This paper includes a self-contained account of the relation between the soliton theory (KP hierarchy), Grassmann manifolds and conformal field theory on Riemann surfaces.
The geometry of the (dressed) moduli space of Riemann surfaces and its embedding to the universal Grassmann manifold are described in algebro-geometrical way.
The theory of KP hierarchy, including the bosonization formalism, is completely reviewed. The vacuum state are given as the $$\tau$$-function for the compact solution (essentially Riemann’s theta function).
The correlation functions and the Ward identities are described. As the main result of this paper, it is proved that the Ward identities and some automorphy properties uniquely characterize the $$\tau$$-function.
Reviewer: N.Kawamoto

##### MSC:
 35Q51 Soliton equations 35A30 Geometric theory, characteristics, transformations in context of PDEs 58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds 81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics 37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
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