FORMULATION OF THE FOUR DIMENSIONAL CONFORMAL FIELD THEORY BASED ON THE MODULI OF STABLE SHEAVES ON ALGEBRAIC SURFACES

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The purpose of the present research has been to establish a mathematical theory which extends the two dimensional conformal field theory to four dimension. We adopted as our model the four dimensional Wess-Zumino-Witten theory. The biggest results was that we gave a mathematically rigorous definition of the space of conformal blocks and computed their dimension in the case of Hirzebruh surfaces. It has been achieved by constructing the determinant line bundle on the Gieseker compactification of the moduli of stable bundles on an algebraic surface. The space of conformal blocks is defined to be the space of global section of the line bundle. Although the relation of the space with representation theory was not clarified sufficiently, in the course of our research we obtained several results which fall in two categories.The first category concerns with the existence of stable sheaves and the geometry of their moduli spaces on an algebraic surface. We introduced the concept of stable bund … More les of degree one and in the case of regular surfaces determined the condition for their existence and the birational types of their moduli spaces. We also proved an existence theorem for stable bundles with the first Chern class zero (I.e. instantons) on a K3 surface by a deformation theoretic method. By the same method we clarified the relationship of Mukai's reflection functor and the T-duality of K3 surfaces which appears in string theory.The second category treats vector bundles on higher dimensional varieties. For varieties defined over a field of positive characteristic, we obtained an effective lower bound for the degree of divisors for which the stability of a bundle is preserved under restriction. It follows that the restriction map induces an em bedding of the moduli of stable sheaves into the moduli of sheaves on a divisor. We also studied the geometry of stable bundles on varieties which has a fibration over a curve and proved that the quantum cohomology of their moduli spaces can be identified with the Gromov-Witten invariant of the product with a curve. Less