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

本研究的目的是建立一种将二维共形场理论扩展到四维的数学理论。我们采用了四维Wess-Zumino-Witten理论作为我们的模型。最大的结果是，我们给出了共形块空间的严格的数学定义，并在Hirzebreh曲面的情况下计算了它们的维度。这是通过构造代数曲面上稳定丛的模的Gieseker紧化上的行列式线丛来实现的。将共形块空间定义为线丛的整体截面空间。虽然空间与表示理论的关系没有得到充分的阐明，但在我们的研究过程中，我们得到了两类结果。第一类是关于代数曲面上稳定层的存在性及其模空间的几何。我们引入了稳定外滩…的概念更多的一次大涡模拟，在规则曲面的情况下，决定了它们存在的条件和它们的模空间的双调类型。我们还用形变理论的方法证明了K3曲面上具有第一类零(即瞬子)的稳定丛的一个存在定理。用同样的方法，我们阐明了Mukai反射函子与弦理论中出现的K3曲面的T-对偶之间的关系。第二类讨论高维簇上的向量丛。对于定义在正特征域上的簇，我们得到了在约束下保持丛稳定性的因子次数的一个有效下界。由此推论，限制映射将稳定层的模导成除数上的层的模。我们还研究了在曲线上有纤化的簇上的稳定丛的几何，证明了它们的模空间的量子上同调可以用曲线乘积的Gromov-Witten不变量来确定。较少

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