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

基于代数面上稳定滑轮模的四维共形场理论的表述

基本信息

批准号：
09640053
负责人：
NAKASHIMA Tohru
金额：
$ 1.86万
依托单位：
TOKYO METOROPOLITAN UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1999
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640053/
关键词：
CONFORMAL FIELD THEORY STABLE VECTOR BUNDLE MODULI SPACE K3曲面カラビーヤウ多様体

项目摘要

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

本研究的目的是建立一个将二维共形场论推广到四维的数学理论。我们采用四维Wess-Zumino-维滕理论作为我们的模型。最大的结果是，我们给出了一个严格的数学定义的空间的共形块，并计算其尺寸的情况下，Hirzebruh曲面。它是通过构造代数曲面上稳定丛模的Gieseker紧化上的行列式线丛而实现的。共形块空间定义为线丛的整体截空间。虽然空间与表示论的关系还没有充分阐明，但在我们的研究过程中，我们得到了两类结果：第一类涉及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 ...更多信息在正则曲面的情况下，一次和二次模空间确定了它们存在的条件和它们的模空间的双有理类型。利用形变论的方法证明了K3曲面上第一类为零的稳定丛（即瞬子）的存在性定理。用同样的方法，我们阐明了Mukai的反射函子与弦论中出现的K_3曲面的T-对偶的关系。第二类是处理高维簇上的向量丛。对于定义在正特征域上的簇，我们得到了在约束条件下保持丛稳定性的因子次数的有效下界。由此可见，限制映射将稳定层的模嵌入到因子上的层的模中。我们还研究了在曲线上具有纤维化的簇上的稳定丛的几何，并证明了它们的模空间的量子上同调可以用与曲线的乘积的Gromov-Witten不变量来表示.少