Representation theory of vertex operator algobras and contormal tield theory

通过建立协不变量束和共形块束的概念，给出了属紧曲线上的共形场论，并研究了它们的数学结构。我们分析这些物体的工具之一是包含简并情况的椭圆曲线变形理论。事实上，我们已经使用了亚纯函数空间的结构理论，该空间只在原点处具有可能的奇点。我们研究的第一步是定义在协变量捆上的连接的规范构造。这些联系描述了轮轴对椭圆曲线变形参数的依赖关系。那么Weierstrass的zeta函数自然会成为这个结构的关键成分之一。其中最重要的问题之一是共形砌块的水平部分的施工。我们也用变形理论来解微分方程，我们发现任何解都可以从与顶点算子代数相关的代数上的线性泛函中得到。因此，我们能够用有限维代数的轨迹来表示任何水平截面。更一般地说，水平部分的空间与。代数上的线性泛函空间。

We have formulated conformal field theory over genus one compact curve by establishing notions of sheaves of coinvariants and sheaves of conformal blocks, and have studied their mathematical structures. One of our tools to analyze these objects is the deformation theory of elliptic curve including the degenerate case. In fact we have used the structure theory of the space of meromorphic functions over elliptic curves with possible singularities only at the origin. The first step of our research is a canonical construction of connections defined on sheaves of coinvariants. These connections describe the dependence of sheaves on the deformation parameter of elliptic curves. Then Weierstrass' zeta function naturally appears as one of key ingredients in this construction. One of the most important problems is the construction of horizontal sections of sheaves of conformal blocks. We also have used the deformation theory to solve the differential equations and we have found that any solutions can be obtained from linear functional over an algebra which is associated to a vertex operator algebra. Consequently we are able to express any horizontal section in terms of traces of the finite-dimensional algebra. More generally there is a one to one correspondence between the space of horizontal sections and. the space of linear functional over the algebra.