Geomatric structure of Solvable Models

我们主要研究二维可积量子场论的几何结构。期望在零能级的KZ方程是高斯-曼宁系统的一个子系统，它与一些代数曲线族有关。事实上，在与S/N相关的方程中，这是正确的。在这些情况下，为了研究可解模型的结构，代数曲线的黎曼关系应该发挥重要作用。我们得到了关于扭曲同调与上同调之间代数曲线的黎曼关系的一些有趣的结果。我们还期望KZ方程在零能级的解可以用常数表示。如果我们通过引入来自雅可比变量变形的新参数来变形这些常数，我们可能会发现二维可积量子场论与相应的经典可积系统之间的某种关系。本研究与模形式、常数、阿贝尔积分及其经典关系密切相关。虽然我们还不能得到明确的结果，但是我们在大量纲理性品种的Fano品种上得到了一些结果，这可能与这一领域有关，希望对这一领域有所贡献。作为我们研究的下一个阶段，我们必须在代数几何的基础上进一步研究二维可积量子场论的可解模型的结构。

We mainly study Geometric structure of two dimensional integrable quantumn field theory.It is expected that KZ equation at level zero should be a subsystem of Gauss-Maninn system associated with some family of algebraic curves. In fact, it is true in the case of equations associated with S/N.In these cases, the Riemann relations of algebraic curves should play essential roles in order to study the structure of solvable models. We obtain some interesting results concerning with the Riemann relations of algebraic curves between the twisted homologies and cohomologies.It is also expected that the solutions of KZ equation at level zero can be expressed in terms of theta constants. If we deform these theta constants by introducing new parameters coming from the deformation of Jacobi varieties, we possibly find some relation between two dimensional integrable quantum field theory and the corresponding classical integrable systems. This study is closely related to modular forms, theta constants, Abel integrals and their classical relations. Though we cannot get any definite results yet, we get some results on Fano varieties with large dimensional rational varieties, which may have something to do with this field, and hope to contribute to these areas.As the next stage of our study, we must further investigate the structure of solvable models of two dimensional integrable quantum field theory on a basis of results of algebraic geometry such as ones we obtained.