Geomatric structure of Solvable Models

我们主要研究二维可积量子场论的几何结构，期望零级KZ方程是伴随着某些代数曲线族的Gauss-Maninn系统的一个子系统。事实上，对于与S/n有关的方程也是如此，在这种情况下，为了研究可解模型的结构，代数曲线的黎曼关系应该起到至关重要的作用。我们得到了一些关于扭曲同调和上同调之间的代数曲线的Riemann关系的有趣结果，也期望KZ方程在零点处的解可以用theta常数来表示。如果我们通过引入来自Jacobi系形变的新参数来形变这些theta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联系。这项研究与模形式、西塔常数、Abel积分及其经典关系密切相关。虽然我们还不能得到任何确定的结果，但我们得到了一些关于大维有理变种的Fano簇的结果，这可能与这一领域有关，并希望对这些领域做出贡献。作为我们下一阶段的研究，我们必须在已有的代数几何结果的基础上，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.