Integrable Structure in Quantum Cohomology

Givental-Kim（Comm.Math Phys.1995）在有限维全旗流形上发现了户田格的量子上同调现象，而M.A.Guest（Comm.Math.Phys.217，475 - 487，2001）在无限维（即仿射）旗流形上也发现了类似的现象.在那篇文章中，我们只利用有限维旗流形上有理曲线的模空间的几何性质，导出了量子上同调中因子类的关系，而没有涉及到无限维仿射旗流形的几何性质，如果我们试图研究这类流形上的量子Schubert演算，就需要研究无限维几何性质。我们希望有钥匙来研究无限维流形中曲线的模空间，并决定从辛几何中学习工具。Hofer-Z ...更多信息 Ehnder容量看起来值得研究，因为它的推广到一些无限维辛流形最近被Kuksin等人考虑.在获得的基础aflfine旗流形的量子上同调，我们认为，我们应该研究它的代数结构假设的基本性质，应预期相比，有限维情况下。我们发现某些仿射旗流形的量子积不满足势性质（即，量子乘积的结构常数由某个生成函数的三阶导数给出），这对于有限维辛流形是成立的。除此之外，我们还研究了仿射旗流形的量子上同调的D-模结构（与A. L. Mare和M. A. Guest的工作正在进行中），我们观察到我们可以在这种量子上同调的小子空间上引入适当的D-模结构。然而，我们既没有得到一个线索，以扩大到一个更大的子空间，也没有几何的理解，上述麻烦。少

Our first motivation for the present research project was to understand why the integrable system, the Toda lattice, appears in the quantum cohomology of flag manifolds.For the finite dimensional full flag manifolds, this phenomenon was found by Givental-Kim (Comm.Math Phys.1995), and an analogous one was observed for the infinite dimensional (i.e.affine) flag manifolds by the head investigator with M.A.Guest (Comm.Math.Phys.217,475-487,2001). In that work we derived the relations for divisor classes in the quantum cohomology, using only the geometry of the moduli space of rational curves in finite dimensional flag manifolds, and did not touch the infinite dimensional geometry of affine flag manifolds essentially.If we try to investigate quantum Schubert calculus for such manifolds, we need to look into the infinite dimensional geometry. We wanted to have keys to study moduli spaces of curves in infinite dimensinal manifolds, and decided to learn tools from symplectic geometry. Hofer-Z … More ehnder capacity looked worth studying, because its generalization to some infinite dimensional symplectic manifolds is considered recently by Kuksin and others.Before obtaining the foundation for quantum cohomology of aflfine flag manifolds, we thought that we should investigate its algebraic structure assuming basic properties that should be expected in comparison with the finite dimensional case. We found that the quantum product for some affine flag manifolds does not satisfy the potential property (i.e., the structure constantsfor quantum product are given by third derivatives of certain generating function), which is known to hold for finite dimensional symplectic manifolds. Besides this, we worked on D-module structures on quantum cohomology of affine flag manifolds (a work with A.L.Mare and M.A.Guest, in progress).We observed that we can introduce an appropriate D-module structure on a small subspace of such quantum cohomology. However we have obtained neither a clue to extend it to a larger subspace, nor a geometric understanding of the trouble above. Less