Equivariant quantum differential equation and $qKZ$ equations for a projective space: Stokes bases as exceptional collections, Stokes matrices as Gram matrices, and B-Theorem.
Equivariant quantum differential equation and $qKZ$ equations for a projective space: Stokes bases as exceptional collections, Stokes matrices as Gram matrices, and B-Theorem.
复制标题
射影空间的等变量子微分方程和 $qKZ$ 方程：作为异常集合的斯托克斯基、作为格拉姆矩阵的斯托克斯矩阵和 B 定理。
DOI：
--
复制
发表时间：
2019
期刊：
影响因子：
--
通讯作者：
A. Varchenko
中科院分区：
文献类型：
--
作者：
G. Cotti;A. Varchenko
在arXiv：1901.02990v1中考虑了射影空间的等变量子微分方程（$qDE$），并引入了相容的差分方程组$qKZ$;将$qDE$和$qKZ$方程的联合系统的解空间与射影空间的等变$K$-理论代数的空间等同起来;将解空间中的Stokes基与等变$K$-理论代数中的例外基等同起来。本文是arXiv：1901.02990v1的延续。
在本文中，我们描述的$qDE$和$qKZ$方程的联合系统的解决方案和拓扑枚举的解决方案之间的关系只有$qDE$，定义为一个生成函数的等变后裔Gromov-Witten不变量。这种关系是根据等变$K$-理论代数上的等变分次陈特征标、射影空间的等变Gamma类和射影空间切丛的第一类陈特征标建立的。
我们还表明，在解空间中，Stokes基上的Stokes矩阵等于等变Grothendieck-Euler-Poincare对的Gram矩阵与等变$K$-理论代数中相应的例外基的Gram矩阵.
在射影空间上的凝聚层的等变导出范畴中，我们确定了具有显式全例外集合的解空间中的Stokes基，其中这些例外集合的元素是射影空间上的线丛和射影空间的切丛的外幂.
这些陈述是G.科蒂，B。Dubrovin，D. Guzzetti和S. Galkin，V. Golyshev，H.入谷
In arXiv:1901.02990v1 the equivariant quantum differential equation ($qDE$) for a projective space was considered and a compatible system of difference $qKZ$ equations was introduced; the space of solutions to the joint system of the $qDE$ and $qKZ$ equations was identified with the space of the equivariant $K$-theory algebra of the projective space; Stokes bases in the space of solutions were identified with exceptional bases in the equivariant $K$-theory algebra. This paper is a continuation of arXiv:1901.02990v1.
In this paper we describe the relation between solutions to the joint system of the $qDE$ and $qKZ$ equations and the topological-enumerative solution to the $qDE$ only, defined as a generating function of equivariant descendant Gromov-Witten invariants. The relation is in terms of the equivariant graded Chern character on the equivariant $K$-theory algebra, the equivariant gamma class of the projective space, and the first Chern class of the tangent bundle of the projective space.
We also show that the Stokes matrix assigned to a Stokes basis in the space of solutions equals the Gram matrix of the equivariant Grothendieck-Euler-Poincare pairing wrt to the corresponding exceptional basis in the equivariant $K$-theory algebra.
We identify the Stokes bases in the space of solutions with explicit full exceptional collections in the equivariant derived category of coherent sheaves on the projective space, where the elements of those exceptional collections are just line bundles on the projective space and exterior powers of the tangent bundle of the projective space.
These statement are equivariant analogs of results of G. Cotti, B. Dubrovin, D. Guzzetti, and S. Galkin, V. Golyshev, H. Iritani.