Quasimaps, straightening laws, and quantum cohomology for the Lagrangian Grassmannian

Double $J$-function of stable quasimap invariants for complete intersection in Grassmannian

Using localization methods on the moduli space of stable quasimaps to Grassmannian, we give explicit formulas of $J$-function and double $J$-function of stable quasimaps for complete intersection in Grassmannian.

格拉斯曼流形的动机拟映射壁穿越

Motivic quasimap wall-crossing for Grassmannians

研究格拉斯曼流形的动机拟映射壁穿越，通过定义对称函数环的\(\mathbb{Q}\)-代数自同构给出公式，其\(\varepsilon \to 0\)极限有特定交换，利用相对商概型几何及对称函数理论证明壁穿越公式。

Quantum K-theory of grassmannians and non-abelian localization

In the example of complex grassmannians, we demonstrate various techniques available for computing genus-0 K-theoretic GW-invariants of flag manifolds and more general quiver varieties. In particular, we address explicit reconstruction of all such invariants using finite-difference operators, the role of the q-hypergeometric series arising in the context of quasimap compactifications of spaces of rational curves in such varieties, the theory of twisted GW-invariants including level structures, as well as the Jackson-type integrals playing the role of equivariant K-theoretic mirrors.

The Abelian-Nonabelian Correspondence for $I$-functions

We prove the abelian-nonabelian correspondence for quasimap $I$-functions. That is, if $Z$ is an affine l.c.i. variety with an action by a complex reductive group $G$, we prove an explicit formula relating the quasimap $I$-functions of the GIT quotients $Z//_θ G$ and $Z//_θ T$ where $T$ is a maximal torus of $G$. We apply the formula to compute the $J$-functions of some Grassmannian bundles on Grassmannian varieties and Calabi-Yau hypersurfaces in them.

Semiinfinite Flags. II. Local and Global Intersection Cohomology of Quasimaps' Spaces

For a simple algebraic group $G$ we study the space $Q$ of Quasimaps from the projective line $C$ to the flag variety of $G$. We prove that the global Intersection Cohomology of $Q$ carries a natural pure Tate Hodge structure, and compute its generating function. We define an action of the Langlands dual Lie algebra $g^L$ on this cohomology. We present a new geometric construction of the universal enveloping algebra $U(n^L_+)$ of the nilpotent subalgebra of $g^L$. It is realized in the Ext-groups of certain perverse sheaves on Quasimaps' spaces, and it is equipped with a canonical basis number…

A sagbi basis for the quantum Grassmannian

The maximal minors of a p by (m + p) matrix of univariate polynomials of degree n with indeterminate coefficients are themselves polynomials of degree np. The subalgebra generated by their coefficients is the coordinate ring of the quantum Grassmannian, a singular compactification of the space of rational curves of degree np in the Grassmannian of p-planes in (m + p)-space. These subalgebra generators are shown to form a sagbi basis. The resulting flat deformation from the quantum Grassmannian to a toric variety gives a new `Gröbner basis style' proof of the Ravi-Rosenthal-Wang formulas in qua…