Toric Degenerations and the Laurent polynomials related to Givental's Landau-Ginzburg models

箭图旗零轨迹的洛朗多项式镜像

Laurent polynomial mirrors for quiver flag zero loci

本文针对高维法诺簇分类问题，将Gelfand–Cetlin退化推广到Y形箭图旗簇，提出构造法诺箭图旗零轨迹洛朗多项式镜像的方法，并验证了99个四维情形的猜想镜像。

On Landau-Ginzburg models for quadrics and flat sections of Dubrovin connections

This paper proves a version of mirror symmetry expressing the (small) Dubrovin connection for even-dimensional quadrics in terms of a mirror-dual Landau-Ginzburg model (Xcan,W). Here Xcan is the complement of an anticanonical divisor in a Langlands dual quadric. The superpotential W is a regular function on Xcan and is written in terms of coordinates which are naturally identified with a cohomology basis of the original quadric. This superpotential is shown to extend the earlier Landau-Ginzburg model of Givental, and to be isomorphic to the Lie-theoretic mirror introduced by Rietsch. We also i…

On parabolic Whittaker functions II

We derive a Givental-type stationary phase integral representation for the specified $\Gr_{m,N}$-Whittaker function introduced in \cite{GLO2}, which presumably describes the $S^1\times U_N$-equivariant Gromov-Witten invariants of Grassmann variety $\Gr_{m,N}$. Our main tool is a generalization of Whittaker model for principal series $\CU(\frak{gl}_N)$-modules. In particular, our construction includes a representation theory interpretation of the Batyrev--Ciocan-Fontanine--Kim--van Straten toric degeneration of Grassmannian, providing a direct connection between this toric degeneration of $\Gr_…

Mutation equivalence of toric Landau-Ginzburg models

Given a Fano complete intersection defined by sections of a collection nef line bundles $L_1,\ldots, L_c$ on a Fano toric manifold $Y$, a construction of Givental/Hori-Vafa provides a mirror-dual Landau-Ginzburg model. This construction depends on a choice of suitable nef partition; that is, a partition of the rays of the fan determined by $Y$. We show that toric Landau-Ginzburg models constructed from different nef partitions representing the same complete intersection are related by a volume preserving birational map. In particular, various Laurent polynomial mirrors which may be obtained fr…

C型Richardson边界与奇维射影空间的Newton-Okounkov退化

Type C Richardson Boundaries and Newton-Okounkov Degenerations of Odd-Dimensional Projective Space

该研究对比奇维射影空间的两种镜像理论退化图景，计算C型边界与对应Laurent多项式，构造Newton–Okounkov体并验证半群等同性，还给出连接两类边界的秩一权退化。

SAGBI基与球面簇退化为环面簇

SAGBI bases and Degeneration of Spherical Varieties to Toric Varieties

设X \subset Proj（V）是投射球面G-簇，其中V是有限维G-模，G = SP（2n，C）.本文证明了X可以通过平面变形而变形为对应于凸多胞体Δ（X）的复曲面簇。多胞形Δ（X）是以Gelfand-Cetlin多胞形为纤维的X的矩多胞形上的多胞形。我们证明了这一点，如果X是一个horosphere品种，例如旗品种和Grassmanians，齐次坐标环的X可以嵌入在一个Laurent多项式代数，并有一个SAGBI基础上的自然项顺序。此外，我们还证明了初始项的半群在变量作线性变量变换后，是多胞形Δ（X）上的锥上的整点的半群.本文的结果对其他典型群也成立，只要A. Okounkov关于SP（2n，C）的表示论的结果对其他典型群也成立.

Laurent多项式空间、$\mathbb{P}^1$ -orbifolds空间和可积族

The spaces of Laurent polynomials, $\mathbb{P}^1$-orbifolds, and integrable hierarchies

设$M_{k,m}$为单变量Laurent多项式空间$x^k + t_1 x^{k-1}+... t_{k+m}x^{-m},$，其中$k,m\geq 1$为固定整数，$t_{k+m}\neq 0$为固定整数.根据B. Dubrovin \cite{D}、$M_{k,m}$可以配备半简单的Frobenius结构。本文证明了$M_{k,m}$（由A. Givental）满足Hirota二次方程（简称HQE）。假设$\mathcal{C}_{k,m}$是从$\mathbb{P}^1$获得的眶褶，通过切割围绕PDMATHAMV和PDMATHAMW的小圆盘PDMATHAMT和$D_2\simeq\{|z^{-1}|\leq ε\}$，并以明显的方式将眶褶$D_1/\mathbb{Z}_k$和$D_2/\mathbb{Z}_m$粘回。证明了$\mathcal{C}_{k,m}$的轨道量子上同调与$M_{k,m}$作为Frobenius流形是一致的.模一些尚未被澄清的细节，这意味着后代（分别祖先）潜在的$M_{k,m}$是一个生成函数的后代（分别祖先）orbifold格罗莫夫-威滕不变量的$\mathcal{C}_{k,m}$。我们的HQE与G. PDMATHAMBMZ中的Carlet。因此，我们的HQE描述了这个层次的τ-函数，我们期望扩展的双阶Toda层次支配$\mathcal{C}_{k…

量子周期、 toric 退化与内禀镜像对称性

Quantum periods, toric degenerations and intrinsic mirror symmetry

本文证明了 Fano 簇 $X$ 的正则化量子周期可由仿射对数 Calabi-Yau 簇 $U$ 的内禀镜像代数元素的经典周期给出，推导出 Fano 镜像对称性的推论，并构造了相关多面体族与 toric 退化，明确了格拉斯曼流形的超势。

Hirzebruch Surfaces, Tyurin Degenerations and Toric Mirrors: Bridging Generalized Calabi-Yau Constructions

There is a large number of different ways of constructing Calabi-Yau manifolds, as well as related non-geometric formulations, relevant in string compactifications. Showcasing this diversity, we discuss explicit deformation families of discretely distinct Hirzebruch hypersurfaces in $\mathbb{P}^n \times \mathbb{P}^1$ and identify their toric counterparts in detail. This precise isomorphism is then used to investigate some of their special divisors of interest, and in particular the secondary deformation family of their Calabi-Yau subspaces. In particular, most of the above so called Hirzebruch…

A comparison of Landau-Ginzburg models for odd dimensional Quadrics

In [Rie08], the second author defined a Landau-Ginzburg model for homogeneous spaces G/P. In this paper, we reformulate this LG model in the case of the odd-dimensional quadric X=Q_{2m-1}. Namely we introduce a regular function Wcan on a variety Xcan x C*, where Xcan is the complement of a particular anticanonical divisor in the the projective space CP^{2m-1}=P(H*(X,C)*). Firstly we prove that the Jacobi ring associated to Wcan is isomorphic to the quantum cohomology ring of the quadric, and that this isomorphism is compatible with the identification of homogeneous coordinates on Xcan with ele…

Grassmannian Schubert 簇的超势

A superpotential for Grassmannian Schubert varieties

本文为奇异的 Grassmannian Schubert 簇构造推广 Marsh-Rietsch 超势的 $W^λ$，并借助 cluster seed 与 Newton-Okounkov 体证明多面体镜像定理及环面退化和部分 Fano 解消。