A simple construction of Grassmannian polylogarithms

Complete symplectic quadrics and Kontsevich spaces of conics in Lagrangian Grassmannians

A wonderful compactification of an orbit under the action of a semi-simple and simply connected group is a smooth projective variety containing the orbit as a dense open subset, and where the added boundary divisor is simple normal crossing. We construct the wonderful compactification of the space of symmetric and symplectic matrices, and investigate its geometry. As an application, we describe the birational geometry of the Kontsevich spaces parametrizing conics in Lagrangian Grassmannians.

Affine Grassmannians of group schemes and exotic principal bundles over A^1

Let G be a simple simply-connected group scheme over a regular local scheme U. Let E be a principal G-bundle over A^1_U trivial away from a subscheme finite over U. We show that E is not necessarily trivial and give some criteria of triviality. To this end we define affine Grassmannians for group schemes and study their Bruhat decompositions for semi-simple group schemes. We also give examples of principal G-bundles over A^1_U with split G such that the bundles are not isomorphic to pull-backs from U.

Quantum cohomology of the odd symplectic Grassmannian of lines

Odd symplectic Grassmannians are a generalization of symplectic Grassmannians to odd-dimensional spaces. Here we compute the classical and quantum cohomology of the odd symplectic Grassmannian of lines. Although these varieties are non homogeneous, we obtain Pieri and Giambelli formulas that are very similar to the symplectic case. We notice that their quantum cohomology is semi-simple, which enables us to check Dubrovin's conjecture for this case.

Equivariant Pieri Rules For Isotropic Grassmannians

We give a Pieri rule for the torus-equivariant cohomology of (submaximal) Grassmannians of Lie types B, C, and D. To the authors' best knowledge, our rule is the first manifestly positive formula, beyond the equivariant Chevalley formula. We also give a simple proof of the equivariant Pieri rule for the ordinary (type A) Grassmannian.

Torus quotient of Richardson varieties in Orthogonal and Symplectic Grassmannians

For any simple, simply connected algebraic group $G$ of type $B,C$ and $D$ and for any maximal parabolic subgroup $P$ of $G$, we provide a criterion for a Richardson variety in $G/P$ to admit semistable points for the action of a maximal torus $T$ with respect to an ample line bundle on $G/P$.

Window shifts, flop equivalences and Grassmannian twists

We introduce a new class of autoequivalences that act on the derived categories of certain vector bundles over Grassmannians. These autoequivalences arise from Grassmannian flops: they generalize Seidel-Thomas spherical twists, which can be seen as arising from standard flops. We first give a simple algebraic construction, which is well-suited to explicit computations. We then give a geometric construction using spherical functors which we prove is equivalent.

格拉斯曼流形在一维环面作用下的GIT商

On the GIT quotient of Grassmannians by one dimensional torus

本文研究格拉斯曼流形在对应单根的一维环面作用下的GIT商结构，利用外尔群组合刻画半稳定轨迹，分析轨道结构并计算相关几何不变量。

Minimal bandwidth $\mathbb{C}^*$-actions on generalized Grassmannians

The bandwidth of a $\mathbb{C}^*$-action of a polarized pair $(X,L)$ is a natural measure of its complexity. In this paper we study $\mathbb{C}^*$-actions on rational homogeneous spaces, determining which provide minimal bandwidth. We prove that the minimal bandwidth is linked to the smallest coefficient of the fundamental weight, in a base of simple roots, which describes the variety as a marked Dynkin diagram. As a direct application of the results we study the Chow ring of the Cayley plane $\mathrm{E}_6(6)$.

定义对称簇和仿射Grassmannian的方程

Equations defining symmetric varieties and affine Grassmannians

设$σ$是代数半单群$G$的单对合，$H$是$G$的由$σ$不动点构成的子群.如果限制根系是$A$、$C$或$BC$型的，并且$G$是单连通的，或者如果限制根系是$B$型的，并且$G$是伴随的，那么我们利用标准单项式理论和适当的（可能是无限维的）格拉斯曼簇的Plücker关系描述了坐标环$k[G/H]$的标准单项式理论和方程。

仿射Grassmannian中的仿射Demazure模和$T$ -不动点子格式

Affine Demazure modules and $T$-fixed point subschemes in the affine Grassmannian

设$G$是定义在$\C$上的$A$或$D$型单代数群，$T$是$G$的极大环面.对于$G$的支配余权$λ$，仿射格拉斯曼$Gr_G$中舒伯特簇$\bar{Gr}_G^λ$的$T$ -不动点子格式$(\bar{Gr}_G^λ)^T$是有限格式.证明了$λ$的一级仿射Demazure模的对偶与$(\bar{Gr}_G^λ)^T$的函数环（由$Gr_G$上的某个线丛扭曲）之间存在自然同构.利用这一事实给出了$A,D,E$型仿射代数的基本表示与格顶点代数之间的Frenkel-Kac-Segal同构的几何证明。

关于单李代数的Cachazo-Douglas-Seiberg-Witten猜想

On Cachazo-Douglas-Seiberg-Witten conjecture for simple Lie algebras

利用拓扑学和几何学的方法对Cachazo-Douglas-Seiberg-Witten猜想的一个重要部分给出了统一的证明。

Scattering Amplitudes and the Positive Grassmannian

We establish a direct connection between scattering amplitudes in planar four-dimensional theories and a remarkable mathematical structure known as the positive Grassmannian. The central physical idea is to focus on on-shell diagrams as objects of fundamental importance to scattering amplitudes. We show that the all-loop integrand in N=4 SYM is naturally represented in this way. On-shell diagrams in this theory are intimately tied to a variety of mathematical objects, ranging from a new graphical representation of permutations to a beautiful stratification of the Grassmannian G(k,n) which gene…

Iterated Elliptic and Hypergeometric Integrals for Feynman Diagrams

We calculate 3-loop master integrals for heavy quark correlators and the 3-loop QCD corrections to the $ρ$-parameter. They obey non-factorizing differential equations of second order with more than three singularities, which cannot be factorized in Mellin-$N$ space either. The solution of the homogeneous equations is possible in terms of convergent close integer power series as $_2F_1$ Gauß hypergeometric functions at rational argument. In some cases, integrals of this type can be mapped to complete elliptic integrals at rational argument. This class of functions appears to be the next one ari…