Kac行列式与Ding-Iohara-Miki代数N级表示中的奇异向量

Ding-Iohara-Miki代数中的反射态与D型流形的膜网

Reflection states in Ding-Iohara-Miki algebra and brane-web for D-type quiver

在Ding-Iohara-Miki（DIM）代数（量子环形$\mathfrak{gl}_1$）的垂直和水平模块中引入反射态。DIM表示的网对应于IIB型弦论的$(p,q)$ -网图，根据Awata，Feigin和Shiraishi的代数交织器与改进的拓扑顶点的识别。将对应关系扩展到垂直反射态，就有可能设计D型（具有么正规范群）的$\mathcal{N}=1$量子规范理论。这样，Nekrasov瞬子配分函数就从缠绕子的期望值的估计中再现出来。这种计算导致了垂直反射态与弦论的定向折叠平面的一致性。我们还提供了一个翻译的Iqbal-Kozcaz-Vafa细化拓扑顶点形式主义的建设。

关于Ding-Iohara代数和AGT猜想的注记

Notes on Ding-Iohara algebra and AGT conjecture

研究了Ding-Iohara代数$\calU$的表示理论，找到了Alday-Gaiotto-Tachikawa（AGT）关系的$q$ -类似.引入了Ding-Iohara代数的自同态$T(u,v)$，它有两个参数$u$和$v$ .我们定义了顶点算子$Φ(w)$，通过指定与Ding-Iohara生成元$x^\pm(z)$和$ψ^\pm(z)$的置换关系，在$T(u,v)$。对于一级表示，顶点算子相对于麦克唐纳多项式的所有矩阵元素被因式分解，并按照AGT关系中的$K$理论配分函数的Nekrasov因子来写。对于更高水平的$m=2,3,...$，我们提出了一些示意图，这意味着存在的$q$ -类似物的AGT关系。

自由场对Ding-Iohara代数在一般层次上的实现

Free field realization of the Ding-Iohara algebra at general levels

本文提出一种统一的自由场实现方法，用于Ding-Iohara代数在任意层次上的表示，满足广义的Serre关系。通过六自由玻色场构造该实现，并基于此发展了Ding-Iohara代数的 intertwining 操作符。

A dynamical analogue of Ding-Iohara quantum algebras

We introduce a family of dynamical Hopf algebroids $U_{q,p}(g,X_l)$ depending on a complex parameter $q$, a formal parameter $p$, a set $g$ of structure functions satisfying the so-called Ding-Iohara condition, and a finite root system of type $X_l$. If $g$ is set to be certain theta functions, then our family recovers the elliptic algebras $U_{q,p}(\widehat{\mathfrak{g}})$ for untwisted affine Lie algebras $\widehat{\mathfrak{g}}$ studied by Konno (1998, 2009), Jimbo-Konno-Odake-Shiraishi (1999) and Farghly-Konno-Oshima (2014). Also, taking the limit $p \to 0$ in the case $X_l=A_l$, we recove…

M-弦的Awata-Feigin-Shiraishi型椭圆顶点

An Elliptic Vertex of Awata-Feigin-Shiraishi type for M-strings

给出了椭圆型Ding-Iohara-Miki代数的一个垂直表示，并构造了Awata，Feigin和Shiraishi的精化拓扑顶点的椭圆型表示.我们明确表明，这个顶点再现椭圆亏格的M-弦，它是一个交织的代数。

The MacMahon R-matrix

We introduce an $R$-matrix acting on the tensor product of MacMahon representations of Ding-Iohara-Miki (DIM) algebra $U_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_1)$. This $R$-matrix acts on pairs of $3d$ Young diagrams and retains the nice symmetry of the DIM algebra under the permutation of three deformation parameters $q$, $t^{-1}$ and $\frac{t}{q}$. We construct the intertwining operator for a tensor product of the horizontal Fock representation and the vertical MacMahon representation and show that the intertwiners are permuted using the MacMahon $R$-matrix.

Generalized Macdonald Functions on Fock Tensor Spaces and Duality Formula for Changing Preferred Direction

An explicit formula is obtained for the generalized Macdonald functions on the $N$-fold Fock tensor spaces, calculating a certain matrix element of a composition of several screened vertex operators. As an application, we prove the factorization property of the arbitrary matrix elements of the multi-valent intertwining operator (or refined topological vertex operator) associated with the Ding--Iohara--Miki algebra (DIM algebra) with respect to the generalized Macdonald functions, which was conjectured by Awata, Feigin, Hoshino, Kanai, Yanagida and one of the authors. Our proof is based on the …

DIM代数中的交换族、可积多体系统与$q,t$矩阵模型

Commutative families in DIM algebra, integrable many-body systems and $q,t$ matrix models

本文将$W_{1+\infty}$代数的交换子代数推广到DIM代数，利用Miki自同构统一处理交换射线，在多体和Fock表示中推导出生成元$e_{n,m}$的显式公式，并讨论相关矩阵模型的$q,t$-形变。