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Discrete Quantum Integrability, Quantum Q-Systems, and Generalized Macdonald Operators

离散量子可积性、量子 Q 系统和广义麦克唐纳算子

基本信息

批准号：
1802044
负责人：
Rinat Kedem
金额：
$ 26.5万
依托单位：
University of Illinois at Urbana-Champaign
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-05-01 至 2023-04-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1802044&HistoricalAwards=false
关键词：
Discrete Quantum Integrability Systems Generalized

项目摘要

This research project investigates questions at the intersection of mathematics and theoretical physics. The project is focused on questions that originate in statistical mechanics, which describes the average behavior of systems with large numbers of components. The algebraic study of these systems, and their symmetries, leads to new algebras, whose structure is in turn encoded combinatorially by collections of functions that arise as solutions to systems of equations. These new algebras, their generalizations, and the resulting combinatorics are the main topics to be studied. The research will be conducted in association with students at the graduate and undergraduate level, and will enhance our understanding of how these fields of study connect to each other.More precisely, the project concerns graded tensor products of cyclic current algebra modules, specifically Kirillov-Reshetikhin modules. The combinatorial framework is the quantum cluster algebra of the associated Q-systems, the solutions of which can be expressed in terms of plane tilings and non-intersecting path models. By investigating the discrete integrability of these systems, the investigators expect to produce conserved quantities, so far achieved only in type A. In turn, these should lead to difference equations for the graded characters of tensor product product, which generalize the difference Toda equations. Moreover, the solutions of these equations can be expressed as iterated action of difference creation operators, closely related to generalized Macdonald operators defined within the context of polynomial representations of double affine Hecke algebras. Relations to quantum affine and elliptic Hall algebras will also be investigated, as well as more combinatorial questions such as the existence of some (quantum) non-commutative counterpart to the Gessel-Viennot determinant for non-intersecting path enumeration.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

这个研究项目调查数学和理论物理交叉的问题。该项目的重点是起源于统计力学的问题，它描述了具有大量组件的系统的平均行为。对这些系统及其对称性的代数学研究导致了新的代数，其结构又由作为方程组的解而出现的函数集合组合编码。这些新的代数，它们的推广，以及由此产生的组合学是要研究的主要课题。该研究将与研究生和本科生一起进行，并将增强我们对这些研究领域如何相互联系的理解。更确切地说，该项目涉及循环电流代数模的分级张量积，特别是Kirillov-Reshetikhin模。组合框架是相关Q系统的量子簇代数，其解可以用平面拼接和非相交路径模型表示。通过研究这些系统的离散可积性，研究者们期望得到守恒量，迄今为止只有在A型系统中才能得到。进而得到张量积的分次特征标的差分方程，它推广了差分户田方程。此外，这些方程的解可以表示为差分生成算子的迭代作用，与双仿射Hecke代数的多项式表示中定义的广义Macdonald算子密切相关。与量子仿射和椭圆霍尔代数的关系也将被调查，以及更多的组合问题，如存在一些（量子）非交换对应的Gessel-Viennot行列式的非相交路径enumeration.This奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

项目成果

期刊论文数量（8）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Macdonald Operators and Quantum Q-Systems for Classical Types

经典类型的麦克唐纳算子和量子 Q 系统

DOI：
发表时间：
2021
期刊：
and Integrable Systems
影响因子：
0
作者：
Di Francesco, Philippe;Kedem, Rinat
通讯作者：
Kedem, Rinat

Exponents for Hamiltonian paths on random bicubic maps and KPZ

随机双三次映射和 KPZ 上哈密顿路径的指数

DOI：
10.1016/j.nuclphysb.2023.116084
发表时间：
2023
期刊：
Nuclear Physics B
影响因子：
2.8
作者：
Di Francesco, Philippe;Duplantier, Bertrand;Golinelli, Olivier;Guitter, Emmanuel
通讯作者：
Guitter, Emmanuel

Arctic Curves of the Twenty-Vertex Model with Domain Wall Boundaries

具有磁畴壁边界的二十顶点模型的北极曲线