结及其相关$q$-系列

IF 0.9 3区 物理与天体物理 Q2 MATHEMATICS
Stavros Garoufalidis, Don Zagier
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引用次数: 17

摘要

讨论了在实数上的矩阵值分段解析函数空间上定义${\rm PSL}_2({\mathbb Z})$-环的结补的Andersen-Kashaev状态积分分解得到的上下半平面上的周期全纯函数矩阵。我们通过[arXiv:2111.06645]的精细量子模性猜想识别出相应的环与结的Kashaev不变量(及其矩阵值扩展)的环,并将矩阵值不变量与Dimofte-Gaiotto-Gukov的3d指数联系起来。循环也具有解析可扩展性,这导致了矩阵值全纯量子模形式的概念。这是一个由几个独立的发现组成的故事,既有经验上的发现,也有理论上的发现,所有这些发现都可以用三个最简单的双曲结来说明。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Knots and Their Related $q$-Series
We discuss a matrix of periodic holomorphic functions in the upper and lower half-plane which can be obtained from a factorization of an Andersen-Kashaev state integral of a knot complement with remarkable analytic and asymptotic properties that defines a ${\rm PSL}_2({\mathbb Z})$-cocycle on the space of matrix-valued piecewise analytic functions on the real numbers. We identify the corresponding cocycle with the one coming from the Kashaev invariant of a knot (and its matrix-valued extension) via the refined quantum modularity conjecture of [arXiv:2111.06645] and also relate the matrix-valued invariant with the 3D-index of Dimofte-Gaiotto-Gukov. The cocycle also has an analytic extendability property that leads to the notion of a matrix-valued holomorphic quantum modular form. This is a tale of several independent discoveries, both empirical and theoretical, all illustrated by the three simplest hyperbolic knots.
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来源期刊
CiteScore
1.80
自引率
0.00%
发文量
87
审稿时长
4-8 weeks
期刊介绍: Scope Geometrical methods in mathematical physics Lie theory and differential equations Classical and quantum integrable systems Algebraic methods in dynamical systems and chaos Exactly and quasi-exactly solvable models Lie groups and algebras, representation theory Orthogonal polynomials and special functions Integrable probability and stochastic processes Quantum algebras, quantum groups and their representations Symplectic, Poisson and noncommutative geometry Algebraic geometry and its applications Quantum field theories and string/gauge theories Statistical physics and condensed matter physics Quantum gravity and cosmology.
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