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The Linear Span of Uniform Matrix Product States 一致矩阵积态的线性张成
IF 0.9 3区 物理与天体物理 Q2 Mathematics Pub Date : 2022-04-21 DOI: 10.3842/SIGMA.2022.099
Claudia De Lazzari, Harshit J. Motwani, Tim Seynnaeve
The variety of uniform matrix product states arises both in algebraic geometry as a natural generalization of the Veronese variety, and in quantum many-body physics as a model for a translation-invariant system of sites placed on a ring. Using methods from linear algebra, representation theory, and invariant theory of matrices, we study the linear span of this variety.
一致矩阵积态的变化在代数几何中作为维罗内塞变化的自然推广而出现,在量子多体物理中作为放置在环上的位置的平移不变系统的模型而出现。利用线性代数、表示理论和矩阵不变理论的方法,研究了这一种类的线性张成空间。
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引用次数: 4
Noncolliding Macdonald Walks with an Absorbing Wall 非碰撞的麦克唐纳与可吸收的墙壁一起行走
IF 0.9 3区 物理与天体物理 Q2 Mathematics Pub Date : 2022-04-20 DOI: 10.3842/SIGMA.2022.079
L. Petrov
The branching rule is one of the most fundamental properties of the Macdonald symmetric polynomials. It expresses a Macdonald polynomial as a nonnegative linear combination of Macdonald polynomials with smaller number of variables. Taking a limit of the branching rule under the principal specialization when the number of variables goes to infinity, we obtain a Markov chain of $m$ noncolliding particles with negative drift and an absorbing wall at zero. The chain depends on the Macdonald parameters $(q,t)$ and may be viewed as a discrete deformation of the Dyson Brownian motion. The trajectory of the Markov chain is equivalent to a certain Gibbs ensemble of plane partitions with an arbitrary cascade front wall. In the Jack limit $t=q^{beta/2}to 1$ the absorbing wall disappears, and the Macdonald noncolliding walks turn into the $beta$-noncolliding random walks studied by Huang [Int. Math. Res. Not. 2021 (2021), 5898-5942, arXiv:1708.07115]. Taking $q=0$ (Hall-Littlewood degeneration) and further sending $tto 1$, we obtain a continuous time particle system on $mathbb{Z}_{ge 0}$ with inhomogeneous jump rates and absorbing wall at zero.
分支规则是麦克唐纳对称多项式最基本的性质之一。它将麦克唐纳多项式表示为变量数量较少的麦克唐纳多项式的非负线性组合。当变量数量无穷大时,在主专门化下取分支规则的一个极限,我们得到了一个具有负漂移和零吸收壁的$m$非碰撞粒子的马尔可夫链。该链取决于麦克唐纳参数$(q,t)$,并且可以被视为戴森布朗运动的离散变形。马尔可夫链的轨迹等价于具有任意级联前壁的平面分区的某个吉布斯系综。在Jack极限$t=q^{beta/2}为1$时,吸收壁消失,Macdonald非碰撞行走转变为Huang研究的$beta$非碰撞随机行走[Int.Math.Res.Not.2021(2021),5898-5942,arXiv:170807115]。取$q=0$(Hall-Littlewood退化)并进一步将$t发送到1$,我们在$mathbb上获得了一个连续时间粒子系统{Z}_{ge 0}$,具有不均匀的跳跃率和零处的吸收壁。
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引用次数: 1
Higher Braidings of Diagonal Type 对角线型高编织
IF 0.9 3区 物理与天体物理 Q2 Mathematics Pub Date : 2022-04-12 DOI: 10.3842/SIGMA.2023.019
M. Cuntz, Tobias Ohrmann
Heckenberger introduced the Weyl groupoid of a finite-dimensional Nichols algebra of diagonal type. We replace the matrix of its braiding by a higher tensor and present a construction which yields further Weyl groupoids. Abelian cohomology theory gives evidence for the existence of a higher braiding associated to such a tensor.
Heckenberger引入了有限维对角型Nichols代数的Weyl群。我们用一个高张量来代替它的编织矩阵,并给出了一个进一步生成Weyl群的构造。阿贝尔上同调理论给出了与这样一个张量相关的更高编织存在的证据。
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引用次数: 0
Markovianity and the Thompson Group F 马尔可夫性和汤普森群F
IF 0.9 3区 物理与天体物理 Q2 Mathematics Pub Date : 2022-04-07 DOI: 10.3842/SIGMA.2022.083
Claus Kostler, Arundhathi Krishnan
We show that representations of the Thompson group F in the automorphisms of a noncommutative probability space yield a large class of bilateral stationary noncommutative Markov processes. As a partial converse, bilateral stationary Markov processes in tensor dilation form yield representations of F. As an application, and building on a result of Kümmerer, we canonically associate a representation of F to a bilateral stationary Markov process in classical probability.
我们证明了Thompson群F在非对易概率空间的自同构中的表示产生了一大类双边平稳非对易Markov过程。作为部分逆,张量扩张形式的双边平稳马尔可夫过程产生了F的表示。作为一个应用,并在Kümmerer的结果的基础上,我们将F的表示与经典概率中的双边平稳Markov过程经典地关联起来。
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引用次数: 3
Shifted Substitution in Non-Commutative Multivariate Power Series with a View Toward Free Probability 从自由概率的角度看非交换多元幂级数的移位代换
IF 0.9 3区 物理与天体物理 Q2 Mathematics Pub Date : 2022-04-04 DOI: 10.3842/SIGMA.2023.038
K. Ebrahimi-Fard, F. Patras, N. Tapia, L. Zambotti
We study a particular group law on formal power series in non-commuting variables induced by their interpretation as linear forms on a suitable graded connected word Hopf algebra. This group law is left-linear and is therefore associated to a pre-Lie structure on formal power series. We study these structures and show how they can be used to recast in a group theoretic form various identities and transformations on formal power series that have been central in the context of non-commutative probability theory, in particular in Voiculescu's theory of free probability.
我们研究了非交换变量中形式幂级数的一个特殊群律,该群律是由它们在适当的分次连接词Hopf代数上的线性形式解释引起的。这个群定律是左线性的,因此与形式幂级数上的前李结构有关。我们研究了这些结构,并展示了它们如何被用来以群论的形式重塑形式幂级数上的各种恒等式和变换,这些恒等式和转换在非交换概率论的背景下一直是核心,特别是在Voiculescu的自由概率论中。
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引用次数: 1
Cohomology of sl3 and gl3 with Coefficients in Simple Modules and Weyl Modules in Positive Characteristics 单模中系数为sl3和gl3的上同调与正特性的Weyl模
IF 0.9 3区 物理与天体物理 Q2 Mathematics Pub Date : 2022-03-30 DOI: 10.3842/SIGMA.2022.026
S. Ibraev
. We calculate the cohomology of sl 3 ( k ) over an algebraically closed field k of characteristic p > 3 with coefficients in simple modules and Weyl modules. We also give descriptions of the corresponding cohomology of gl 3 ( k ).
.我们计算了在简单模和Weyl模中具有系数的特征p>3的代数封闭域k上的sl 3(k)的上同调。我们还给出了gl3(k)的相应上同调的描述。
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引用次数: 1
Field Calculus: Quantum and Statistical Field Theory without the Feynman Diagrams 场演算:没有费曼图的量子和统计场理论
IF 0.9 3区 物理与天体物理 Q2 Mathematics Pub Date : 2022-03-17 DOI: 10.3842/SIGMA.2022.044
J. Gough
For a given base space $M$ (spacetime), we consider the Guichardet space over the Guichardet space over $M$. Here we develop a ''field calculus'' based on the Guichardet integral. This is the natural setting in which to describe Green function relations for Boson systems. Here we can follow the suggestion of Schwinger and develop a differential (local field) approach rather than the integral one pioneered by Feynman. This is helped by a DEFG (Dyson-Einstein-Feynman-Guichardet) shorthand which greatly simplifies expressions. This gives a convenient framework for the formal approach of Schwinger and Tomonaga as opposed to Feynman diagrams. The Dyson-Schwinger is recast in this language with the help of bosonic creation/annihilation operators. We also give the combinatorial approach to tree-expansions.
对于给定的基空间$M$(时空),我们考虑$M$上的Guichardet空间上的Guichardet空间。在这里,我们基于Guichardet积分开发了一个“场演算”。这是描述玻色子系统格林函数关系的自然环境。在这里,我们可以遵循Schwinger的建议,发展一种微分(局部场)方法,而不是费曼开创的积分方法。这得益于DEFG (Dyson-Einstein-Feynman-Guichardet)速记法,它极大地简化了表达式。这为Schwinger和Tomonaga的形式化方法提供了一个方便的框架,而不是费曼图。在玻色子创造/湮灭算子的帮助下,我们用这种语言重新定义了戴森-施温格。我们还给出了树展开的组合方法。
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引用次数: 0
From Twistor-Particle Models to Massive Amplitudes 从扭曲粒子模型到质量振幅
IF 0.9 3区 物理与天体物理 Q2 Mathematics Pub Date : 2022-03-15 DOI: 10.3842/SIGMA.2022.045
G. Albonico, Yvonne Geyer, L. Mason
In his twistor-particle programme of the 1970's, Roger Penrose introduced a representation of the massive particle phase space in terms of a pair of twistors subject to an internal symmetry group. Here we use this representation to introduce a chiral string whose target is a complexification of this space, extended so as to incorporate supersymmetry. We show that the gauge anomalies associated to the internal symmetry group vanish only for maximal supersymmetry, and that correlators in these string models describe amplitudes involving massive particles with manifest supersymmetry. The models and amplitude formulae exhibit a double copy structure from gauge theory on the Coulomb branch to gravity, although the graviton remains massless. The formulae are closely related to those obtained earlier by the authors expressed in terms of the polarised scattering equations.
在20世纪70年代的龙卷风粒子计划中,罗杰·彭罗斯引入了一种大质量粒子相空间的表示,即一对内部对称群的龙卷风。在这里,我们使用这个表示来引入一个手性串,它的目标是这个空间的络合,扩展以包含超对称性。我们证明,与内部对称群相关的规范异常仅在最大超对称性时消失,并且这些串模型中的相关器描述了涉及具有明显超对称性的大质量粒子的振幅。模型和振幅公式表现出从库仑分支上的规范理论到重力的双重复制结构,尽管引力子仍然是无质量的。这些公式与作者早先用偏振散射方程表示的公式密切相关。
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引用次数: 6
Quantum Curves, Resurgence and Exact WKB 量子曲线、复活与精确WKB
IF 0.9 3区 物理与天体物理 Q2 Mathematics Pub Date : 2022-03-15 DOI: 10.3842/SIGMA.2023.009
M. Alim, Lotte Hollands, Iván Tulli
We study the non-perturbative quantum geometry of the open and closed topological string on the resolved conifold and its mirror. Our tools are finite difference equations in the open and closed string moduli and the resurgence analysis of their formal power series solutions. In the closed setting, we derive new finite difference equations for the refined partition function as well as its Nekrasov-Shatashvili (NS) limit. We write down a distinguished analytic solution for the refined difference equation that reproduces the expected non-perturbative content of the refined topological string. We compare this solution to the Borel analysis of the free energy in the NS limit. We find that the singularities of the Borel transform lie on infinitely many rays in the Borel plane and that the Stokes jumps across these rays encode the associated Donaldson-Thomas invariants of the underlying Calabi-Yau geometry. In the open setting, the finite difference equation corresponds to a canonical quantization of the mirror curve. We analyze this difference equation using Borel analysis and exact WKB techniques and identify the 5d BPS states in the corresponding exponential spectral networks. We furthermore relate the resurgence analysis in the open and closed setting. This guides us to a five-dimensional extension of the Nekrasov-Rosly-Shatashvili proposal, in which the NS free energy is computed as a generating function of q-difference opers in terms of a special set of spectral coordinates. Finally, we examine two spectral problems describing the corresponding quantum integrable system.
研究了在已分解的折叠及其镜像上的开闭拓扑弦的非摄动量子几何。我们的工具是开弦模和闭弦模的有限差分方程及其形式幂级数解的重现分析。在封闭条件下,导出了精化配分函数的新的有限差分方程及其Nekrasov-Shatashvili (NS)极限。我们写出了精炼差分方程的解析解,它再现了精炼拓扑弦的期望非微扰内容。我们将此解与NS极限下自由能的Borel分析进行了比较。我们发现Borel变换的奇异点存在于Borel平面上的无限多条射线上,并且Stokes跳越这些射线编码了底层Calabi-Yau几何的相关Donaldson-Thomas不变量。在开放条件下,有限差分方程对应于镜像曲线的正则量化。我们使用Borel分析和精确WKB技术分析了该差分方程,并在相应的指数谱网络中识别了5d BPS状态。我们进一步将开放和封闭环境下的复苏分析联系起来。这将我们引向nekrasov - rosley - shatashvili建议的五维扩展,其中NS自由能是根据一组特殊的光谱坐标作为q差算子的生成函数来计算的。最后,我们研究了描述相应量子可积系统的两个谱问题。
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引用次数: 12
A Representation-Theoretic Approach to qq-Characters qq字符的表示理论方法
IF 0.9 3区 物理与天体物理 Q2 Mathematics Pub Date : 2022-03-14 DOI: 10.3842/SIGMA.2022.090
Henry Liu
We raise the question of whether (a slightly generalized notion of) $qq$-characters can be constructed purely representation-theoretically. In the main example of the quantum toroidal $mathfrak{gl}_1$ algebra, geometric engineering of adjoint matter produces an explicit vertex operator $mathsf{RR}$ which computes certain $qq$-characters, namely Hirzebruch $chi_y$-genera, completely analogously to how the R-matrix $mathsf{R}$ computes $q$-characters. We give a geometric proof of the independence of preferred direction for the refined vertex in this and more general non-toric settings.
我们提出了一个问题,即$qq$-字符(一个稍微广义的概念)是否可以在理论上被构造为纯粹的表示。在量子超环面$mathfrak的主要例子中{gl}_1$代数,伴随物质的几何工程产生了一个显式顶点算子$mathsf{RR}$,它计算某些$qq$-字符,即Hirzebruch$chi_y$-属,完全类似于R-矩阵$mathsf{R}$计算$q$-字符的方式。在这个和更一般的非复曲面设置中,我们给出了精化顶点的优选方向独立性的几何证明。
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引用次数: 1
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Symmetry Integrability and Geometry-Methods and Applications
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