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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
Symmetry Integrability and Geometry-Methods and Applications Pub Date : 2022-04-21 DOI: 10.3842/SIGMA.2022.099
Claudia De Lazzari, Harshit J. Motwani, Tim Seynnaeve
{"title":"The Linear Span of Uniform Matrix Product States","authors":"Claudia De Lazzari, Harshit J. Motwani, Tim Seynnaeve","doi":"10.3842/SIGMA.2022.099","DOIUrl":"https://doi.org/10.3842/SIGMA.2022.099","url":null,"abstract":"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.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2022-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41290771","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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
Symmetry Integrability and Geometry-Methods and Applications Pub Date : 2022-04-20 DOI: 10.3842/SIGMA.2022.079
L. Petrov
{"title":"Noncolliding Macdonald Walks with an Absorbing Wall","authors":"L. Petrov","doi":"10.3842/SIGMA.2022.079","DOIUrl":"https://doi.org/10.3842/SIGMA.2022.079","url":null,"abstract":"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.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2022-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41249652","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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
Symmetry Integrability and Geometry-Methods and Applications Pub Date : 2022-04-12 DOI: 10.3842/SIGMA.2023.019
M. Cuntz, Tobias Ohrmann
{"title":"Higher Braidings of Diagonal Type","authors":"M. Cuntz, Tobias Ohrmann","doi":"10.3842/SIGMA.2023.019","DOIUrl":"https://doi.org/10.3842/SIGMA.2023.019","url":null,"abstract":"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.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2022-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42398921","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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
Symmetry Integrability and Geometry-Methods and Applications Pub Date : 2022-04-07 DOI: 10.3842/SIGMA.2022.083
Claus Kostler, Arundhathi Krishnan
{"title":"Markovianity and the Thompson Group F","authors":"Claus Kostler, Arundhathi Krishnan","doi":"10.3842/SIGMA.2022.083","DOIUrl":"https://doi.org/10.3842/SIGMA.2022.083","url":null,"abstract":"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.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2022-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48712889","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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
Symmetry Integrability and Geometry-Methods and Applications Pub Date : 2022-04-04 DOI: 10.3842/SIGMA.2023.038
K. Ebrahimi-Fard, F. Patras, N. Tapia, L. Zambotti
{"title":"Shifted Substitution in Non-Commutative Multivariate Power Series with a View Toward Free Probability","authors":"K. Ebrahimi-Fard, F. Patras, N. Tapia, L. Zambotti","doi":"10.3842/SIGMA.2023.038","DOIUrl":"https://doi.org/10.3842/SIGMA.2023.038","url":null,"abstract":"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.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2022-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49311822","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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
Symmetry Integrability and Geometry-Methods and Applications Pub Date : 2022-03-30 DOI: 10.3842/SIGMA.2022.026
S. Ibraev
{"title":"Cohomology of sl3 and gl3 with Coefficients in Simple Modules and Weyl Modules in Positive Characteristics","authors":"S. Ibraev","doi":"10.3842/SIGMA.2022.026","DOIUrl":"https://doi.org/10.3842/SIGMA.2022.026","url":null,"abstract":". 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 ).","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2022-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43757512","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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
Symmetry Integrability and Geometry-Methods and Applications Pub Date : 2022-03-17 DOI: 10.3842/SIGMA.2022.044
J. Gough
{"title":"Field Calculus: Quantum and Statistical Field Theory without the Feynman Diagrams","authors":"J. Gough","doi":"10.3842/SIGMA.2022.044","DOIUrl":"https://doi.org/10.3842/SIGMA.2022.044","url":null,"abstract":"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.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2022-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46589220","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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
Symmetry Integrability and Geometry-Methods and Applications Pub Date : 2022-03-15 DOI: 10.3842/SIGMA.2022.045
G. Albonico, Yvonne Geyer, L. Mason
{"title":"From Twistor-Particle Models to Massive Amplitudes","authors":"G. Albonico, Yvonne Geyer, L. Mason","doi":"10.3842/SIGMA.2022.045","DOIUrl":"https://doi.org/10.3842/SIGMA.2022.045","url":null,"abstract":"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.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2022-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42234601","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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
Symmetry Integrability and Geometry-Methods and Applications Pub Date : 2022-03-15 DOI: 10.3842/SIGMA.2023.009
M. Alim, Lotte Hollands, Iván Tulli
{"title":"Quantum Curves, Resurgence and Exact WKB","authors":"M. Alim, Lotte Hollands, Iván Tulli","doi":"10.3842/SIGMA.2023.009","DOIUrl":"https://doi.org/10.3842/SIGMA.2023.009","url":null,"abstract":"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.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2022-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46781478","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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
Symmetry Integrability and Geometry-Methods and Applications Pub Date : 2022-03-14 DOI: 10.3842/SIGMA.2022.090
Henry Liu
{"title":"A Representation-Theoretic Approach to qq-Characters","authors":"Henry Liu","doi":"10.3842/SIGMA.2022.090","DOIUrl":"https://doi.org/10.3842/SIGMA.2022.090","url":null,"abstract":"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.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2022-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44488293","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
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