This work is motivated by the relation between the KP and BKP integrable hierarchies, whose $tau$-functions may be viewed as sections of dual determinantal and Pfaffian line bundles over infinite dimensional Grassmannians. In finite dimensions, we show how to relate the Cartan map which, for a vector space $V$ of dimension $N$, embeds the Grassmannian ${mathrm {Gr}}^0_V(V+V^*)$ of maximal isotropic subspaces of $V+ V^*$, with respect to the natural scalar product, into the projectivization of the exterior space $Lambda(V)$, and the Plucker map, which embeds the Grassmannian ${mathrm {Gr}}_V(V+ V^*)$ of all $N$-planes in $V+ V^*$ into the projectivization of $Lambda^N(V + V^*)$. The Plucker coordinates on ${mathrm {Gr}}^0_V(V+V^*)$ are expressed bilinearly in terms of the Cartan coordinates, which are holomorphic sections of the dual Pfaffian line bundle ${mathrm {Pf}}^* rightarrow {mathrm {Gr}}^0_V(V+V^*, Q)$. In terms of affine coordinates on the big cell, this is equivalent to an identity of Cauchy-Binet type, expressing the determinants of square submatrices of a skew symmetric $N times N$ matrix as bilinear sums over the Pfaffians of their principal minors.
{"title":"Isotropic Grassmannians, Plücker and Cartan maps","authors":"F. Balogh, J. Harnad, J. Hurtubise","doi":"10.1063/5.0021269","DOIUrl":"https://doi.org/10.1063/5.0021269","url":null,"abstract":"This work is motivated by the relation between the KP and BKP integrable hierarchies, whose $tau$-functions may be viewed as sections of dual determinantal and Pfaffian line bundles over infinite dimensional Grassmannians. In finite dimensions, we show how to relate the Cartan map which, for a vector space $V$ of dimension $N$, embeds the Grassmannian ${mathrm {Gr}}^0_V(V+V^*)$ of maximal isotropic subspaces of $V+ V^*$, with respect to the natural scalar product, into the projectivization of the exterior space $Lambda(V)$, and the Plucker map, which embeds the Grassmannian ${mathrm {Gr}}_V(V+ V^*)$ of all $N$-planes in $V+ V^*$ into the projectivization of $Lambda^N(V + V^*)$. The Plucker coordinates on ${mathrm {Gr}}^0_V(V+V^*)$ are expressed bilinearly in terms of the Cartan coordinates, which are holomorphic sections of the dual Pfaffian line bundle ${mathrm {Pf}}^* rightarrow {mathrm {Gr}}^0_V(V+V^*, Q)$. In terms of affine coordinates on the big cell, this is equivalent to an identity of Cauchy-Binet type, expressing the determinants of square submatrices of a skew symmetric $N times N$ matrix as bilinear sums over the Pfaffians of their principal minors.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88028539","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Hyunjin Ahn, Seung‐Yeal Ha, Hansol Park, Woojoo Shim
We study emergent behaviors of Cucker-Smale(CS) flocks on the hyperboloid $mathbb{H}^d$ in any dimensions. In a recent work cite{H-H-K-K-M}, a first-order aggregation model on the hyperboloid was proposed and its emergent dynamics was analyzed in terms of initial configuration and system parameters. In this paper, we are interested in the second-order modeling of Cucker-Smale flocks on the hyperboloid. For this, we derive our second-order model from the abstract CS model on complete and smooth Riemannian manifolds by explicitly calculating the geodesic and parallel transport. Velocity alignment has been shown by combining general {velocity alignment estimates} for the abstract CS model on manifolds and verifications of a priori estimate of second derivative of energy functional. For the two-dimensional case $mathbb{H}^2$, similar to the recent result in cite{A-H-S}, asymptotic flocking admits only two types of asymptotic scenarios, either convergence to a rest state or a state lying on the same plane (coplanar state). We also provide several numerical simulations to illustrate an aforementioned dichotomy on the asymptotic dynamics of the hyperboloid CS model on $mathbb{H}^2$.
研究了任意维双曲面$mathbb{H}^d$上cucker - small (CS)群的涌现行为。在最近的工作cite{H-H-K-K-M}中，提出了双曲面上的一阶聚集模型，并从初始构型和系统参数的角度分析了其紧急动力学。本文主要研究双曲面上cucker - small群的二阶建模问题。为此，我们从完全光滑黎曼流形上的抽象CS模型出发，通过显式计算测地线和平行移动，推导出二阶模型。结合流形上抽象CS模型的一般{速度对准估计}和对能量泛函二阶导数的先验估计的验证，证明了速度对准。对于二维情况$mathbb{H}^2$，类似于cite{A-H-S}中最近的结果，渐近群集只允许两种类型的渐近场景，收敛到静止状态或位于同一平面上的状态(共面状态)。我们还提供了几个数值模拟来说明上述二分法的双曲面CS模型的渐近动力学在$mathbb{H}^2$上。
{"title":"Emergent behaviors of Cucker–Smale flocks on the hyperboloid","authors":"Hyunjin Ahn, Seung‐Yeal Ha, Hansol Park, Woojoo Shim","doi":"10.1063/5.0020923","DOIUrl":"https://doi.org/10.1063/5.0020923","url":null,"abstract":"We study emergent behaviors of Cucker-Smale(CS) flocks on the hyperboloid $mathbb{H}^d$ in any dimensions. In a recent work cite{H-H-K-K-M}, a first-order aggregation model on the hyperboloid was proposed and its emergent dynamics was analyzed in terms of initial configuration and system parameters. In this paper, we are interested in the second-order modeling of Cucker-Smale flocks on the hyperboloid. For this, we derive our second-order model from the abstract CS model on complete and smooth Riemannian manifolds by explicitly calculating the geodesic and parallel transport. Velocity alignment has been shown by combining general {velocity alignment estimates} for the abstract CS model on manifolds and verifications of a priori estimate of second derivative of energy functional. For the two-dimensional case $mathbb{H}^2$, similar to the recent result in cite{A-H-S}, asymptotic flocking admits only two types of asymptotic scenarios, either convergence to a rest state or a state lying on the same plane (coplanar state). We also provide several numerical simulations to illustrate an aforementioned dichotomy on the asymptotic dynamics of the hyperboloid CS model on $mathbb{H}^2$.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85589338","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-07-05DOI: 10.1016/j.geomphys.2020.103997
A. Duyunova, V. Lychagin, S. Tychkov
{"title":"Symmetry classification of viscid flows on space curves","authors":"A. Duyunova, V. Lychagin, S. Tychkov","doi":"10.1016/j.geomphys.2020.103997","DOIUrl":"https://doi.org/10.1016/j.geomphys.2020.103997","url":null,"abstract":"","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72943344","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the Hamiltonian describing two anyons moving in a plane in presence of an external magnetic field and identify a one-parameter family of self-adjoint realizations of the corresponding Schrodinger operator. We also discuss the associated model describing a quantum particle immersed in a magnetic field with a local Aharonov-Bohm singularity. For a special class of magnetic potentials, we provide a complete classification of all possible self-adjoint extensions.
{"title":"Magnetic perturbations of anyonic and Aharonov–Bohm Schrödinger operators","authors":"M. Correggi, Davide Fermi","doi":"10.1063/5.0018933","DOIUrl":"https://doi.org/10.1063/5.0018933","url":null,"abstract":"We study the Hamiltonian describing two anyons moving in a plane in presence of an external magnetic field and identify a one-parameter family of self-adjoint realizations of the corresponding Schrodinger operator. We also discuss the associated model describing a quantum particle immersed in a magnetic field with a local Aharonov-Bohm singularity. For a special class of magnetic potentials, we provide a complete classification of all possible self-adjoint extensions.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75225165","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The integration problem of a C-bracket and a Vaisman (metric, pre-DFT) algebroid which are geometric structures of double field theory (DFT) is analyzed. We introduce a notion of a pre-rackoid as a global group-like object for an infinitesimal algebroid structure. We propose that several realizations of pre-rackoid structures. One realization is that elements of a pre-rackoid are defined by cotangent paths along doubled foliations in a para-Hermitian manifold. Another realization is proposed as a formal exponential map of the algebroid of DFT. We show that the pre-rackoid reduces to a rackoid that is the integration of the Courant algebroid when the strong constraint of DFT is imposed. Finally, for a physical application, we exhibit an implementation of the (pre-)rackoid in a three-dimensional topological sigma model.
{"title":"Global aspects of doubled geometry and pre-rackoid","authors":"N. Ikeda, S. Sasaki","doi":"10.1063/5.0020127","DOIUrl":"https://doi.org/10.1063/5.0020127","url":null,"abstract":"The integration problem of a C-bracket and a Vaisman (metric, pre-DFT) algebroid which are geometric structures of double field theory (DFT) is analyzed. We introduce a notion of a pre-rackoid as a global group-like object for an infinitesimal algebroid structure. We propose that several realizations of pre-rackoid structures. One realization is that elements of a pre-rackoid are defined by cotangent paths along doubled foliations in a para-Hermitian manifold. Another realization is proposed as a formal exponential map of the algebroid of DFT. We show that the pre-rackoid reduces to a rackoid that is the integration of the Courant algebroid when the strong constraint of DFT is imposed. Finally, for a physical application, we exhibit an implementation of the (pre-)rackoid in a three-dimensional topological sigma model.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":"40 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82709841","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
States of Low Energy (SLE) are exact Hadamard states defined on arbitrary Friedmann-Lemaitre spacetimes. They are constructed from a fiducial state by minimizing the Hamiltonian's expectation value after averaging with a temporal window function. We show the SLE to be expressible solely in terms of the (state independent) commutator function. They also admit a convergent series expansion in powers of the spatial momentum, both for massive and for massless theories. In the massless case the leading infrared behavior is found to be Minkowski-like for all scale factors. This provides a new cure for the infrared divergences in Friedmann-Lemaitre spacetimes with accelerated expansion. In consequence, massless SLE are viable candidates for pre-inflationary vacua and in a soluble model are shown to entail a qualitatively correct primordial power spectrum.
{"title":"Bonus properties of states of low energy","authors":"R. Banerjee, M. Niedermaier","doi":"10.1063/5.0019311","DOIUrl":"https://doi.org/10.1063/5.0019311","url":null,"abstract":"States of Low Energy (SLE) are exact Hadamard states defined on arbitrary Friedmann-Lemaitre spacetimes. They are constructed from a fiducial state by minimizing the Hamiltonian's expectation value after averaging with a temporal window function. We show the SLE to be expressible solely in terms of the (state independent) commutator function. They also admit a convergent series expansion in powers of the spatial momentum, both for massive and for massless theories. In the massless case the leading infrared behavior is found to be Minkowski-like for all scale factors. This provides a new cure for the infrared divergences in Friedmann-Lemaitre spacetimes with accelerated expansion. In consequence, massless SLE are viable candidates for pre-inflationary vacua and in a soluble model are shown to entail a qualitatively correct primordial power spectrum.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84280864","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This work provides a first step towards the construction of a noncommutative geometry for the Quantum Hall Effect in the continuous. Taking inspiration from the ideas developed by Bellissard during the 80's we build a spectral triple for the $C^*$-algebra of continuous magnetic operators based on a Dirac operator with compact resolvent. The metric aspects of this spectral triple are studied, and an important piece of Bellissard's theory (the so-called first Connes' formula) is proved.
{"title":"The Noncommutative Geometry of the Landau Hamiltonian: Metric Aspects","authors":"G. Nittis, M. Sandoval","doi":"10.3842/sigma.2020.146","DOIUrl":"https://doi.org/10.3842/sigma.2020.146","url":null,"abstract":"This work provides a first step towards the construction of a noncommutative geometry for the Quantum Hall Effect in the continuous. Taking inspiration from the ideas developed by Bellissard during the 80's we build a spectral triple for the $C^*$-algebra of continuous magnetic operators based on a Dirac operator with compact resolvent. The metric aspects of this spectral triple are studied, and an important piece of Bellissard's theory (the so-called first Connes' formula) is proved.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87136691","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
An asymptotic expression of the orthonormal polynomials $mathcal{P}_{N}(z)$ as $Nrightarrowinfty$, associated with the singularly perturbed Laguerre weight $w_{alpha}(x;t)=x^{alpha}{rm e}^{-x-frac{t}{x}},~xin[0,infty),~alpha>-1,~tgeq0$ is derived. Based on this, we establish the asymptotic behavior of the smallest eigenvalue, $lambda_{N}$, of the Hankel matrix generated by the weight $w_{alpha}(x;t)$.
{"title":"The smallest eigenvalue of large Hankel matrices generated by a singularly perturbed Laguerre weight","authors":"Mengkun Zhu, Yang Chen, Chuanzhong Li","doi":"10.1063/1.5140079","DOIUrl":"https://doi.org/10.1063/1.5140079","url":null,"abstract":"An asymptotic expression of the orthonormal polynomials $mathcal{P}_{N}(z)$ as $Nrightarrowinfty$, associated with the singularly perturbed Laguerre weight $w_{alpha}(x;t)=x^{alpha}{rm e}^{-x-frac{t}{x}},~xin[0,infty),~alpha>-1,~tgeq0$ is derived. Based on this, we establish the asymptotic behavior of the smallest eigenvalue, $lambda_{N}$, of the Hankel matrix generated by the weight $w_{alpha}(x;t)$.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":"30 1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84062551","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We investigate eigenvalue moments of matrices from Circular Orthogonal Ensemble multiplicatively perturbed by a permutation matrix. More precisely we investigate variance of the sum of the eigenvalues raised to power $k$, for arbitrary but fixed $k$ and in the limit of large matrix size. We find that when the permutation defining the perturbed ensemble has only long cycles, the answer is universal and approaches the corresponding moment of the Circular Unitary Ensemble with a particularly fast rate: the error is of order $1/N^3$ and the terms of orders $1/N$ and $1/N^2$ disappear due to cancellations. We prove this rate of convergence using Weingarten calculus and classifying the contributing Weingarten functions first in terms of a graph model and then algebraically.
{"title":"Convergence of moments of twisted COE matrices","authors":"G. Berkolaiko, Laura Booton","doi":"10.1063/5.0018927","DOIUrl":"https://doi.org/10.1063/5.0018927","url":null,"abstract":"We investigate eigenvalue moments of matrices from Circular Orthogonal Ensemble multiplicatively perturbed by a permutation matrix. More precisely we investigate variance of the sum of the eigenvalues raised to power $k$, for arbitrary but fixed $k$ and in the limit of large matrix size. We find that when the permutation defining the perturbed ensemble has only long cycles, the answer is universal and approaches the corresponding moment of the Circular Unitary Ensemble with a particularly fast rate: the error is of order $1/N^3$ and the terms of orders $1/N$ and $1/N^2$ disappear due to cancellations. We prove this rate of convergence using Weingarten calculus and classifying the contributing Weingarten functions first in terms of a graph model and then algebraically.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82406368","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-06-09DOI: 10.1142/s0129055x22500222
W. A. Z'uniga-Galindo
We construct in a rigorous mathematical way interacting quantum field theories on a p-adic spacetime. The main result is the construction of a measure on a function space which allows a rigorous definition of the partition function. The calculation of the correlation functions is carried out in the standard form. In the case of $varphi^{4}$-theories, we show the existence of systems admitting spontaneous symmetry breaking.
{"title":"Non-Archimedean Statistical Field Theory","authors":"W. A. Z'uniga-Galindo","doi":"10.1142/s0129055x22500222","DOIUrl":"https://doi.org/10.1142/s0129055x22500222","url":null,"abstract":"We construct in a rigorous mathematical way interacting quantum field theories on a p-adic spacetime. The main result is the construction of a measure on a function space which allows a rigorous definition of the partition function. The calculation of the correlation functions is carried out in the standard form. In the case of $varphi^{4}$-theories, we show the existence of systems admitting spontaneous symmetry breaking.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":"115 2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78015546","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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