Shankar Balasubramanian, Sarang Gopalakrishnan, Alexey Khudorozhkov, Ethan Lake
We introduce a family of local models of dynamics based on ``word problems'' from computer science and group theory, for which we can place rigorous lower bounds on relaxation timescales. These models can be regarded either as random circuit or local Hamiltonian dynamics, and include many familiar examples of constrained dynamics as special cases. The configuration space of these models splits into dynamically disconnected sectors, and for initial states to relax, they must ``work out'' the other states in the sector to which they belong. When this problem has a high time complexity, relaxation is slow. In some of the cases we study, this problem also has high space complexity. When the space complexity is larger than the system size, an unconventional type of jamming transition can occur, whereby a system of a fixed size is not ergodic, but can be made ergodic by appending a large reservoir of sites in a trivial product state. This manifests itself in a new type of Hilbert space fragmentation that we call fragile fragmentation. We present explicit examples where slow relaxation and jamming strongly modify the hydrodynamics of conserved densities. In one example, density modulations of wavevector $q$ exhibit almost no relaxation until times $O(exp(1/q))$, at which point they abruptly collapse. We also comment on extensions of our results to higher dimensions.
{"title":"Glassy word problems: ultraslow relaxation, Hilbert space jamming, and computational complexity","authors":"Shankar Balasubramanian, Sarang Gopalakrishnan, Alexey Khudorozhkov, Ethan Lake","doi":"arxiv-2312.04562","DOIUrl":"https://doi.org/arxiv-2312.04562","url":null,"abstract":"We introduce a family of local models of dynamics based on ``word problems''\u0000from computer science and group theory, for which we can place rigorous lower\u0000bounds on relaxation timescales. These models can be regarded either as random\u0000circuit or local Hamiltonian dynamics, and include many familiar examples of\u0000constrained dynamics as special cases. The configuration space of these models\u0000splits into dynamically disconnected sectors, and for initial states to relax,\u0000they must ``work out'' the other states in the sector to which they belong.\u0000When this problem has a high time complexity, relaxation is slow. In some of\u0000the cases we study, this problem also has high space complexity. When the space\u0000complexity is larger than the system size, an unconventional type of jamming\u0000transition can occur, whereby a system of a fixed size is not ergodic, but can\u0000be made ergodic by appending a large reservoir of sites in a trivial product\u0000state. This manifests itself in a new type of Hilbert space fragmentation that\u0000we call fragile fragmentation. We present explicit examples where slow\u0000relaxation and jamming strongly modify the hydrodynamics of conserved\u0000densities. In one example, density modulations of wavevector $q$ exhibit almost\u0000no relaxation until times $O(exp(1/q))$, at which point they abruptly\u0000collapse. We also comment on extensions of our results to higher dimensions.","PeriodicalId":501275,"journal":{"name":"arXiv - PHYS - Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138555542","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 stability of two-dimensional buoyancy-driven convection in a vertical porous slot, wherein a plane Couette flow is additionally present, is studied. This complex fluid flow scenario is examined under the influence of Robin-type boundary conditions, which are applied to perturbations in both velocity and temperature. The inclusion of a time-derivative velocity term within the Darcy momentum equation notably introduces intricacies to the study. The stability of the basic natural convection flow is primarily governed by several key parameters namely, the P'eclet number, the Prandtl-Darcy number, the Biot number and a non-negative parameter that dictates the nature of the vertical boundaries. Through numerical analysis, the stability eigenvalue problem is solved for a variety of combinations of boundary conditions. The outcomes of this analysis reveal the critical threshold values that signify the onset of instability. Furthermore, a detailed examination of the stability of the system has provided insights into both its commonalities and distinctions under different conditions. It is observed that, except for the scenario featuring impermeable-isothermal boundaries, the underlying base flow exhibits instability when subjected to various other configurations of perturbed velocity and temperature boundary conditions. This underscores the notion that the presence of Couette flow alone does not suffice to induce instability within the system. The plots depicting neutral stability curves show either bi-modal or uni-modal characteristics, contingent upon specific parameter values that influence the onset of instability.
{"title":"Stability of buoyant-Couette flow in a vertical porous slot","authors":"B. M. Shankar, I. S. Shivakumara","doi":"arxiv-2312.04270","DOIUrl":"https://doi.org/arxiv-2312.04270","url":null,"abstract":"The stability of two-dimensional buoyancy-driven convection in a vertical\u0000porous slot, wherein a plane Couette flow is additionally present, is studied.\u0000This complex fluid flow scenario is examined under the influence of Robin-type\u0000boundary conditions, which are applied to perturbations in both velocity and\u0000temperature. The inclusion of a time-derivative velocity term within the Darcy\u0000momentum equation notably introduces intricacies to the study. The stability of\u0000the basic natural convection flow is primarily governed by several key\u0000parameters namely, the P'eclet number, the Prandtl-Darcy number, the Biot\u0000number and a non-negative parameter that dictates the nature of the vertical\u0000boundaries. Through numerical analysis, the stability eigenvalue problem is\u0000solved for a variety of combinations of boundary conditions. The outcomes of\u0000this analysis reveal the critical threshold values that signify the onset of\u0000instability. Furthermore, a detailed examination of the stability of the system\u0000has provided insights into both its commonalities and distinctions under\u0000different conditions. It is observed that, except for the scenario featuring\u0000impermeable-isothermal boundaries, the underlying base flow exhibits\u0000instability when subjected to various other configurations of perturbed\u0000velocity and temperature boundary conditions. This underscores the notion that\u0000the presence of Couette flow alone does not suffice to induce instability\u0000within the system. The plots depicting neutral stability curves show either\u0000bi-modal or uni-modal characteristics, contingent upon specific parameter\u0000values that influence the onset of instability.","PeriodicalId":501275,"journal":{"name":"arXiv - PHYS - Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138555272","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}
Partially motivated by the fact that the grand partition function of the ABJM theory or its generalization is expressed by a spectral operator enjoying symmetries of the Weyl group, it was found that the grand partition function satisfies the q-Painleve equation, which is constructed from the affine Weyl group. In this paper we clarify the affine symmetries of the grand partition function. With the affine symmetries, we find that the grand partition function extends naturally outside the fundamental domain of duality cascades and once the Painleve equation holds in the fundamental domain, so does it outside.
{"title":"Affine Symmetries for ABJM Partition Function and its Generalization","authors":"Sanefumi Moriyama, Tomoki Nosaka","doi":"arxiv-2312.04206","DOIUrl":"https://doi.org/arxiv-2312.04206","url":null,"abstract":"Partially motivated by the fact that the grand partition function of the ABJM\u0000theory or its generalization is expressed by a spectral operator enjoying\u0000symmetries of the Weyl group, it was found that the grand partition function\u0000satisfies the q-Painleve equation, which is constructed from the affine Weyl\u0000group. In this paper we clarify the affine symmetries of the grand partition\u0000function. With the affine symmetries, we find that the grand partition function\u0000extends naturally outside the fundamental domain of duality cascades and once\u0000the Painleve equation holds in the fundamental domain, so does it outside.","PeriodicalId":501275,"journal":{"name":"arXiv - PHYS - Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138555283","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 give several quantum dynamical analogs of the classical Kronecker-Weyl theorem, which says that the trajectory of free motion on the torus along almost every direction tends to equidistribute. As a quantum analog, we study the quantum walk $exp(-i t Delta) psi$ starting from a localized initial state $psi$. Then the flow will be ergodic if this evolved state becomes equidistributed as time goes on. We prove that this is indeed the case for evolutions on the flat torus, provided we start from a point mass, and we prove discrete analogs of this result for crystal lattices. On some periodic graphs, the mass spreads out non-uniformly, on others it stays localized. Finally, we give examples of quantum evolutions on the sphere which do not equidistribute.
我们给出了经典的克朗内克尔-韦尔定理（Kronecker-Weyltheorem）的几个量子动力学类比，这个定理说的是环上自由运动的轨迹沿着几乎每个方向都趋于等分布。作为量子类比，我们研究了从局部初始状态 $psi$ 开始的量子行走 $exp(-i t Delta) psi$。如果这个演化状态随着时间的推移变得液态分布，那么这个流动就是遍历性的。我们证明，只要我们从一个点质量出发，平面环面上的旋转确实如此，我们还证明了这一结果在晶格上的离散类比。在某些周期图上，质量非均匀分布，而在另一些周期图上，质量则保持局部。最后，我们举例说明了球面上不等分布的量子演化。
{"title":"Ergodic theorems for continuous-time quantum walks on crystal lattices and the torus","authors":"Anne Boutet de Monvel, Mostafa Sabri","doi":"arxiv-2312.04492","DOIUrl":"https://doi.org/arxiv-2312.04492","url":null,"abstract":"We give several quantum dynamical analogs of the classical Kronecker-Weyl\u0000theorem, which says that the trajectory of free motion on the torus along\u0000almost every direction tends to equidistribute. As a quantum analog, we study\u0000the quantum walk $exp(-i t Delta) psi$ starting from a localized initial\u0000state $psi$. Then the flow will be ergodic if this evolved state becomes\u0000equidistributed as time goes on. We prove that this is indeed the case for\u0000evolutions on the flat torus, provided we start from a point mass, and we prove\u0000discrete analogs of this result for crystal lattices. On some periodic graphs,\u0000the mass spreads out non-uniformly, on others it stays localized. Finally, we\u0000give examples of quantum evolutions on the sphere which do not equidistribute.","PeriodicalId":501275,"journal":{"name":"arXiv - PHYS - Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138555414","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 introduce a class of non-commutative geometries, loosely referred to as para-spaces, which are manifolds equipped with sheaves of non-commutative algebras called para-algebras. A differential analysis on para-spaces is investigated, which is reminiscent of that on super manifolds and can be readily applied to model physical problems, for example, by using para-space analogues of differential equations. Two families of examples, the affine para-spaces $mathbb{K}^{m|n}(p)$ and para-projective spaces $mathbb{KP}^{m|n}(p)$, with $mathbb{K}$ being $mathbb{R}$ and $mathbb{C}$, are treated in detail for all positive integers $p$. As an application of such non-commutative geometries, we interpret Green's theory of parafermions in terms of para-spaces on a point. Other potential applications in quantum field theory are also commented upon.
{"title":"Para-spaces, their differential analysis and an application to Green's quantisation","authors":"Ruibin Zhang","doi":"arxiv-2312.04250","DOIUrl":"https://doi.org/arxiv-2312.04250","url":null,"abstract":"We introduce a class of non-commutative geometries, loosely referred to as\u0000para-spaces, which are manifolds equipped with sheaves of non-commutative\u0000algebras called para-algebras. A differential analysis on para-spaces is\u0000investigated, which is reminiscent of that on super manifolds and can be\u0000readily applied to model physical problems, for example, by using para-space\u0000analogues of differential equations. Two families of examples, the affine\u0000para-spaces $mathbb{K}^{m|n}(p)$ and para-projective spaces\u0000$mathbb{KP}^{m|n}(p)$, with $mathbb{K}$ being $mathbb{R}$ and $mathbb{C}$,\u0000are treated in detail for all positive integers $p$. As an application of such\u0000non-commutative geometries, we interpret Green's theory of parafermions in\u0000terms of para-spaces on a point. Other potential applications in quantum field\u0000theory are also commented upon.","PeriodicalId":501275,"journal":{"name":"arXiv - PHYS - Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138557041","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 developed a statistical theory of zero-count-detector (ZCD), which is defined as a zero-class Poisson under conditions outlined in the paper. ZCD is often encountered in the studies of rare events in physics, health physics, and many other fields where counting of events occurs. We found no acceptable solution to ZCD in classical statistics and affirmed the need for the Bayesian statistics. Several uniform and reference priors were studied and we derived Bayesian posteriors, point estimates, and upper limits. It was showed that the maximum-entropy prior, containing the most information, resulted in the smallest bias and the lowest risk, making it the most admissible and acceptable among the priors studied. We also investigated application of zero-inflated Poisson and Negative-binomial distributions to ZCD. It was showed using Bayesian marginalization that, under limited information, these distributions reduce to the Poisson distribution.
{"title":"Zero-Class Poisson for Rare-Event Studies","authors":"Thomas M. Semkow","doi":"arxiv-2312.03894","DOIUrl":"https://doi.org/arxiv-2312.03894","url":null,"abstract":"We developed a statistical theory of zero-count-detector (ZCD), which is\u0000defined as a zero-class Poisson under conditions outlined in the paper. ZCD is\u0000often encountered in the studies of rare events in physics, health physics, and\u0000many other fields where counting of events occurs. We found no acceptable\u0000solution to ZCD in classical statistics and affirmed the need for the Bayesian\u0000statistics. Several uniform and reference priors were studied and we derived\u0000Bayesian posteriors, point estimates, and upper limits. It was showed that the\u0000maximum-entropy prior, containing the most information, resulted in the\u0000smallest bias and the lowest risk, making it the most admissible and acceptable\u0000among the priors studied. We also investigated application of zero-inflated\u0000Poisson and Negative-binomial distributions to ZCD. It was showed using\u0000Bayesian marginalization that, under limited information, these distributions\u0000reduce to the Poisson distribution.","PeriodicalId":501275,"journal":{"name":"arXiv - PHYS - Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138555428","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}
In this article, we follow an idea that is opposite to the idea of Hopf and Cole: we use transformations in order to transform simpler linear or nonlinear differential equations (with known solutions) to more complicated nonlinear differential equations. In such a way, we can obtain numerous exact solutions of nonlinear differential equations. We apply this methodology to the classical parabolic differential equation (the wave equation), to the classical hyperbolic differential equation (the heat equation), and to the classical elliptic differential equation (Laplace equation). In addition, we use the methodology to obtain exact solutions of nonlinear ordinary differential equations by means of the solutions of linear differential equations and by means of the solutions of the nonlinear differential equations of Bernoulli and Riccati. Finally, we demonstrate the capacity of the methodology to lead to exact solutions of nonlinear partial differential equations on the basis of known solutions of other nonlinear partial differential equations. As an example of this, we use the Korteweg--de Vries equation and its solutions. Traveling wave solutions of nonlinear differential equations are of special interest in this article. We demonstrate the existence of the following phenomena described by some of the obtained solutions: (i) occurrence of the solitary wave--solitary antiwave from the solution, which is zero at the initial moment (analogy of an occurrence of particle and antiparticle from the vacuum); (ii) splitting of a nonlinear solitary wave into two solitary waves (analogy of splitting of a particle into two particles); (iii) soliton behavior of some of the obtained waves; (iv) existence of solitons which move with the same velocity despite the different shape and amplitude of the solitons.
{"title":"On the Obtaining Solutions of Nonlinear Differential Equations by Means of the Solutions of Simpler Linear or Nonlinear Differential Equations","authors":"Nikolay K. Vitanov","doi":"arxiv-2312.03621","DOIUrl":"https://doi.org/arxiv-2312.03621","url":null,"abstract":"In this article, we follow an idea that is opposite to the idea of Hopf and\u0000Cole: we use transformations in order to transform simpler linear or nonlinear\u0000differential equations (with known solutions) to more complicated nonlinear\u0000differential equations. In such a way, we can obtain numerous exact solutions\u0000of nonlinear differential equations. We apply this methodology to the classical\u0000parabolic differential equation (the wave equation), to the classical\u0000hyperbolic differential equation (the heat equation), and to the classical\u0000elliptic differential equation (Laplace equation). In addition, we use the\u0000methodology to obtain exact solutions of nonlinear ordinary differential\u0000equations by means of the solutions of linear differential equations and by\u0000means of the solutions of the nonlinear differential equations of Bernoulli and\u0000Riccati. Finally, we demonstrate the capacity of the methodology to lead to\u0000exact solutions of nonlinear partial differential equations on the basis of\u0000known solutions of other nonlinear partial differential equations. As an\u0000example of this, we use the Korteweg--de Vries equation and its solutions.\u0000Traveling wave solutions of nonlinear differential equations are of special\u0000interest in this article. We demonstrate the existence of the following\u0000phenomena described by some of the obtained solutions: (i) occurrence of the\u0000solitary wave--solitary antiwave from the solution, which is zero at the\u0000initial moment (analogy of an occurrence of particle and antiparticle from the\u0000vacuum); (ii) splitting of a nonlinear solitary wave into two solitary waves\u0000(analogy of splitting of a particle into two particles); (iii) soliton behavior\u0000of some of the obtained waves; (iv) existence of solitons which move with the\u0000same velocity despite the different shape and amplitude of the solitons.","PeriodicalId":501275,"journal":{"name":"arXiv - PHYS - Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138545963","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}
Thomas Christopoulos, Odysseas Tsilipakos, Emmanouil E. Kriezis
Temporal coupled-mode theory (CMT) is an acclaimed and widely used theoretical framework for modeling the continuous wave (CW) response and temporal dynamics of any integrated or free-space photonic resonant structure. It was initially employed to understand how energy is coupled into and out of a cavity and how it is exchanged between different resonant modes. In the 30 years that followed its establishment, CMT has been expanded to describe a broad range of nonlinear interactions as well (self- and cross-phase modulation, saturable absorption, frequency generation, gain, etc.). In this tutorial, we thoroughly present the basic principles and the evolution of CMT throughout the years, showcasing its immense capabilities for the analysis and design of linear and nonlinear resonant photonic systems. Importantly, we focus on examples of modern, open nanophotonic resonators incorporating contemporary bulk or sheet (2D) materials that may be lossy and dispersive. For each linear/nonlinear effect under study we follow a meticulous, step-by-step approach, starting from an accurate model of the physical phenomenon and proceeding to its introduction in the CMT framework all the way to the efficient solution of the resulting system of equations. Our work highlights the merits of CMT as an efficient, accurate, and versatile theoretical tool. We envision that it can serve both as an introductory reference for any reader, as well as a comprehensive handbook on how to incorporate a broad range of linear and nonlinear effects in the CMT framework.
{"title":"Temporal coupled-mode theory in nonlinear resonant photonics: From basic principles to contemporary systems with 2D materials, dispersion, loss, and gain","authors":"Thomas Christopoulos, Odysseas Tsilipakos, Emmanouil E. Kriezis","doi":"arxiv-2312.03539","DOIUrl":"https://doi.org/arxiv-2312.03539","url":null,"abstract":"Temporal coupled-mode theory (CMT) is an acclaimed and widely used\u0000theoretical framework for modeling the continuous wave (CW) response and\u0000temporal dynamics of any integrated or free-space photonic resonant structure.\u0000It was initially employed to understand how energy is coupled into and out of a\u0000cavity and how it is exchanged between different resonant modes. In the 30\u0000years that followed its establishment, CMT has been expanded to describe a\u0000broad range of nonlinear interactions as well (self- and cross-phase\u0000modulation, saturable absorption, frequency generation, gain, etc.). In this\u0000tutorial, we thoroughly present the basic principles and the evolution of CMT\u0000throughout the years, showcasing its immense capabilities for the analysis and\u0000design of linear and nonlinear resonant photonic systems. Importantly, we focus\u0000on examples of modern, open nanophotonic resonators incorporating contemporary\u0000bulk or sheet (2D) materials that may be lossy and dispersive. For each\u0000linear/nonlinear effect under study we follow a meticulous, step-by-step\u0000approach, starting from an accurate model of the physical phenomenon and\u0000proceeding to its introduction in the CMT framework all the way to the\u0000efficient solution of the resulting system of equations. Our work highlights\u0000the merits of CMT as an efficient, accurate, and versatile theoretical tool. We\u0000envision that it can serve both as an introductory reference for any reader, as\u0000well as a comprehensive handbook on how to incorporate a broad range of linear\u0000and nonlinear effects in the CMT framework.","PeriodicalId":501275,"journal":{"name":"arXiv - PHYS - Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138545962","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 construct vertex algebras $mathbb{V}(Y,S)$ from divisors $S$ on toric Calabi-Yau threefolds $Y$, satisfying conjectures of Gaiotto-Rapcak and Feigin-Gukov, as the kernel of screening operators on lattice vertex algebras determined by the GKM graph of $Y$ and a filtration on $mathcal{O}_S$. We prove that there are representations of $mathbb{V}(Y,S)$ on the homology groups of various moduli spaces of coherent sheaves on $Y$ supported on $S$ constructed in a companion paper with Rapcak, defined by certain Hecke modifications of these sheaves along points and curve classes in the divisor $S$. This generalizes the common mathematical formulation of a conjecture of Alday-Gaiotto-Tachikawa, the special case in which $Y=mathbb{C}^3$ and $S=r[mathbb{C}^2]$, to toric threefolds and divisors as proposed by Gaiotto-Rapcak. We outline an approach to the general conjecture and prove many special cases and partial results using tools developed in the companion paper, following the proof of the original conjecture by Schiffmann-Vasserot and its generalization to divisors in $mathbb{C}^3$ by Rapcak-Soibelman-Yang-Zhao. The vertex algebras $mathbb{V}(Y,S)$ conjecturally include $W$-superalgebras $ W_{f_0,f_1}^kappa(mathfrak{gl}_{m|n})$ and genus zero class $mathcal{S}$ chiral algebras $mathbb{V}^{mathcal{S}}_{text{Gl}_m;f_1,...,f_k}$, each for general nilpotents $f_i$. By definition, this implies the existence of a family of compatible free field realizations of these vertex algebras, relevant to their parabolic induction and inverse quantum Hamiltonian reduction. We prove these conjectures in the examples of lowest non-trivial rank for each case, and outline the proof in general for some cases.
{"title":"Vertex algebras from divisors on Calabi-Yau threefolds","authors":"Dylan Butson","doi":"arxiv-2312.03648","DOIUrl":"https://doi.org/arxiv-2312.03648","url":null,"abstract":"We construct vertex algebras $mathbb{V}(Y,S)$ from divisors $S$ on toric\u0000Calabi-Yau threefolds $Y$, satisfying conjectures of Gaiotto-Rapcak and\u0000Feigin-Gukov, as the kernel of screening operators on lattice vertex algebras\u0000determined by the GKM graph of $Y$ and a filtration on $mathcal{O}_S$. We\u0000prove that there are representations of $mathbb{V}(Y,S)$ on the homology\u0000groups of various moduli spaces of coherent sheaves on $Y$ supported on $S$\u0000constructed in a companion paper with Rapcak, defined by certain Hecke\u0000modifications of these sheaves along points and curve classes in the divisor\u0000$S$. This generalizes the common mathematical formulation of a conjecture of\u0000Alday-Gaiotto-Tachikawa, the special case in which $Y=mathbb{C}^3$ and\u0000$S=r[mathbb{C}^2]$, to toric threefolds and divisors as proposed by\u0000Gaiotto-Rapcak. We outline an approach to the general conjecture and prove many\u0000special cases and partial results using tools developed in the companion paper,\u0000following the proof of the original conjecture by Schiffmann-Vasserot and its\u0000generalization to divisors in $mathbb{C}^3$ by Rapcak-Soibelman-Yang-Zhao. The vertex algebras $mathbb{V}(Y,S)$ conjecturally include $W$-superalgebras\u0000$ W_{f_0,f_1}^kappa(mathfrak{gl}_{m|n})$ and genus zero class $mathcal{S}$\u0000chiral algebras $mathbb{V}^{mathcal{S}}_{text{Gl}_m;f_1,...,f_k}$, each for\u0000general nilpotents $f_i$. By definition, this implies the existence of a family\u0000of compatible free field realizations of these vertex algebras, relevant to\u0000their parabolic induction and inverse quantum Hamiltonian reduction. We prove\u0000these conjectures in the examples of lowest non-trivial rank for each case, and\u0000outline the proof in general for some cases.","PeriodicalId":501275,"journal":{"name":"arXiv - PHYS - Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138545814","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}
Ken Furukawa, Yoshikazu Giga, Matthias Hieber, Amru Hussein, Takahito Kashiwabara, Marc Wrona
The primitive equations are derived from the $3D$-Navier-Stokes equations by the hydrostatic approximation. Formally, assuming an $varepsilon$-thin domain and anisotropic viscosities with vertical viscosity $nu_z=mathcal{O}(varepsilon^gamma)$ where $gamma=2$, one obtains the primitive equations with full viscosity as $varepsilonto 0$. Here, we take two more limit equations into consideration: For $gamma<2$ the $2D$-Navier-Stokes equations are obtained. For $gamma>2$ the primitive equations with only horizontal viscosity $-Delta_H$ as $varepsilonto 0$. Thus, there are three possible limits of the hydrostatic approximation depending on the assumption on the vertical viscosity. The latter convergence has been proven recently by Li, Titi, and Yuan using energy estimates. Here, we consider more generally $nu_z=varepsilon^2 delta$ and show how maximal regularity methods and quadratic inequalities can be an efficient approach to the same end for $varepsilon,deltato 0$. The flexibility of our methods is also illustrated by the convergence for $deltato infty$ and $varepsilonto 0$ to the $2D$-Navier-Stokes equations.
{"title":"The three limits of the hydrostatic approximation","authors":"Ken Furukawa, Yoshikazu Giga, Matthias Hieber, Amru Hussein, Takahito Kashiwabara, Marc Wrona","doi":"arxiv-2312.03418","DOIUrl":"https://doi.org/arxiv-2312.03418","url":null,"abstract":"The primitive equations are derived from the $3D$-Navier-Stokes equations by\u0000the hydrostatic approximation. Formally, assuming an $varepsilon$-thin domain\u0000and anisotropic viscosities with vertical viscosity\u0000$nu_z=mathcal{O}(varepsilon^gamma)$ where $gamma=2$, one obtains the\u0000primitive equations with full viscosity as $varepsilonto 0$. Here, we take\u0000two more limit equations into consideration: For $gamma<2$ the\u0000$2D$-Navier-Stokes equations are obtained. For $gamma>2$ the primitive\u0000equations with only horizontal viscosity $-Delta_H$ as $varepsilonto 0$.\u0000Thus, there are three possible limits of the hydrostatic approximation\u0000depending on the assumption on the vertical viscosity. The latter convergence\u0000has been proven recently by Li, Titi, and Yuan using energy estimates. Here, we\u0000consider more generally $nu_z=varepsilon^2 delta$ and show how maximal\u0000regularity methods and quadratic inequalities can be an efficient approach to\u0000the same end for $varepsilon,deltato 0$. The flexibility of our methods is\u0000also illustrated by the convergence for $deltato infty$ and $varepsilonto\u00000$ to the $2D$-Navier-Stokes equations.","PeriodicalId":501275,"journal":{"name":"arXiv - PHYS - Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138545906","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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