You may have seen the words "topological recursion" mentioned in papers on matrix models, Hurwitz theory, Gromov-Witten theory, topological string theory, knot theory, topological field theory, JT gravity, cohomological field theory, free probability theory, gauge theories, to name a few. The goal of these lecture notes is certainly not to explain all these applications of the topological recursion framework. Rather, the intention is to provide a down-to-earth (and hopefully accessible) introduction to topological recursion itself, so that when you see these words mentioned, you can understand what it is all about. These lecture notes accompanied a series of lectures at the Les Houches school "Quantum Geometry (Mathematical Methods for Gravity, Gauge Theories and Non-Perturbative Physics)" in Summer 2024.
{"title":"Les Houches lecture notes on topological recursion","authors":"Vincent Bouchard","doi":"arxiv-2409.06657","DOIUrl":"https://doi.org/arxiv-2409.06657","url":null,"abstract":"You may have seen the words \"topological recursion\" mentioned in papers on\u0000matrix models, Hurwitz theory, Gromov-Witten theory, topological string theory,\u0000knot theory, topological field theory, JT gravity, cohomological field theory,\u0000free probability theory, gauge theories, to name a few. The goal of these\u0000lecture notes is certainly not to explain all these applications of the\u0000topological recursion framework. Rather, the intention is to provide a\u0000down-to-earth (and hopefully accessible) introduction to topological recursion\u0000itself, so that when you see these words mentioned, you can understand what it\u0000is all about. These lecture notes accompanied a series of lectures at the Les\u0000Houches school \"Quantum Geometry (Mathematical Methods for Gravity, Gauge\u0000Theories and Non-Perturbative Physics)\" in Summer 2024.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216506","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}
Flat bands correspond to the spatial localization of a quantum particle moving in a field with discrete or continuous translational invariance. The canonical example is the flat Landau levels in a homogeneous magnetic field. Several significant problems -- including flat bands in moir'e structures -- are related to the problem of a particle moving in an inhomogeneous magnetic field with zero total flux. We demonstrate that while perfectly flat bands in such cases are impossible, the introduction of a "non-Abelian component" -- a spin field with zero total curvature -- can lead to perfect localization. Several exactly solvable models are constructed: (i) a half-space up/down field with a sharp 1D boundary; (ii) an alternating up/down field periodic in one direction on a cylinder; and (iii) a doubly periodic alternating field on a torus. The exact solution on the torus is expressed in terms of elliptic functions. It is shown that flat bands are only possible for certain magic values of the field corresponding to a quantized flux through an individual tile. These exact solutions clarify the simple structure underlying flat bands in moir'e materials and provide a springboard for constructing a novel class of fractional quantum Hall states.
{"title":"Zero Flux Localization: Magic Revealed","authors":"Alireza Parhizkar, Victor Galitski","doi":"arxiv-2409.05942","DOIUrl":"https://doi.org/arxiv-2409.05942","url":null,"abstract":"Flat bands correspond to the spatial localization of a quantum particle\u0000moving in a field with discrete or continuous translational invariance. The\u0000canonical example is the flat Landau levels in a homogeneous magnetic field.\u0000Several significant problems -- including flat bands in moir'e structures --\u0000are related to the problem of a particle moving in an inhomogeneous magnetic\u0000field with zero total flux. We demonstrate that while perfectly flat bands in\u0000such cases are impossible, the introduction of a \"non-Abelian component\" -- a\u0000spin field with zero total curvature -- can lead to perfect localization.\u0000Several exactly solvable models are constructed: (i) a half-space up/down field\u0000with a sharp 1D boundary; (ii) an alternating up/down field periodic in one\u0000direction on a cylinder; and (iii) a doubly periodic alternating field on a\u0000torus. The exact solution on the torus is expressed in terms of elliptic\u0000functions. It is shown that flat bands are only possible for certain magic\u0000values of the field corresponding to a quantized flux through an individual\u0000tile. These exact solutions clarify the simple structure underlying flat bands\u0000in moir'e materials and provide a springboard for constructing a novel class\u0000of fractional quantum Hall states.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"53 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216550","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}
Somnath Maity, Vivek Kumar Singh, Pramod Padmanabhan, Vladimir Korepin
Classifying Yang-Baxter operators is an essential first step in the study of the simulation of integrable quantum systems on quantum computers. One of the earliest initiatives was taken by Hietarinta in classifying constant Yang-Baxter solutions for a two-dimensional local Hilbert space (qubit representation). He obtained 11 families of invertible solutions, including the one generated by the permutation operator. While these methods work well for 4 by 4 solutions, they become cumbersome for higher dimensional representations. In this work, we overcome this restriction by constructing the constant Yang-Baxter solutions in a representation independent manner by using ans"{a}tze from algebraic structures. We use four different algebraic structures that reproduce 10 of the 11 Hietarinta families when the qubit representation is chosen. The methods include a set of commuting operators, Clifford algebras, Temperley-Lieb algebras, and partition algebras. We do not obtain the $(2,2)$ Hietarinta class with these methods.
{"title":"Hietarinta's classification of $4times 4$ constant Yang-Baxter operators using algebraic approach","authors":"Somnath Maity, Vivek Kumar Singh, Pramod Padmanabhan, Vladimir Korepin","doi":"arxiv-2409.05375","DOIUrl":"https://doi.org/arxiv-2409.05375","url":null,"abstract":"Classifying Yang-Baxter operators is an essential first step in the study of\u0000the simulation of integrable quantum systems on quantum computers. One of the\u0000earliest initiatives was taken by Hietarinta in classifying constant\u0000Yang-Baxter solutions for a two-dimensional local Hilbert space (qubit\u0000representation). He obtained 11 families of invertible solutions, including the\u0000one generated by the permutation operator. While these methods work well for 4\u0000by 4 solutions, they become cumbersome for higher dimensional representations.\u0000In this work, we overcome this restriction by constructing the constant\u0000Yang-Baxter solutions in a representation independent manner by using\u0000ans\"{a}tze from algebraic structures. We use four different algebraic\u0000structures that reproduce 10 of the 11 Hietarinta families when the qubit\u0000representation is chosen. The methods include a set of commuting operators,\u0000Clifford algebras, Temperley-Lieb algebras, and partition algebras. We do not\u0000obtain the $(2,2)$ Hietarinta class with these methods.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216552","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}
Daniele Bielli, Christian Ferko, Liam Smith, Gabriele Tartaglino-Mazzucchelli
We generalize the auxiliary field deformations of the principal chiral model (PCM) introduced in arXiv:2405.05899 and arXiv:2407.16338 to sigma models whose target manifolds are symmetric or semi-symmetric spaces, including a Wess-Zumino term in the latter case. This gives rise to a new infinite family of classically integrable $mathbb{Z}_2$ and $mathbb{Z}_4$ coset models of the form which are of interest in applications of integrability to worldsheet string theory and holography. We demonstrate that every theory in this infinite class admits a zero-curvature representation for its equations of motion by exhibiting a Lax connection.
{"title":"Auxiliary Field Deformations of (Semi-)Symmetric Space Sigma Models","authors":"Daniele Bielli, Christian Ferko, Liam Smith, Gabriele Tartaglino-Mazzucchelli","doi":"arxiv-2409.05704","DOIUrl":"https://doi.org/arxiv-2409.05704","url":null,"abstract":"We generalize the auxiliary field deformations of the principal chiral model\u0000(PCM) introduced in arXiv:2405.05899 and arXiv:2407.16338 to sigma models whose\u0000target manifolds are symmetric or semi-symmetric spaces, including a\u0000Wess-Zumino term in the latter case. This gives rise to a new infinite family\u0000of classically integrable $mathbb{Z}_2$ and $mathbb{Z}_4$ coset models of the\u0000form which are of interest in applications of integrability to worldsheet\u0000string theory and holography. We demonstrate that every theory in this infinite\u0000class admits a zero-curvature representation for its equations of motion by\u0000exhibiting a Lax connection.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"59 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216551","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}
By combining multiple copies of noisy coherent states of light (or other bosonic systems), it is possible to obtain a single mode in a state with lesser noise, a process known as distillation or purification of coherent states. We investigate the distillation of coherent states from coherent thermal states under general phase-insensitive operations, and find a distillation protocol that is optimal in the asymptotic regime, i.e., when the number of input copies is much greater than 1. Remarkably, we find that in this regime, the error -- as quantified by infidelity (one minus the fidelity) of the output state with the desired coherent state -- is proportional to the inverse of the purity of coherence of the input state, a quantity obtained from the Right-Logarithmic-Derivative (RLD) Fisher information metric, hence revealing an operational interpretation of this quantity. The heart of this protocol is a phase-insensitive channel that optimally converts an input coherent thermal state with high amplitude, into an output with significantly lower amplitude and temperature. Under this channel, the purity of coherence remains asymptotically conserved. While both the input and desired output are Gaussian states, we find that the optimal protocol cannot be a Gaussian channel. Among Gaussian phase-insensitive channels, the optimal distillation protocol is a simple linear optical scheme that can be implemented with beam splitters.
{"title":"Optimal Distillation of Coherent States with Phase-Insensitive Operations","authors":"Shiv Akshar Yadavalli, Iman Marvian","doi":"arxiv-2409.05974","DOIUrl":"https://doi.org/arxiv-2409.05974","url":null,"abstract":"By combining multiple copies of noisy coherent states of light (or other\u0000bosonic systems), it is possible to obtain a single mode in a state with lesser\u0000noise, a process known as distillation or purification of coherent states. We\u0000investigate the distillation of coherent states from coherent thermal states\u0000under general phase-insensitive operations, and find a distillation protocol\u0000that is optimal in the asymptotic regime, i.e., when the number of input copies\u0000is much greater than 1. Remarkably, we find that in this regime, the error --\u0000as quantified by infidelity (one minus the fidelity) of the output state with\u0000the desired coherent state -- is proportional to the inverse of the purity of\u0000coherence of the input state, a quantity obtained from the\u0000Right-Logarithmic-Derivative (RLD) Fisher information metric, hence revealing\u0000an operational interpretation of this quantity. The heart of this protocol is a\u0000phase-insensitive channel that optimally converts an input coherent thermal\u0000state with high amplitude, into an output with significantly lower amplitude\u0000and temperature. Under this channel, the purity of coherence remains\u0000asymptotically conserved. While both the input and desired output are Gaussian\u0000states, we find that the optimal protocol cannot be a Gaussian channel. Among\u0000Gaussian phase-insensitive channels, the optimal distillation protocol is a\u0000simple linear optical scheme that can be implemented with beam splitters.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"39 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216520","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}
Claudia García, Martina Magliocca, Nicolas Meunier
Cell motility is connected to the spontaneous symmetry breaking of a circular shape. In https://doi.org/10.1103/PhysRevLett.110.078102, Blanch-Mercader and Casademunt perfomed a nonlinear analysis of the minimal model proposed by Callan and Jones https://doi.org/10.1103/PhysRevLett.100.258106 and numerically conjectured the existence of traveling solutions once that symmetry is broken. In this work, we prove analytically that conjecture by means of nonlinear bifurcation techniques.
细胞运动与圆形的自发对称破缺有关。在 https://doi.org/10.1103/PhysRevLett.110.078102，Blanch-Mercader 和 Casademunt 对 Callan 和 Jones 提出的最小模型进行了非线性分析 https://doi.org/10.1103/PhysRevLett.100.258106，并从数值上猜想，一旦对称性被打破，就会存在行进解。在这项工作中，我们通过非线性分岔技术分析证明了这一猜想。
{"title":"Traveling Motility of Actin Lamellar Fragments Under spontaneous symmetry breaking","authors":"Claudia García, Martina Magliocca, Nicolas Meunier","doi":"arxiv-2409.05762","DOIUrl":"https://doi.org/arxiv-2409.05762","url":null,"abstract":"Cell motility is connected to the spontaneous symmetry breaking of a circular\u0000shape. In https://doi.org/10.1103/PhysRevLett.110.078102, Blanch-Mercader and\u0000Casademunt perfomed a nonlinear analysis of the minimal model proposed by\u0000Callan and Jones https://doi.org/10.1103/PhysRevLett.100.258106 and numerically\u0000conjectured the existence of traveling solutions once that symmetry is broken.\u0000In this work, we prove analytically that conjecture by means of nonlinear\u0000bifurcation techniques.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216523","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}
Charles Collot, Tej-Eddine Ghoul, Nader Masmoudi, Van Tien Nguyen
It is well-known that the two-dimensional Keller-Segel system admits finite time blowup solutions, which is the case if the initial density has a total mass greater than $8pi$ and a finite second moment. Several constructive examples of such solutions have been obtained, where for all of them a perturbed stationary state undergoes scale instability and collapses at a point, resulting in a $8pi$-mass concentration. It was conjectured that singular solutions concentrating simultaneously more than one solitons could exist. We construct rigorously such a new blowup mechanism, where two stationary states are simultaneously collapsing and colliding, resulting in a $16pi$-mass concentration at a single blowup point, and with a new blowup rate which corresponds to the formal prediction by Seki, Sugiyama and Vel'azquez. We develop for the first time a robust framework to construct rigorously such blowup solutions involving simultaneously the non-radial collision and concentration of several solitons, which we expect to find applications to other evolution problems.
{"title":"Singularity formed by the collision of two collapsing solitons in interaction for the 2D Keller-Segel system","authors":"Charles Collot, Tej-Eddine Ghoul, Nader Masmoudi, Van Tien Nguyen","doi":"arxiv-2409.05363","DOIUrl":"https://doi.org/arxiv-2409.05363","url":null,"abstract":"It is well-known that the two-dimensional Keller-Segel system admits finite\u0000time blowup solutions, which is the case if the initial density has a total\u0000mass greater than $8pi$ and a finite second moment. Several constructive\u0000examples of such solutions have been obtained, where for all of them a\u0000perturbed stationary state undergoes scale instability and collapses at a\u0000point, resulting in a $8pi$-mass concentration. It was conjectured that\u0000singular solutions concentrating simultaneously more than one solitons could\u0000exist. We construct rigorously such a new blowup mechanism, where two\u0000stationary states are simultaneously collapsing and colliding, resulting in a\u0000$16pi$-mass concentration at a single blowup point, and with a new blowup rate\u0000which corresponds to the formal prediction by Seki, Sugiyama and Vel'azquez.\u0000We develop for the first time a robust framework to construct rigorously such\u0000blowup solutions involving simultaneously the non-radial collision and\u0000concentration of several solitons, which we expect to find applications to\u0000other evolution problems.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"50 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216555","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 the present paper we consider global solutions of a class of non-linear wave equations of the form begin{equation*} Box u= N(x,t,u)u, end{equation*} where the nonlinearity~$ N(x,t,u)u$ is assumed to satisfy appropriate boundedness assumptions. Under these appropriate assumptions we prove that the free channel wave operator exists. Moreover, if the interaction term~$N(x,t,u)u$ is localised, then we prove that the global solution of the full nonlinear equation can be decomposed into a `free' part and a `localised' part. The present work can be seen as an extension of the scattering results of~cite{SW20221} for the Schr"odinger equation.
{"title":"Decomposition of global solutions for a class of nonlinear wave equations","authors":"Georgios Mavrogiannis, Avy Soffer, Xiaoxu Wu","doi":"arxiv-2409.05272","DOIUrl":"https://doi.org/arxiv-2409.05272","url":null,"abstract":"In the present paper we consider global solutions of a class of non-linear\u0000wave equations of the form begin{equation*} Box u= N(x,t,u)u, end{equation*} where the nonlinearity~$ N(x,t,u)u$ is\u0000assumed to satisfy appropriate boundedness assumptions. Under these appropriate assumptions we prove that the free channel wave\u0000operator exists. Moreover, if the interaction term~$N(x,t,u)u$ is localised,\u0000then we prove that the global solution of the full nonlinear equation can be\u0000decomposed into a `free' part and a `localised' part. The present work can be seen as an extension of the scattering results\u0000of~cite{SW20221} for the Schr\"odinger equation.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216556","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 work, we show that the completeness relation for the eigenvectors, which is an essential assumption of quantum mechanics, remains true if the initial Hamiltonian, having a discrete spectrum, is modified by a delta potential (to be made precise by a renormalization scheme) supported at a point in two and three-dimensional compact manifolds or Euclidean spaces. The formulation can be easily extended to $N$ center case, and the case where delta interaction is supported on curves in the plane or space.
{"title":"Completeness Relation in Renormalized Quantum Systems","authors":"Fatih Erman, O. Teoman Turgut","doi":"arxiv-2409.05372","DOIUrl":"https://doi.org/arxiv-2409.05372","url":null,"abstract":"In this work, we show that the completeness relation for the eigenvectors,\u0000which is an essential assumption of quantum mechanics, remains true if the\u0000initial Hamiltonian, having a discrete spectrum, is modified by a delta\u0000potential (to be made precise by a renormalization scheme) supported at a point\u0000in two and three-dimensional compact manifolds or Euclidean spaces. The\u0000formulation can be easily extended to $N$ center case, and the case where delta\u0000interaction is supported on curves in the plane or space.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"472 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216554","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 develop a comprehensive framework for realizing anyon condensation of topological orders within the string-net model by constructing a Hamiltonian that bridges the parent string-net model before and the child string-net model after anyon condensation. Our approach classifies all possible types of bosonic anyon condensation in any parent string-net model and identifies the basic degrees of freedom in the corresponding child models. Compared with the traditional UMTC perspective of topological orders, our method offers a finer categorical description of anyon condensation at the microscopic level. We also explicitly represent relevant UMTC categorical entities characterizing anyon condensation through our model-based physical quantities, providing practical algorithms for calculating these categorical data.
{"title":"Nonabelian Anyon Condenstion in 2+1d topological orders: A String-Net Model Realization","authors":"Yu Zhao, Yidun Wan","doi":"arxiv-2409.05852","DOIUrl":"https://doi.org/arxiv-2409.05852","url":null,"abstract":"We develop a comprehensive framework for realizing anyon condensation of\u0000topological orders within the string-net model by constructing a Hamiltonian\u0000that bridges the parent string-net model before and the child string-net model\u0000after anyon condensation. Our approach classifies all possible types of bosonic\u0000anyon condensation in any parent string-net model and identifies the basic\u0000degrees of freedom in the corresponding child models. Compared with the\u0000traditional UMTC perspective of topological orders, our method offers a finer\u0000categorical description of anyon condensation at the microscopic level. We also\u0000explicitly represent relevant UMTC categorical entities characterizing anyon\u0000condensation through our model-based physical quantities, providing practical\u0000algorithms for calculating these categorical data.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216522","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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