Taha Ameen, Kalle Kytölä, S. C. Park, David Radnell
{"title":"Slit-Strip Ising Boundary Conformal Field Theory 1: Discrete and Continuous Function Spaces","authors":"Taha Ameen, Kalle Kytölä, S. C. Park, David Radnell","doi":"10.1007/s11040-022-09442-5","DOIUrl":"10.1007/s11040-022-09442-5","url":null,"abstract":"<div><p>This is the first in a series of articles about recovering the full algebraic structure of a boundary conformal field theory (CFT) from the scaling limit of the critical Ising model in slit-strip geometry. Here, we introduce spaces of holomorphic functions in continuum domains as well as corresponding spaces of discrete holomorphic functions in lattice domains. We find distinguished sets of functions characterized by their singular behavior in the three infinite directions in the slit-strip domains, and note in particular that natural subsets of these functions span analogues of Hardy spaces. We prove convergence results of the distinguished discrete holomorphic functions to the continuum ones. In the subsequent articles, the discrete holomorphic functions will be used for the calculation of the Ising model fusion coefficients (as well as for the diagonalization of the Ising transfer matrix), and the convergence of the functions is used to prove the convergence of the fusion coefficients. It will also be shown that the vertex operator algebra of the boundary conformal field theory can be recovered from the limit of the fusion coefficients via geometric transformations involving the distinguished continuum functions.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11040-022-09442-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4204234","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Bispectrality of (AG_2) Calogero–Moser–Sutherland System","authors":"Misha Feigin, Martin Vrabec","doi":"10.1007/s11040-022-09440-7","DOIUrl":"10.1007/s11040-022-09440-7","url":null,"abstract":"<div><p>We consider the generalised Calogero–Moser–Sutherland quantum integrable system associated to the configuration of vectors <span>(AG_2)</span>, which is a union of the root systems <span>(A_2)</span> and <span>(G_2)</span>. We establish the existence of and construct a suitably defined Baker–Akhiezer function for the system, and we show that it satisfies bispectrality. We also find two corresponding dual difference operators of rational Macdonald–Ruijsenaars type in an explicit form.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11040-022-09440-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"5577218","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Stability of the Classical Catenoid and Darboux–Pöschl–Teller Potentials","authors":"Jens Hoppe, Per Moosavi","doi":"10.1007/s11040-022-09437-2","DOIUrl":"10.1007/s11040-022-09437-2","url":null,"abstract":"<div><p>We revisit the stability (instability) of the outer (inner) catenoid connecting two concentric circular rings and give an explicit new construction of the unstable mode of the inner catenoid by studying the spectrum of an exactly solvable one-dimensional Schrödinger operator with an asymmetric Darboux–Pöschl–Teller potential.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11040-022-09437-2.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"5131760","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
David A. Croydon, Makiko Sasada, Satoshi Tsujimoto
{"title":"Bi-infinite Solutions for KdV- and Toda-Type Discrete Integrable Systems Based on Path Encodings","authors":"David A. Croydon, Makiko Sasada, Satoshi Tsujimoto","doi":"10.1007/s11040-022-09435-4","DOIUrl":"10.1007/s11040-022-09435-4","url":null,"abstract":"<div><p>We define bi-infinite versions of four well-studied discrete integrable models, namely the ultra-discrete KdV equation, the discrete KdV equation, the ultra-discrete Toda equation, and the discrete Toda equation. For each equation, we show that there exists a unique solution to the initial value problem when the given data lies within a certain class, which includes the support of many shift ergodic measures. Our unified approach, which is also applicable to other integrable systems defined locally via lattice maps, involves the introduction of a path encoding (that is, a certain antiderivative) of the model configuration, for which we are able to describe the dynamics more generally than in previous work on finite size systems, periodic systems and semi-infinite systems. In particular, in each case we show that the behaviour of the system is characterized by a generalization of the classical ‘Pitman’s transformation’ of reflection in the past maximum, which is well-known to probabilists. The picture presented here also provides a means to identify a natural ‘carrier process’ for configurations within the given class, and is convenient for checking that the systems we discuss are all-time reversible. Finally, we investigate links between the different systems, such as showing that bi-infinite all-time solutions for the ultra-discrete KdV (resp. Toda) equation may appear as ultra-discretizations of corresponding solutions for the discrete KdV (resp. Toda) equation.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11040-022-09435-4.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4876694","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Quasilinear Systems of First Order PDEs with Nonlocal Hamiltonian Structures","authors":"Pierandrea Vergallo","doi":"10.1007/s11040-022-09438-1","DOIUrl":"10.1007/s11040-022-09438-1","url":null,"abstract":"<div><p>In this paper we investigate whether a quasilinear system of PDEs of first order admits Hamiltonian formulation with local and nonlocal operators. By using the theory of differential coverings, we find differential-geometric conditions necessary to write a given system with one of the three Hamiltonian operators investigated.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4631998","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On Bose–Einstein condensates in the Thomas–Fermi regime","authors":"Daniele Dimonte, Emanuela L. Giacomelli","doi":"10.1007/s11040-022-09439-0","DOIUrl":"10.1007/s11040-022-09439-0","url":null,"abstract":"<div><p>We study a system of <i>N</i> trapped bosons in the Thomas–Fermi regime with an interacting pair potential of the form <span>( g_N N^{3beta -1} V(N^beta x) )</span>, for some <span>( beta in (0,1/3) )</span> and <span>( g_N )</span> diverging as <span>( N rightarrow infty )</span>. We prove that there is complete Bose–Einstein condensation at the level of the ground state and, furthermore, that, if <span>( beta in (0,1/6) )</span>, condensation is preserved by the time evolution.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11040-022-09439-0.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4882957","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Limit Theorems for Multi-group Curie–Weiss Models via the Method of Moments","authors":"Werner Kirsch, Gabor Toth","doi":"10.1007/s11040-022-09433-6","DOIUrl":"10.1007/s11040-022-09433-6","url":null,"abstract":"<div><p>We study a multi-group version of the mean-field or Curie–Weiss spin model. For this model, we show how, analogously to the classical (single-group) model, the three temperature regimes are defined. Then we use the method of moments to determine for each regime how the vector of the group magnetisations behaves asymptotically. Some possible applications to social or political sciences are discussed.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11040-022-09433-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4954524","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Conjugate Frobenius Manifold and Inversion Symmetry","authors":"Zainab Al-Maamari, Yassir Dinar","doi":"10.1007/s11040-022-09436-3","DOIUrl":"10.1007/s11040-022-09436-3","url":null,"abstract":"<div><p>We give a conjugacy relation on certain type of Frobenius manifold structures using the theory of flat pencils of metrics. It leads to a geometric interpretation for the inversion symmetry of solutions to Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48415836","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Frobenius Manifolds on Orbits Spaces","authors":"Zainab Al-Maamari, Yassir Dinar","doi":"10.1007/s11040-022-09434-5","DOIUrl":"10.1007/s11040-022-09434-5","url":null,"abstract":"<div><p>The orbits space of an irreducible linear representation of a finite group is a variety whose coordinate ring is the ring of invariant polynomials. Boris Dubrovin proved that the orbits space of the standard reflection representation of an irreducible finite Coxeter group <span>({mathcal {W}})</span> acquires a natural polynomial Frobenius manifold structure. We apply Dubrovin’s method on various orbits spaces of linear representations of finite groups. We find some of them has non or several natural Frobenius manifold structures. On the other hand, these Frobenius manifold structures include rational and trivial structures which are not known to be related to the invariant theory of finite groups.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11040-022-09434-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43518701","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Equivariant Spectral Triples for Homogeneous Spaces of the Compact Quantum Group (U_q(2))","authors":"Satyajit Guin, Bipul Saurabh","doi":"10.1007/s11040-022-09432-7","DOIUrl":"10.1007/s11040-022-09432-7","url":null,"abstract":"<div><p>In this article, we study homogeneous spaces <span>(U_q(2)/_phi mathbb {T})</span> and <span>(U_q(2)/_psi mathbb {T})</span> of the compact quantum group <span>(U_q(2),,qin {mathbb {C}}setminus {0})</span>. The homogeneous space <span>(U_q(2)/_phi mathbb {T})</span> is shown to be the braided quantum group <span>(SU_q(2))</span>. The homogeneous space <span>(U_q(2)/_psi mathbb {T})</span> is established as a universal <span>(C^*)</span>-algebra given by a finite set of generators and relations. Its <span>({mathcal {K}})</span>-groups are computed and two families of finitely summable odd spectral triples, one is <span>(U_q(2))</span>-equivariant and the other is <span>(mathbb {T}^2)</span>-equivariant, are constructed. Using the index pairing, it is shown that the induced Fredholm modules for these families of spectral triples give each element in the <span>({mathcal {K}})</span>-homology group <span>(K^1(C(U_q(2)/_psi mathbb {T})))</span>.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134878272","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}