{"title":"On the geometry of Lagrangian one-forms","authors":"Vincent Caudrelier, Derek Harland","doi":"10.1007/s11005-025-01925-0","DOIUrl":"10.1007/s11005-025-01925-0","url":null,"abstract":"<div><p>Lagrangian multiform theory is a variational framework for integrable systems. In this article, we introduce a new formulation which is based on symplectic geometry and which treats position, momentum and time coordinates of a finite-dimensional integrable hierarchy on an equal footing. This formulation allows a streamlined one-step derivation of both the multi-time Euler–Lagrange equations and the closure relation (encoding integrability). We argue that any Lagrangian one-form for a finite-dimensional system can be recast in our new framework. This framework easily extends to non-commuting flows, and we show that the equations characterising (infinitesimal) Hamiltonian Lie group actions are variational in character. We reinterpret these equations as a system of compatible non-autonomous Hamiltonian equations.</p></div>","PeriodicalId":685,"journal":{"name":"Letters in Mathematical Physics","volume":"115 2","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11005-025-01925-0.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143726632","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Decomposition of global solutions for a class of nonlinear wave equations","authors":"Georgios Mavrogiannis, Avy Soffer, Xiaoxu Wu","doi":"10.1007/s11005-025-01924-1","DOIUrl":"10.1007/s11005-025-01924-1","url":null,"abstract":"<div><p>In the present paper, we consider global solutions of a class of nonlinear wave equations of the form </p><div><div><span>$$begin{aligned} Box u= N(x,t,u)u, end{aligned}$$</span></div></div><p>where the nonlinearity <i>N</i>(<i>x</i>, <i>t</i>, <i>u</i>)<i>u</i> 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 <i>N</i>(<i>x</i>, <i>t</i>, <i>u</i>)<i>u</i> is localized, then we prove that the global solution of the full nonlinear equation can be decomposed into a ‘free’ part and a ‘localized’ part.\u0000</p></div>","PeriodicalId":685,"journal":{"name":"Letters in Mathematical Physics","volume":"115 2","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11005-025-01924-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143716810","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Anyons on M5-probes of Seifert 3-orbifoldsvia flux quantization","authors":"Hisham Sati, Urs Schreiber","doi":"10.1007/s11005-025-01918-z","DOIUrl":"10.1007/s11005-025-01918-z","url":null,"abstract":"<div><p>We observe that there is a rigorous derivation of (abelian) anyonic quantum states, hence of “topological order”, on the 1+2-dimensional fixed locus of M5-probes wrapped over a trivially Seifert-fibered 3-orbifold singularity. Similar statements have previously been conjectured by appeal to the unknown dynamics of “coincident” M5-branes, but neglecting effects of flux-quantization that, as we highlight, entail anyonic solitons already in the rigorously tractable case of single M5-brane probes. This is possible after globally completing the “self-dual” tensor field on probe M5-branes by flux-quantization in the non-abelian cohomology theory called <i>equivariant twistorial Cohomotopy</i>, which is admissible by recent results.\u0000</p></div>","PeriodicalId":685,"journal":{"name":"Letters in Mathematical Physics","volume":"115 2","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143688609","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Yang–Mills algebra and symmetry transformations of vertex operators","authors":"Andrei Mikhailov","doi":"10.1007/s11005-025-01922-3","DOIUrl":"10.1007/s11005-025-01922-3","url":null,"abstract":"<div><p>Linearized solutions of SUGRA equations of motion are described in the pure spinor formalism by vertex operators. Under supersymmetry transformations, they transform covariantly only up to BRST exact terms. We identify the cohomology class which is the obstacle for exact covariance. Computations are simplified by using the formalism of quadratic algebras.</p></div>","PeriodicalId":685,"journal":{"name":"Letters in Mathematical Physics","volume":"115 2","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143676480","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Supersymmetric Klein–Gordon and Dirac oscillators","authors":"Alexander D. Popov","doi":"10.1007/s11005-025-01927-y","DOIUrl":"10.1007/s11005-025-01927-y","url":null,"abstract":"<div><p>We have recently shown that the space of initial data (covariant phase space) of the relativistic oscillator in Minkowski space <span>(mathbb {R}^{3,1})</span> is a homogeneous Kähler–Einstein manifold <span>(Z_6=textrm{AdS}_7/textrm{U}(1) =textrm{U}(3,1)/textrm{U}(3)times textrm{U}(1))</span>. It was also shown that the energy eigenstates of the quantum relativistic oscillator form a direct sum of two weighted Bergman spaces of holomorphic (particles) and antiholomorphic (antiparticles) square-integrable functions on the covariant phase space <span>(Z_6)</span> of the classical oscillator. Here we show that the covariant phase space of the supersymmetric version of the relativistic oscillator (oscillating spinning particle) is the odd tangent bundle of the space <span>(Z_6)</span>. Quantizing this model yields a Dirac oscillator equation on the phase space whose solution space is a direct sum of two spinor spaces parametrized by holomorphic and antiholomorphic functions on the odd tangent bundle of <span>(Z_6)</span>. After expanding the general solution in Grassmann variables, we obtain components of the spinor field that are holomorphic and antiholomorphic functions from Bergman spaces on <span>(Z_6)</span> with different weight functions. Thus, the supersymmetric model under consideration is exactly solvable, Lorentz covariant and unitary.</p></div>","PeriodicalId":685,"journal":{"name":"Letters in Mathematical Physics","volume":"115 2","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11005-025-01927-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143668386","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the thermodynamic limit of interacting fermions in the continuum","authors":"Oliver Siebert","doi":"10.1007/s11005-025-01919-y","DOIUrl":"10.1007/s11005-025-01919-y","url":null,"abstract":"<div><p>We study the dynamics of non-relativistic fermions in <span>(mathbb {R}^d)</span> interacting through a pair potential. Employing methods developed by Buchholz in the framework of resolvent algebras, we consider an extension of the CAR algebra where the dynamics acts as a group of <span>(*)</span>-automorphisms, which are continuous in time in all sectors for fixed particle numbers. In addition, we construct a <span>(C^*)</span>-dynamical system by identifying a suitable dense subalgebra. Finally, we briefly discuss how this framework could be used to construct KMS states in the future.\u0000</p></div>","PeriodicalId":685,"journal":{"name":"Letters in Mathematical Physics","volume":"115 2","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11005-025-01919-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143667988","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Cédric Bény, Jason Crann, Hun Hee Lee, Sang-Jun Park, Sang-Gyun Youn
{"title":"GAUSSIAN QUANTUM INFORMATION OVER GENERAL QUANTUM KINEMATICAL SYSTEMS I: GAUSSIAN STATES","authors":"Cédric Bény, Jason Crann, Hun Hee Lee, Sang-Jun Park, Sang-Gyun Youn","doi":"10.1007/s11005-025-01908-1","DOIUrl":"10.1007/s11005-025-01908-1","url":null,"abstract":"<div><p>We develop a theory of Gaussian states over general quantum kinematical systems with finitely many degrees of freedom. The underlying phase space is described by a locally compact abelian (LCA) group <i>G</i> with a symplectic structure determined by a 2-cocycle on <i>G</i>. We use the concept of Gaussian distributions on LCA groups in the sense of Bernstein to define Gaussian states and completely characterize Gaussian states over 2-regular LCA groups of the form <span>(G= Ftimes widehat{F})</span> endowed with a canonical normalized 2-cocycle. This covers, in particular, the case of <i>n</i>-bosonic modes, <i>n</i>-qudit systems with odd <span>(dge 3)</span>, and <i>p</i>-adic quantum systems. Our characterization reveals a topological obstruction to Gaussian state entanglement when we decompose the quantum kinematical system into the Euclidean part and the remaining part (whose phase space admits a compact open subgroup). We then generalize the discrete Hudson theorem (Gross in J Math Phys 47(12):122107, 2006) to the case of totally disconnected 2-regular LCA groups. We also examine angle-number systems with phase space <span>(mathbb {T}^ntimes mathbb {Z}^n)</span> and fermionic/hard-core bosonic systems with phase space <span>(mathbb {Z}^{2n}_2)</span> (which are not 2-regular) and completely characterize their Gaussian states.</p></div>","PeriodicalId":685,"journal":{"name":"Letters in Mathematical Physics","volume":"115 2","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11005-025-01908-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143655362","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Petz–Rényi relative entropy in QFT from modular theory","authors":"Markus B. Fröb, Leonardo Sangaletti","doi":"10.1007/s11005-025-01923-2","DOIUrl":"10.1007/s11005-025-01923-2","url":null,"abstract":"<div><p>We consider the generalization of the Araki–Uhlmann formula for relative entropy to Petz–Rényi relative entropy. We compute this entropy for a free scalar field in the Minkowski wedge between the vacuum and a coherent state, as well as for the free chiral current in a thermal state. In contrast to the relative entropy which in these cases only depends on the symplectic form and thus reduces to the classical entropy of a wave packet, the Petz–Rényi relative entropy also depends on the symmetric part of the two-point function and is thus genuinely quantum. We also consider the relation with standard subspaces, where we define the Rényi entropy of a vector and show that it admits an upper bound given by the entropy of the vector.</p></div>","PeriodicalId":685,"journal":{"name":"Letters in Mathematical Physics","volume":"115 2","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11005-025-01923-2.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143632562","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Stabilization against collapse of 2D attractive Bose–Einstein condensates with repulsive, three-body interactions","authors":"Dinh-Thi Nguyen, Julien Ricaud","doi":"10.1007/s11005-025-01897-1","DOIUrl":"10.1007/s11005-025-01897-1","url":null,"abstract":"<div><p>We consider a trapped Bose gas of <i>N</i> identical bosons in two-dimensional space with both an attractive, two-body, scaled interaction and a repulsive, three-body, scaled interaction of the form <span>(-aN^{2alpha -1} U(N^alpha cdot ))</span> and <span>(bN^{4beta -2} W(N^beta cdot , N^beta cdot )))</span>, respectively, where <span>(a,b,alpha ,beta >0)</span> and <span>(int _{mathbb R^2}U(x) {text {d}} x = 1 = iint _{mathbb R^{4}} W(x,y) {text {d}} x {text {d}} y)</span>. We derive rigorously the cubic–quintic nonlinear Schrödinger semiclassical theory as the mean-field limit of the model and we investigate the behavior of the system in the double-limit <span>(a = a_N rightarrow a_*)</span> and <span>(b = b_N searrow 0)</span>. Moreover, we also consider the homogeneous problem where the trapping potential is removed.</p></div>","PeriodicalId":685,"journal":{"name":"Letters in Mathematical Physics","volume":"115 2","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143638629","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Complex structure on quantum-braided planes","authors":"Edwin Beggs, Shahn Majid","doi":"10.1007/s11005-025-01914-3","DOIUrl":"10.1007/s11005-025-01914-3","url":null,"abstract":"<div><p>We construct a quantum Dolbeault double complex <span>(oplus _{p,q}Omega ^{p,q})</span> on the quantum plane <span>({mathbb {C}}_q^2)</span>. This solves the long-standing problem that the standard differential calculus on the quantum plane is not a <span>(*)</span>-calculus, by embedding it as the holomorphic part of a <span>(*)</span>-calculus. We show in general that any Nichols–Woronowicz algebra or braided plane <span>(B_+(V))</span>, where <i>V</i> is an object in an Abelian <span>({mathbb {C}})</span>-linear braided bar category of real type, is a quantum complex space in this sense of a factorisable Dolbeault double complex. We combine the Chern construction on <span>(Omega ^{1,0})</span> in such a Dolbeault complex for an algebra <i>A</i> with its conjugate to construct a canonical metric-compatible connection on <span>(Omega ^1)</span> associated with a class of quantum metrics, and apply this to the quantum plane. We also apply this to finite groups <i>G</i> with Cayley graph generators split into two halves related by inversion, constructing such a Dolbeault complex <span>(Omega (G))</span> in this case. This construction recovers the quantum Levi-Civita connection for any edge-symmetric metric on the integer lattice with <span>(Omega ({mathbb {Z}}))</span>, now viewed as a quantum complex structure on <span>({mathbb {Z}})</span>. We also show how to build natural quantum metrics on <span>(Omega ^{1,0})</span> and <span>(Omega ^{0,1})</span> separately, where the inner product in the case of the quantum plane, in order to descend to <span>(otimes _A)</span>, is taken with values in an <i>A</i>-bimodule.</p></div>","PeriodicalId":685,"journal":{"name":"Letters in Mathematical Physics","volume":"115 2","pages":""},"PeriodicalIF":1.3,"publicationDate":"2025-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11005-025-01914-3.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143622089","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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