Journal of Geometry and Physics最新文献

{"title":"Partial order and topology of Hermitian matrices and quantum Choquet integrals for density matrices with given expectation values","authors":"A. Vourdas","doi":"10.1016/j.geomphys.2025.105555","DOIUrl":"10.1016/j.geomphys.2025.105555","url":null,"abstract":"<div><div>The set <em>M</em> of <span><math><mi>d</mi><mo>×</mo><mi>d</mi></math></span> Hermitian matrices (observables) is studied as a partially ordered set with the Löwner partial order. Upper and lower sets in it, define the concept of cumulativeness (used mainly with scalar quantities) in the context of Hermitian matrices. Partial order and topology are intimately related to each other and the set <em>M</em> of Hermitian matrices is also studied as a topological space, where open and closed sets are the upper and lower sets. It is shown that the set <em>M</em> of Hermitian matrices is a <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> topological space, and its subset <span><math><mi>D</mi></math></span> of density matrices is Hausdorff totally disconnected topological space. These ideas are a prerequisite for studying quantum Choquet integrals with Hermitian matrices (as opposed to classical Choquet integrals with scalar quantities). Capacities (non-additive probabilities), cumulative quantities that involve Hermitian matrices, and Möbius transforms that remove the overlaps between non-commuting observables, are used in quantum Choquet integrals. An application of the formalism is to find a density matrix, with given expectation values with respect to <em>n</em> (non-commuting) observables. Examples of calculations of such a density matrix (with quantified errors in its expectation values), are presented.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"216 ","pages":"Article 105555"},"PeriodicalIF":1.6,"publicationDate":"2025-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144223185","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":"Hierarchies in relative Picard-Lefschetz theory","authors":"Marko Berghoff ,&nbsp;Erik Panzer","doi":"10.1016/j.geomphys.2025.105539","DOIUrl":"10.1016/j.geomphys.2025.105539","url":null,"abstract":"<div><div>We prove a relative version of the Picard-Lefschetz theorem, describing the variation of relative homology groups <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>∖</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>,</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>∖</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>)</mo></math></span> in the fibers of a smooth fiber bundle <span><math><mi>Y</mi><mo>→</mo><mi>T</mi></math></span> of complex manifolds with <span><math><mi>A</mi><mo>∪</mo><mi>B</mi><mo>⊂</mo><mi>Y</mi></math></span> transverse. From this we derive the vanishing of certain iterated variations, a system of constraints dubbed “hierarchy”.</div><div>As applications, we rederive the known analytic structure of Aomoto polylogarithms and massive one loop Feynman integrals. Moreover, we introduce the “simple type” to prove hierarchy constraints in degenerate cases where the Picard-Lefschetz formula does not apply, e.g. the massless triangle or the ice cream cone Feynman diagram. We compare our findings with a “classical” hierarchy of iterated variations (from 1960's <em>S</em>-matrix theory) and show how our setup not only explains, but also refines the latter. In order to do so, we need to further resolve the geometry of Feynman motives: We boldly blow up what no one has blown up before.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"216 ","pages":"Article 105539"},"PeriodicalIF":1.6,"publicationDate":"2025-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144322411","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":"CY/LG correspondence for Weil-Petersson metrics and tt⁎ structures","authors":"Xinxing Tang ,&nbsp;Junrong Yan","doi":"10.1016/j.geomphys.2025.105545","DOIUrl":"10.1016/j.geomphys.2025.105545","url":null,"abstract":"<div><div>The purpose of this paper is to establish the Calabi-Yau/Landau-Ginzburg (CY/LG) correspondence for the <span><math><mi>t</mi><msup><mrow><mi>t</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> geometry structure, which is a generalized version of the variation of Hodge structures. To begin, consider a homogeneous polynomial <span><math><mi>f</mi><mo>:</mo><msup><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>→</mo><mi>C</mi></math></span> of degree <em>n</em>. We can put a natural Hodge structure on the space of harmonic forms with respect to the twisted Laplacian <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>f</mi></mrow></msub></math></span> (cf. §3.1). Additionally, there exists a natural Hodge structure on the cohomology of the Calabi-Yau hypersurface defined by <em>f</em> in the projective space <span><math><msup><mrow><mi>P</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span>. Naturally, one may question the relationship between these two Hodge structures and, more generally, the connection between the corresponding <span><math><mi>t</mi><msup><mrow><mi>t</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> structures. As an application of our main result, we modify the real structure proposed by Cecotti on the Jacobi ring of <em>f</em> (cf. <span><span>[8, (4.2)]</span></span>). We show that this modified real structure not only aligns with pole-order filtration and Grothendieck residue pairing on the Jacobi ring of <em>f</em>, but it is also preserved by the residue map constructed by Griffiths-Carlson. Finally, it is crucial to emphasize that the CY/LG correspondence for the <span><math><mi>t</mi><msup><mrow><mi>t</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> structures establishes the fundamental basis for studying the CY/LG correspondence for the genus one terms in the B-model.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"216 ","pages":"Article 105545"},"PeriodicalIF":1.6,"publicationDate":"2025-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144223183","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":"Outer billiards in the complex hyperbolic plane","authors":"Yamile Godoy,&nbsp;Marcos Salvai","doi":"10.1016/j.geomphys.2025.105544","DOIUrl":"10.1016/j.geomphys.2025.105544","url":null,"abstract":"<div><div>Given a quadratically convex compact connected oriented hypersurface <em>N</em> of the complex hyperbolic plane, we prove that the characteristic rays of the symplectic form restricted to <em>N</em> determine a double geodesic foliation of the exterior <em>U</em> of <em>N</em>. This induces an outer billiard map <em>B</em> on <em>U</em>. We prove that <em>B</em> is a diffeomorphism (notice that weaker notions of strict convexity may allow the billiard map to be well-defined and invertible, but not smooth) and moreover, a symplectomorphism. These results generalize known geometric properties of the outer billiard maps in the hyperbolic plane and complex Euclidean space.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"215 ","pages":"Article 105544"},"PeriodicalIF":1.6,"publicationDate":"2025-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144194788","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":"Twisting O-operators by (2,3)-cocycle of Hom-Lie-Yamaguti algebras with representations","authors":"Sami Mabrouk ,&nbsp;Sergei Silvestrov ,&nbsp;Fatma Zouaidi","doi":"10.1016/j.geomphys.2025.105546","DOIUrl":"10.1016/j.geomphys.2025.105546","url":null,"abstract":"<div><div>In this paper, we first introduce the notion of twisted <span><math><mi>O</mi></math></span>-operators on a Hom-Lie-Yamaguti algebra by a given <span><math><mo>(</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>)</mo></math></span>-cocycle with coefficients in a representation. We show that a twisted <span><math><mi>O</mi></math></span>-operator induces a Hom-Lie-Yamaguti structure. We also introduce the notion of a weighted Reynolds operator on a Hom-Lie-Yamaguti algebra, which can serve as a special case of twisted <span><math><mi>O</mi></math></span>-operators on Hom-Lie-Yamaguti algebras. Then, we define a cohomology of twisted <span><math><mi>O</mi></math></span>-operator on Hom-Lie-Yamaguiti algebras with coefficients in a representation. Furthermore, we introduce and study the Hom-NS-Lie-Yamaguti algebras as the underlying structure of the twisted <span><math><mi>O</mi></math></span>-operator on Hom-Lie-Yamaguti algebras. Finally, we investigate the twisted <span><math><mi>O</mi></math></span>-operator on Hom-Lie-Yamaguti algebras induced by the twisted <span><math><mi>O</mi></math></span>-operator on Hom-Lie algebras.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"216 ","pages":"Article 105546"},"PeriodicalIF":1.6,"publicationDate":"2025-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144223184","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":"Prequantization of differential characters of Lie groupoids","authors":"Cheng-Yong Du","doi":"10.1016/j.geomphys.2025.105547","DOIUrl":"10.1016/j.geomphys.2025.105547","url":null,"abstract":"<div><div>In this paper, we describe a category <span><math><msubsup><mrow><mi>DC</mi></mrow><mrow><mrow><mi>ex</mi></mrow><mo>,</mo><mn>3</mn><mo>−</mo><mn>1</mn></mrow><mrow><mn>3</mn></mrow></msubsup><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of degree-3 differential characters of a Lie groupoid <span><math><mi>G</mi></math></span> together with a prequantization functor Preq from it to the category <span><math><mi>G</mi><mi>e</mi><mi>r</mi><msub><mrow><mi>b</mi></mrow><mrow><mi>∇</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-central extensions with pseudo-connections over <span><math><mi>G</mi></math></span>, and show that Preq is an equivalence of categories and the isomorphism classes of <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-central extensions with pseudo-connections over <span><math><mi>G</mi></math></span> are classified by the cohomology group <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>(</mo><mi>D</mi><msubsup><mrow><mi>C</mi></mrow><mrow><mrow><mi>ex</mi></mrow><mo>,</mo><mn>3</mn><mo>−</mo><mn>1</mn></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo></math></span> of degree-3 differential characters. As an application, we characterize closed integral 3-forms with prequantization <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-central extensions and pseudo-connections for all Lie groupoids. This generalizes Behrend–Xu's prequantization result of degree 3-context for Lie groupoids satisfying <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><msup><mrow><mi>Ω</mi></mrow><mrow><mn>0</mn></mrow></msup><mo>(</mo><msub><mrow><mi>G</mi></mrow><mrow><mo>•</mo></mrow></msub><mo>)</mo><mo>,</mo><mo>∂</mo><mo>)</mo><mo>=</mo><mn>0</mn></math></span>. Moreover we identify the group of flat <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-central extensions over a Lie groupoid <span><math><mi>G</mi></math></span> with the cohomology group <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><msubsup><mrow><mi>C</mi></mrow><mrow><mi>ex</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>(</mo><mi>G</mi><mo>,</mo><mi>R</mi><mo>/</mo><mi>Z</mi><mo>)</mo><mo>)</mo></math></span> of a modification of the complex of singular cochains with coefficient in <span><math><mi>R</mi><mo>/</mo><mi>Z</mi></math></span>. We also extend these results to differentiable stacks.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"216 ","pages":"Article 105547"},"PeriodicalIF":1.6,"publicationDate":"2025-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144203640","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":"Representations and cohomology of Rota-Baxter Lie conformal algebras","authors":"Jun Zhao ,&nbsp;Bing Sun ,&nbsp;Liangyun Chen","doi":"10.1016/j.geomphys.2025.105542","DOIUrl":"10.1016/j.geomphys.2025.105542","url":null,"abstract":"<div><div>In this paper, we study representations and cohomology of a weighted Rota-Baxter Lie conformal algebra. Given a weighted Rota-Baxter Lie conformal algebra <span><math><mo>(</mo><mi>R</mi><mo>,</mo><mi>T</mi><mo>)</mo></math></span> and its representation <span><math><mo>(</mo><mi>M</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span>, we define its cohomology and discuss the relation with the cohomology of weighted Rota-Baxter associative conformal algebra. As applications of the cohomology theory, we study abelian extensions, formal deformations of a weighted Rota-Baxter Lie conformal algebra.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"216 ","pages":"Article 105542"},"PeriodicalIF":1.6,"publicationDate":"2025-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144203639","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":"Integrability structures of the (2 + 1)-dimensional Euler equation","authors":"I.S. Krasil′shchik,&nbsp;O.I. Morozov","doi":"10.1016/j.geomphys.2025.105543","DOIUrl":"10.1016/j.geomphys.2025.105543","url":null,"abstract":"<div><div>We construct a local variational Poisson structure (a Hamiltonian operator) for the <span><math><mo>(</mo><mn>2</mn><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-dimensional Euler equation in vorticity form. The inverse defines a nonlocal symplectic structure for the equation. We describe the action of this operator on the infinitesimal contact symmetries in terms of differential coverings over the Euler equation. Furthermore, we construct a nonlocal recursion operator for cosymmetries. Finally, we generalize the local variational Poisson structure for the Euler equation in vorticity form on a two-dimensional Riemannian manifold.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"215 ","pages":"Article 105543"},"PeriodicalIF":1.6,"publicationDate":"2025-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144194728","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":"Algebro-geometric initial value problems for integrable nonlinear lattices: Tetragonal curves and Riemann theta function solutions","authors":"Xianguo Geng,&nbsp;Minxin Jia,&nbsp;Ruomeng Li","doi":"10.1016/j.geomphys.2025.105541","DOIUrl":"10.1016/j.geomphys.2025.105541","url":null,"abstract":"<div><div>In this paper, we establish the theory of tetragonal curves and address a series of fundamental problems within this framework, including the construction of a basis for holomorphic Abelian differentials, Abelian differentials of the second and third kinds, Baker-Akhiezer functions, and meromorphic functions. Building on these results, we apply the theory of tetragonal curves to investigate algebro-geometric initial value problems for integrable nonlinear lattice systems. As an illustrative example, we employ the discrete zero-curvature equation and the discrete Lenard equation to derive a hierarchy of coupled Bogoyavlensky lattice equations associated with a discrete <span><math><mn>4</mn><mo>×</mo><mn>4</mn></math></span> matrix spectral problem. By analyzing the characteristic polynomial of the Lax matrix for this hierarchy, we introduce a tetragonal curve and its associated Riemann theta function, exploring the algebro-geometric properties of Baker-Akhiezer functions and a class of meromorphic functions. Using the Abel map and Abelian differentials, we precisely straighten out various flows. Finally, we obtain Riemann theta function solutions for the algebro-geometric initial value problems of the entire coupled Bogoyavlensky lattice hierarchy.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"215 ","pages":"Article 105541"},"PeriodicalIF":1.6,"publicationDate":"2025-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144169540","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":"Closed real plane curves of hyperelliptic solutions of focusing gauged modified KdV equation of genus three","authors":"Shigeki Matsutani","doi":"10.1016/j.geomphys.2025.105540","DOIUrl":"10.1016/j.geomphys.2025.105540","url":null,"abstract":"<div><div>The real and imaginary parts of the focusing modified Korteweg-de Vries (MKdV) equation defined over the complex field <span><math><mi>C</mi></math></span> give rise to the focusing gauged MKdV (FGMKdV) equations. As a generalization of Euler's elastica whose curvature obeys the focusing static MKdV (FSMKdV) equation, we study real plane curves whose curvature obeys the FGMKdV equation since the FSMKdV equation is a special case of the FGMKdV equation. In this paper, we focus on the hyperelliptic curves of genus three. By tuning some moduli parameters and initial conditions, we show closed real plane curves associated with the FGMKdV equation beyond Euler's figure-eight of elastica.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"215 ","pages":"Article 105540"},"PeriodicalIF":1.6,"publicationDate":"2025-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144134550","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}