Pub Date : 2024-12-25DOI: 10.1016/j.geomphys.2024.105413
Oliver Fürst
We calculate the Witten index of a class of (non-Fredholm) Dirac-Schrödinger operators over for odd, and thus generalize known results for the case . For a concrete example of the potential, we give a index formula explicit in terms of the underlying potential, in particular showing that the Witten index assumes any real number on this class of operators.
{"title":"The Witten index of massless (d + 1)-Dirac-Schrödinger operators","authors":"Oliver Fürst","doi":"10.1016/j.geomphys.2024.105413","DOIUrl":"10.1016/j.geomphys.2024.105413","url":null,"abstract":"<div><div>We calculate the Witten index of a class of (non-Fredholm) Dirac-Schrödinger operators over <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi><mo>+</mo><mn>1</mn></mrow></msup></math></span> for <span><math><mi>d</mi><mo>≥</mo><mn>3</mn></math></span> odd, and thus generalize known results for the case <span><math><mi>d</mi><mo>=</mo><mn>1</mn></math></span>. For a concrete example of the potential, we give a index formula explicit in terms of the underlying potential, in particular showing that the Witten index assumes any real number on this class of operators.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"209 ","pages":"Article 105413"},"PeriodicalIF":1.6,"publicationDate":"2024-12-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143096609","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}
Pub Date : 2024-12-25DOI: 10.1016/j.geomphys.2024.105407
Alexander I. Bobenko , Sebastian Heller , Nick Schmitt
We study compact minimal surfaces in the 3-sphere which are constructed by successive reflections from a minimal n-gon — so-called minimal reflection surfaces. The minimal n-gon solves a free boundary problem in a fundamental piece of the respective reflection group. We investigate the combinatorics of the curvature lines of reflection surfaces, and construct new examples of minimal reflection surfaces based on pentagons based on integrable system theory. We end the paper by discussing the area of these minimal surfaces.
{"title":"Minimal reflection surfaces in S3. Curvature line foliations and new examples based on fundamental pentagons","authors":"Alexander I. Bobenko , Sebastian Heller , Nick Schmitt","doi":"10.1016/j.geomphys.2024.105407","DOIUrl":"10.1016/j.geomphys.2024.105407","url":null,"abstract":"<div><div>We study compact minimal surfaces in the 3-sphere which are constructed by successive reflections from a minimal <em>n</em>-gon — so-called minimal reflection surfaces. The minimal <em>n</em>-gon solves a free boundary problem in a fundamental piece of the respective reflection group. We investigate the combinatorics of the curvature lines of reflection surfaces, and construct new examples of minimal reflection surfaces based on pentagons based on integrable system theory. We end the paper by discussing the area of these minimal surfaces.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"209 ","pages":"Article 105407"},"PeriodicalIF":1.6,"publicationDate":"2024-12-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143096616","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}
Pub Date : 2024-12-20DOI: 10.1016/j.geomphys.2024.105408
Kazuhiro Hikami
We construct a generalization of the -type double affine Hecke algebra for the skein algebra on the twice-punctured torus using the Heegaard dual of the Iwahori–Hecke operator recently introduced in our previous article. We show that the automorphisms of our algebra correspond to the Dehn twists about the curves on . We also give the cluster algebraic construction of the classical limit of the skein algebra, where the Dehn twists are given in terms of the cluster mutations.
{"title":"A note on double affine Hecke algebra for skein algebra on twice-punctured torus","authors":"Kazuhiro Hikami","doi":"10.1016/j.geomphys.2024.105408","DOIUrl":"10.1016/j.geomphys.2024.105408","url":null,"abstract":"<div><div>We construct a generalization of the <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>∨</mo></mrow></msup><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-type double affine Hecke algebra for the skein algebra on the twice-punctured torus <span><math><msub><mrow><mi>Σ</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></math></span> using the Heegaard dual of the Iwahori–Hecke operator recently introduced in our previous article. We show that the automorphisms of our algebra correspond to the Dehn twists about the curves on <span><math><msub><mrow><mi>Σ</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></math></span>. We also give the cluster algebraic construction of the classical limit of the skein algebra, where the Dehn twists are given in terms of the cluster mutations.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"209 ","pages":"Article 105408"},"PeriodicalIF":1.6,"publicationDate":"2024-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143096620","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}
Pub Date : 2024-12-20DOI: 10.1016/j.geomphys.2024.105411
Andrea Galasso , Chin-Yu Hsiao
In this paper we study the microlocal properties of the Szegő kernel of a given compact connected orientable CR orbifold, whose Kohn Laplacian has closed range. This last assumption is satisfied if certain geometric conditions hold true, as in the smooth case. Furthermore, we explain how to generalize a CR version of quantization commutes with reduction to orbifolds. As an application, we give a pure analytic proof of Kodaira-Bailey theorem.
{"title":"On the singularities of the Szegő kernels on CR orbifolds","authors":"Andrea Galasso , Chin-Yu Hsiao","doi":"10.1016/j.geomphys.2024.105411","DOIUrl":"10.1016/j.geomphys.2024.105411","url":null,"abstract":"<div><div>In this paper we study the microlocal properties of the Szegő kernel of a given compact connected orientable CR orbifold, whose Kohn Laplacian has closed range. This last assumption is satisfied if certain geometric conditions hold true, as in the smooth case. Furthermore, we explain how to generalize a CR version of quantization commutes with reduction to orbifolds. As an application, we give a pure analytic proof of Kodaira-Bailey theorem.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"209 ","pages":"Article 105411"},"PeriodicalIF":1.6,"publicationDate":"2024-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143135856","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}
Pub Date : 2024-12-20DOI: 10.1016/j.geomphys.2024.105410
And. Morozov , A. Popolitov , A. Sleptsov
We prove that normalized colored Alexander polynomial (the limit of colored HOMFLY-PT polynomial) of a knot evaluated for one-hook (L-shape) representation R possesses scaling property: it is equal to the fundamental Alexander polynomial with the substitution . The proof is simple and direct use of Reshetikhin-Turaev formalism to get all required -matrices.
{"title":"Direct proof of one-hook scaling property for Alexander polynomial from Reshetikhin-Turaev formalism","authors":"And. Morozov , A. Popolitov , A. Sleptsov","doi":"10.1016/j.geomphys.2024.105410","DOIUrl":"10.1016/j.geomphys.2024.105410","url":null,"abstract":"<div><div>We prove that normalized colored Alexander polynomial (the <span><math><mi>A</mi><mo>→</mo><mn>1</mn></math></span> limit of colored HOMFLY-PT polynomial) of a knot <span><math><mi>K</mi></math></span> evaluated for one-hook (L-shape) representation <em>R</em> possesses <em>scaling property</em>: it is equal to the fundamental Alexander polynomial with the substitution <span><math><mi>q</mi><mo>→</mo><msup><mrow><mi>q</mi></mrow><mrow><mo>|</mo><mi>R</mi><mo>|</mo></mrow></msup></math></span>. The proof is simple and direct use of Reshetikhin-Turaev formalism to get all required <span><math><mi>R</mi></math></span>-matrices.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"209 ","pages":"Article 105410"},"PeriodicalIF":1.6,"publicationDate":"2024-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143096618","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}
Pub Date : 2024-12-20DOI: 10.1016/j.geomphys.2024.105409
L. Fehér
We study strongly isochronous Hamiltonians that generate periodic time evolution with the same basic period for a dense set of initial values. We explain that all such Hamiltonians are maximally superintegrable, and show that if the system is subjected to Hamiltonian reduction based on a compact symmetry group and certain conditions are met, then the reduced Hamiltonian is strongly isochronous with the original basic period. We utilize these simple observations for demonstrating the maximal superintegrability of rational spin Calogero–Moser type models in confining harmonic potential.
{"title":"On the maximal superintegrability of strongly isochronous Hamiltonians","authors":"L. Fehér","doi":"10.1016/j.geomphys.2024.105409","DOIUrl":"10.1016/j.geomphys.2024.105409","url":null,"abstract":"<div><div>We study strongly isochronous Hamiltonians that generate periodic time evolution with the same basic period for a dense set of initial values. We explain that all such Hamiltonians are maximally superintegrable, and show that if the system is subjected to Hamiltonian reduction based on a compact symmetry group and certain conditions are met, then the reduced Hamiltonian is strongly isochronous with the original basic period. We utilize these simple observations for demonstrating the maximal superintegrability of rational spin Calogero–Moser type models in confining harmonic potential.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"209 ","pages":"Article 105409"},"PeriodicalIF":1.6,"publicationDate":"2024-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143096619","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}
Pub Date : 2024-12-17DOI: 10.1016/j.geomphys.2024.105403
Bei Li, Dingguo Wang
In this paper, we present the concept of Novikov Poisson bialgebra and establish the equivalence between matched pairs, Manin triples, and Novikov Poisson bialgebras. Specifically, a Novikov Poisson bialgebra can be derived by uniformly solving the associative Yang-Baxter equation and the Novikov Yang-Baxter equation. Furthermore, we introduce the concepts of -operators on Novikov Poisson algebras and pre-Novikov Poisson algebras.
{"title":"Novikov Poisson bialgebra","authors":"Bei Li, Dingguo Wang","doi":"10.1016/j.geomphys.2024.105403","DOIUrl":"10.1016/j.geomphys.2024.105403","url":null,"abstract":"<div><div>In this paper, we present the concept of Novikov Poisson bialgebra and establish the equivalence between matched pairs, Manin triples, and Novikov Poisson bialgebras. Specifically, a Novikov Poisson bialgebra can be derived by uniformly solving the associative Yang-Baxter equation and the Novikov Yang-Baxter equation. Furthermore, we introduce the concepts of <span><math><mi>O</mi></math></span>-operators on Novikov Poisson algebras and pre-Novikov Poisson algebras.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"209 ","pages":"Article 105403"},"PeriodicalIF":1.6,"publicationDate":"2024-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143092367","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}
Pub Date : 2024-12-17DOI: 10.1016/j.geomphys.2024.105405
Yan Wang
Suppose that M is a complete Riemannian manifolds with nonnegative sectional curvature. We prove that for the exponentially harmonic heat flow (3) on bounded regular domain with the Dirichlet initial-boundary value data, there exists a unique global solution. We prove that for any bounded solution of the exponentially harmonic function heat flow on M, there is a gradient estimate. As a consequence of this estimate, we derive the Liouville type theorem for bounded ancient solutions to exponentially harmonic function heat flow on M. We also obtain Liouville type results for the exponentially harmonic functions with finite weighted norms.
{"title":"The exponentially harmonic heat flow on Riemannian manifolds and gradient estimates","authors":"Yan Wang","doi":"10.1016/j.geomphys.2024.105405","DOIUrl":"10.1016/j.geomphys.2024.105405","url":null,"abstract":"<div><div>Suppose that <em>M</em> is a complete Riemannian manifolds with nonnegative sectional curvature. We prove that for the exponentially harmonic heat flow <span><span>(3)</span></span> on bounded regular domain with the Dirichlet initial-boundary value data, there exists a unique global solution. We prove that for any bounded solution of the exponentially harmonic function heat flow on <em>M</em>, there is a gradient estimate. As a consequence of this estimate, we derive the Liouville type theorem for bounded ancient solutions to exponentially harmonic function heat flow on <em>M</em>. We also obtain Liouville type results for the exponentially harmonic functions with finite weighted <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> norms.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"209 ","pages":"Article 105405"},"PeriodicalIF":1.6,"publicationDate":"2024-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143105223","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}
Pub Date : 2024-12-17DOI: 10.1016/j.geomphys.2024.105404
Florio M. Ciaglia , Shuhan Jiang , Jürgen Jost , Lorenz Schwachhöfer
We investigate the canonical pseudo-Riemannian metrics on the Jordan-analogues of the coadjoint superorbits of a unital pseudo-Euclidean Jordan superalgebra with a positive even part.
{"title":"A coadjoint orbit–like construction for Jordan superalgebras","authors":"Florio M. Ciaglia , Shuhan Jiang , Jürgen Jost , Lorenz Schwachhöfer","doi":"10.1016/j.geomphys.2024.105404","DOIUrl":"10.1016/j.geomphys.2024.105404","url":null,"abstract":"<div><div>We investigate the canonical pseudo-Riemannian metrics on the Jordan-analogues of the coadjoint superorbits of a unital pseudo-Euclidean Jordan superalgebra with a positive even part.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"209 ","pages":"Article 105404"},"PeriodicalIF":1.6,"publicationDate":"2024-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143105222","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}
Pub Date : 2024-12-12DOI: 10.1016/j.geomphys.2024.105402
David Rochera
In this paper Zindler curves are studied in elliptic and hyperbolic planes. In some cases, these curves are associated to self-parallel curves through a double-traced closed curve with an odd number of singularities via front tire-track curves and parallel curves. It is shown that similar properties to those of planar Zindler curves are satisfied as well. Moreover, easy explicit parameterizations of these curves can be given through Leichtweiss support functions and some examples are constructed.
{"title":"Zindler curves in non-Euclidean geometry","authors":"David Rochera","doi":"10.1016/j.geomphys.2024.105402","DOIUrl":"10.1016/j.geomphys.2024.105402","url":null,"abstract":"<div><div>In this paper Zindler curves are studied in elliptic and hyperbolic planes. In some cases, these curves are associated to self-parallel curves through a double-traced closed curve with an odd number of singularities via front tire-track curves and parallel curves. It is shown that similar properties to those of planar Zindler curves are satisfied as well. Moreover, easy explicit parameterizations of these curves can be given through Leichtweiss support functions and some examples are constructed.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"209 ","pages":"Article 105402"},"PeriodicalIF":1.6,"publicationDate":"2024-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143096610","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}
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