Pub Date : 2025-12-01Epub Date: 2025-09-30DOI: 10.1016/j.geomphys.2025.105667
Daniel Grady
We identify some of the k-invariants for the Postnikov tower of the stable and unstable 4-sphere. Assuming the stable Hypothesis H of Fiorenza–Sati–Schreiber, we use the resulting obstruction theory to prove that the Chern–Simons term in the effective action of M-theory is well defined. In particular, we do not assume the presence of an -gauge field.
{"title":"Cohomotopy and flux quantization in M-theory","authors":"Daniel Grady","doi":"10.1016/j.geomphys.2025.105667","DOIUrl":"10.1016/j.geomphys.2025.105667","url":null,"abstract":"<div><div>We identify some of the <em>k</em>-invariants for the Postnikov tower of the stable and unstable 4-sphere. Assuming the stable Hypothesis H of Fiorenza–Sati–Schreiber, we use the resulting obstruction theory to prove that the Chern–Simons term in the effective action of M-theory is well defined. In particular, we do not assume the presence of an <span><math><msub><mrow><mi>E</mi></mrow><mrow><mn>8</mn></mrow></msub></math></span>-gauge field.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"218 ","pages":"Article 105667"},"PeriodicalIF":1.2,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145269917","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 : 2025-12-01Epub Date: 2025-09-22DOI: 10.1016/j.geomphys.2025.105650
M. Huzaifa Yaseen , Rida Hashmi , Najla A. Mohammed , Hala A Hejazi
The Lie symmetry method offers a systematic approach for analyzing and solving differential equations by identifying continuous transformations that preserve their structure. In this study, we investigate a general system of two nonlinear second-order elliptic partial differential equations using Lie symmetry techniques. We compute the equivalence transformations for the system, which serve as the foundation for deriving differential invariants. Specifically, we establish both joint differential invariants that are obtained under transformations of dependent and independent variables along with semi-differential invariants, derived solely from transformations of dependent variables. These invariants play a crucial role in reducing the system to its simplest possible form while retaining its essential features. By applying these differential invariants, we present reduced forms of various nonlinear systems of elliptic partial differential equations, demonstrating the effectiveness of the method in simplifying complex equations. Our results highlight the utility of Lie symmetry analysis in deriving invariant structures and facilitating the systematic reduction of coupled nonlinear systems of partial differential equations.
{"title":"Differential invariants of systems of two nonlinear elliptic partial differential equations by Lie symmetry method","authors":"M. Huzaifa Yaseen , Rida Hashmi , Najla A. Mohammed , Hala A Hejazi","doi":"10.1016/j.geomphys.2025.105650","DOIUrl":"10.1016/j.geomphys.2025.105650","url":null,"abstract":"<div><div>The Lie symmetry method offers a systematic approach for analyzing and solving differential equations by identifying continuous transformations that preserve their structure. In this study, we investigate a general system of two nonlinear second-order elliptic partial differential equations using Lie symmetry techniques. We compute the equivalence transformations for the system, which serve as the foundation for deriving differential invariants. Specifically, we establish both joint differential invariants that are obtained under transformations of dependent and independent variables along with semi-differential invariants, derived solely from transformations of dependent variables. These invariants play a crucial role in reducing the system to its simplest possible form while retaining its essential features. By applying these differential invariants, we present reduced forms of various nonlinear systems of elliptic partial differential equations, demonstrating the effectiveness of the method in simplifying complex equations. Our results highlight the utility of Lie symmetry analysis in deriving invariant structures and facilitating the systematic reduction of coupled nonlinear systems of partial differential equations.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"218 ","pages":"Article 105650"},"PeriodicalIF":1.2,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145223060","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 : 2025-12-01Epub Date: 2025-09-24DOI: 10.1016/j.geomphys.2025.105656
Alexey V. Bolsinov , Andrey Yu. Konyaev , Vladimir S. Matveev
We explore the Jordan–Chevalley decomposition problem for an operator field in small dimensions. In dimensions three and four, we find tensorial conditions for an operator field L, similar to a nilpotent Jordan block, to possess local coordinates in which L takes a strictly upper triangular form. We prove the Tempesta–Tondo conjecture for higher order brackets of Frölicher-Nijenhuis type.
{"title":"On the Jordan–Chevalley decomposition problem for operator fields in small dimensions and Tempesta–Tondo conjecture","authors":"Alexey V. Bolsinov , Andrey Yu. Konyaev , Vladimir S. Matveev","doi":"10.1016/j.geomphys.2025.105656","DOIUrl":"10.1016/j.geomphys.2025.105656","url":null,"abstract":"<div><div>We explore the Jordan–Chevalley decomposition problem for an operator field in small dimensions. In dimensions three and four, we find tensorial conditions for an operator field <em>L</em>, similar to a nilpotent Jordan block, to possess local coordinates in which <em>L</em> takes a strictly upper triangular form. We prove the Tempesta–Tondo conjecture for higher order brackets of Frölicher-Nijenhuis type.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"218 ","pages":"Article 105656"},"PeriodicalIF":1.2,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145223063","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 : 2025-12-01Epub Date: 2025-10-08DOI: 10.1016/j.geomphys.2025.105671
Maria Matushko
We propose a trigonometric solution of the associative Yang-Baxter equation related to the queer Lie superalgebra which in its turn satisfies the quantum Yang-Baxter equation.
{"title":"A solution of the associative Yang-Baxter equation related to the queer Lie superalgebra","authors":"Maria Matushko","doi":"10.1016/j.geomphys.2025.105671","DOIUrl":"10.1016/j.geomphys.2025.105671","url":null,"abstract":"<div><div>We propose a trigonometric solution of the associative Yang-Baxter equation related to the queer Lie superalgebra which in its turn satisfies the quantum Yang-Baxter equation.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"218 ","pages":"Article 105671"},"PeriodicalIF":1.2,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145321869","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 : 2025-12-01Epub Date: 2025-09-29DOI: 10.1016/j.geomphys.2025.105663
Yinuo Zhao , Liangyun Chen
In this paper, we show the relationship between skew-symmetric solutions of the Yang-Baxter equation (YBE) and super Kupershmidt operators of Malcev superalgebras. First, we show that a skew-supersymmetric solution of the Yang-Baxter equation on a Malcev superalgebra can be interpreted as an super Kupershmidt operator associated to the coadjoint representation. On this basis, when considering non-degenerate skew-symmetric solutions of the Yang-Baxter equation, this connection can be enhanced with symplectic forms. We also show that super Kupershmidt operators associated with a general representation could give skew-symmetric solutions of the Yang-Baxter equation on certain semi-direct products of Malcev superalgebras. What's more, we reveal that in the case of pre-Malcev superalgebras, We can get similar results between the Yang-Baxter equation and super Kupershmidt operators.
{"title":"Super Kupershmidt operator and the Yang-Baxter equation in Malcev superalgebras","authors":"Yinuo Zhao , Liangyun Chen","doi":"10.1016/j.geomphys.2025.105663","DOIUrl":"10.1016/j.geomphys.2025.105663","url":null,"abstract":"<div><div>In this paper, we show the relationship between skew-symmetric solutions of the Yang-Baxter equation (YBE) and super Kupershmidt operators of Malcev superalgebras. First, we show that a skew-supersymmetric solution of the Yang-Baxter equation on a Malcev superalgebra can be interpreted as an super Kupershmidt operator associated to the coadjoint representation. On this basis, when considering non-degenerate skew-symmetric solutions of the Yang-Baxter equation, this connection can be enhanced with symplectic forms. We also show that super Kupershmidt operators associated with a general representation could give skew-symmetric solutions of the Yang-Baxter equation on certain semi-direct products of Malcev superalgebras. What's more, we reveal that in the case of pre-Malcev superalgebras, We can get similar results between the Yang-Baxter equation and super Kupershmidt operators.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"218 ","pages":"Article 105663"},"PeriodicalIF":1.2,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145223058","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 : 2025-12-01Epub Date: 2025-10-06DOI: 10.1016/j.geomphys.2025.105670
C.J. Lang
Throughout this paper, we comprehensively study instantons with every kind of continuous conformal symmetry. Examples of these objects are hard to come by due to non-linear constraints. However, by applying previous work on moduli spaces, we introduce a linear constraint, whose solution greatly simplifies these non-linear constraints. This simplification not only allows us to easily find a plethora of novel instantons with various continuous conformal symmetries and higher rank structure groups, it also provides a framework for classifying such symmetric objects. We also prove that the basic instanton is essentially the only instanton with two particular kinds of conformal symmetry. Additionally, we discuss the connections between instantons with continuous symmetries and other gauge-theoretic objects: hyperbolic and singular monopoles as well as hyperbolic analogues to Higgs bundles and Nahm data.
{"title":"Instantons with continuous conformal symmetries","authors":"C.J. Lang","doi":"10.1016/j.geomphys.2025.105670","DOIUrl":"10.1016/j.geomphys.2025.105670","url":null,"abstract":"<div><div>Throughout this paper, we comprehensively study instantons with every kind of continuous conformal symmetry. Examples of these objects are hard to come by due to non-linear constraints. However, by applying previous work on moduli spaces, we introduce a linear constraint, whose solution greatly simplifies these non-linear constraints. This simplification not only allows us to easily find a plethora of novel instantons with various continuous conformal symmetries and higher rank structure groups, it also provides a framework for classifying such symmetric objects. We also prove that the basic instanton is essentially the only instanton with two particular kinds of conformal symmetry. Additionally, we discuss the connections between instantons with continuous symmetries and other gauge-theoretic objects: hyperbolic and singular monopoles as well as hyperbolic analogues to Higgs bundles and Nahm data.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"218 ","pages":"Article 105670"},"PeriodicalIF":1.2,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145269931","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 : 2025-12-01Epub Date: 2025-10-02DOI: 10.1016/j.geomphys.2025.105668
Hacen Zelaci
In this note, we show that the space of theta functions of the Pfaffian line bundle over the moduli space of stable -bundles has dimension one. Using the Hitchin system, we construct a dominant rational map from a principally polarized Prym variety to each connected component of this moduli space.
{"title":"Moduli space of orthogonal bundles of even rank and theta functions","authors":"Hacen Zelaci","doi":"10.1016/j.geomphys.2025.105668","DOIUrl":"10.1016/j.geomphys.2025.105668","url":null,"abstract":"<div><div>In this note, we show that the space of theta functions of the Pfaffian line bundle over the moduli space of stable <span><math><msub><mrow><mi>SO</mi></mrow><mrow><mn>2</mn><mi>m</mi></mrow></msub></math></span>-bundles has dimension one. Using the Hitchin system, we construct a dominant rational map from a principally polarized Prym variety to each connected component of this moduli space.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"218 ","pages":"Article 105668"},"PeriodicalIF":1.2,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145269918","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 : 2025-12-01Epub Date: 2025-10-24DOI: 10.1016/j.geomphys.2025.105687
Dmitrii Adler , Valery Gritsenko
We study modular differential equations (MDEs) of high orders and find necessary conditions for weak Jacobi forms to satisfy MDEs of order 3 with respect to the heat operator. We investigate all possible MDEs for weak Jacobi forms of weight 0 and index 3. This is the target space for the elliptic genus of the compact complex manifolds of dimension 6 with trivial first Chern class. We prove that the minimal possible order of MDEs of such Jacobi forms is four. Moreover, we find all such forms and show that only three of them might be the elliptic genus of strict Calabi–Yau six-folds. We describe also a discrete set of Jacobi forms satisfying fifth-order MDEs and the divisor of forms satisfying sixth-order MDEs. Then we prove that a Jacobi form of weight 0 and index 3 which does not belong to a smooth cubic in the space of coefficients satisfies a MDE of order 7. We provide such MDEs for the elliptic genus of 6-dimensional holomorphic symplectic varieties of types , , and OG6.
{"title":"Modular differential equations of minimal orders of the elliptic genus of Calabi–Yau varieties","authors":"Dmitrii Adler , Valery Gritsenko","doi":"10.1016/j.geomphys.2025.105687","DOIUrl":"10.1016/j.geomphys.2025.105687","url":null,"abstract":"<div><div>We study modular differential equations (MDEs) of high orders and find necessary conditions for weak Jacobi forms to satisfy MDEs of order 3 with respect to the heat operator. We investigate all possible MDEs for weak Jacobi forms of weight 0 and index 3. This is the target space for the elliptic genus of the compact complex manifolds of dimension 6 with trivial first Chern class. We prove that the minimal possible order of MDEs of such Jacobi forms is four. Moreover, we find all such forms and show that only three of them might be the elliptic genus of strict Calabi–Yau six-folds. We describe also a discrete set of Jacobi forms satisfying fifth-order MDEs and the divisor of forms satisfying sixth-order MDEs. Then we prove that a Jacobi form of weight 0 and index 3 which does not belong to a smooth cubic in the space of coefficients satisfies a MDE of order 7. We provide such MDEs for the elliptic genus of 6-dimensional holomorphic symplectic varieties of types <span><math><msup><mrow><mtext>Hilb</mtext></mrow><mrow><mo>[</mo><mn>3</mn><mo>]</mo></mrow></msup><mo>(</mo><mi>K</mi><mn>3</mn><mo>)</mo></math></span>, <span><math><msub><mrow><mtext>Kum</mtext></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span>, and OG6.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"218 ","pages":"Article 105687"},"PeriodicalIF":1.2,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145417181","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 : 2025-12-01Epub Date: 2025-09-24DOI: 10.1016/j.geomphys.2025.105654
Esteban Andruchow , Gabriel Larotonda , Lázaro Recht
Let be a component of the Grassmann manifold of a -algebra, presented as the unitary orbit of a given orthogonal projection . There are several natural connections on this manifold, and we first show that they all agree (in the presence of a finite trace in , when we give the Riemannian metric induced by the Killing form, this is the Levi-Civita connection of the metric). We study the cut locus of for the spectral rectifiable distance, and also the conjugate tangent locus of along a geodesic. Furthermore, for each tangent vector V at P, we compute the kernel of the differential of the exponential map of the connection. We exhibit examples where points that are tangent conjugate in the classical setting, fail to be conjugate: in some cases they are not monoconjugate but epinconjugate, and in other cases they are not conjugate at all.
{"title":"Conjugate points in the Grassmann manifold of a C⁎-algebra","authors":"Esteban Andruchow , Gabriel Larotonda , Lázaro Recht","doi":"10.1016/j.geomphys.2025.105654","DOIUrl":"10.1016/j.geomphys.2025.105654","url":null,"abstract":"<div><div>Let <figure><img></figure> be a component of the Grassmann manifold of a <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-algebra, presented as the unitary orbit of a given orthogonal projection <figure><img></figure>. There are several natural connections on this manifold, and we first show that they all agree (in the presence of a finite trace in <span><math><mi>A</mi></math></span>, when we give <figure><img></figure> the Riemannian metric induced by the Killing form, this is the Levi-Civita connection of the metric). We study the cut locus of <figure><img></figure> for the spectral rectifiable distance, and also the conjugate tangent locus of <figure><img></figure> along a geodesic. Furthermore, for each tangent vector <em>V</em> at <em>P</em>, we compute the kernel of the differential of the exponential map of the connection. We exhibit examples where points that are tangent conjugate in the classical setting, fail to be conjugate: in some cases they are not monoconjugate but epinconjugate, and in other cases they are not conjugate at all.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"218 ","pages":"Article 105654"},"PeriodicalIF":1.2,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145223062","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 : 2025-12-01Epub Date: 2025-10-24DOI: 10.1016/j.geomphys.2025.105688
Gianni Manno, Filippo Salis
A classical and long-staying problem addressed, among others, by Calabi and Chern, is that to find a complete list of mutually non-isometric Kähler-Einstein manifolds immersed in a finite-dimensional Kähler space form. We address the same problem in the para-Kähler context and, then, we find a list of mutually non-isometric toric para-Kähler manifolds analytically immersed in a finite-dimensional para-Kähler space form.
{"title":"Toric para-Kähler-Einstein manifolds immersed in para-Kähler space forms","authors":"Gianni Manno, Filippo Salis","doi":"10.1016/j.geomphys.2025.105688","DOIUrl":"10.1016/j.geomphys.2025.105688","url":null,"abstract":"<div><div>A classical and long-staying problem addressed, among others, by Calabi and Chern, is that to find a complete list of mutually non-isometric Kähler-Einstein manifolds immersed in a finite-dimensional Kähler space form. We address the same problem in the para-Kähler context and, then, we find a list of mutually non-isometric toric para-Kähler manifolds analytically immersed in a finite-dimensional para-Kähler space form.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"218 ","pages":"Article 105688"},"PeriodicalIF":1.2,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145417183","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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