Pub Date : 2024-10-16DOI: 10.1016/j.geomphys.2024.105342
Bartłomiej Bąk
It is shown that Maxwell equations for electromagnetic fields generated by a uniformly accelerated charge could be reduced to the Laplace equation in a co-moving frame (represented by the Łobaczewski geometry of the one-sheeted hyperboloid) for a single scalar potential. A full solution of this equation is derived. Then, the famous problem of radiation of a uniformly accelerated particle is revised. Finally, a description of the electromagnetic field on the scri is presented. Both of those approaches produce the same result, which, surprisingly, is slightly different to the well-established Larmor formula for radiation.
{"title":"Description of electromagnetic fields in uniformly accelerated frame. Revision of the radiation problem","authors":"Bartłomiej Bąk","doi":"10.1016/j.geomphys.2024.105342","DOIUrl":"10.1016/j.geomphys.2024.105342","url":null,"abstract":"<div><div>It is shown that Maxwell equations for electromagnetic fields generated by a uniformly accelerated charge could be reduced to the Laplace equation in a co-moving frame (represented by the Łobaczewski geometry of the one-sheeted hyperboloid) for a single scalar potential. A full solution of this equation is derived. Then, the famous problem of radiation of a uniformly accelerated particle is revised. Finally, a description of the electromagnetic field on the scri is presented. Both of those approaches produce the same result, which, surprisingly, is slightly different to the well-established Larmor formula for radiation.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142526878","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 : 2024-10-15DOI: 10.1016/j.geomphys.2024.105341
R. Lávička , V. Souček , W. Wang
Main topic of the paper is a study of properties of massless fields of spin 3/2. A lot of information is available already for massless fields in dimension 4. Here, we concentrate on dimension 6 and we are using the fact that the group is isomorphic with the group . It makes it possible to use tensor formalism for massless fields. Main problems treated in the paper are a description of fields which need to be considered in the spin 3/2 case, a suitable choice of equations they should satisfy, irreducibility of homogeneous solutions of massless field equations, the Fischer decomposition and the Howe duality for such fields.
{"title":"Massless field equations for spin 3/2 in dimension 6","authors":"R. Lávička , V. Souček , W. Wang","doi":"10.1016/j.geomphys.2024.105341","DOIUrl":"10.1016/j.geomphys.2024.105341","url":null,"abstract":"<div><div>Main topic of the paper is a study of properties of massless fields of spin 3/2. A lot of information is available already for massless fields in dimension 4. Here, we concentrate on dimension 6 and we are using the fact that the group <span><math><mi>S</mi><mi>L</mi><mo>(</mo><mn>4</mn><mo>,</mo><mi>C</mi><mo>)</mo></math></span> is isomorphic with the group <span><math><mi>S</mi><mi>p</mi><mi>i</mi><mi>n</mi><mo>(</mo><mn>6</mn><mo>,</mo><mi>C</mi><mo>)</mo></math></span>. It makes it possible to use tensor formalism for massless fields. Main problems treated in the paper are a description of fields which need to be considered in the spin 3/2 case, a suitable choice of equations they should satisfy, irreducibility of homogeneous solutions of massless field equations, the Fischer decomposition and the Howe duality for such fields.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142526877","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 : 2024-10-09DOI: 10.1016/j.geomphys.2024.105337
Elena Poletaeva
Let be the queer Lie superalgebra. We determine conditions under which two 1-dimensional modules over the super-Yangian of can be extended nontrivially. We describe the dual modules of the simple finite-dimensional modules over . We use these results to describe blocks in the subcategory of finite-dimensional -modules admitting the zero generalized central character.
{"title":"On representations of the super-Yangian of the queer Lie superalgebra","authors":"Elena Poletaeva","doi":"10.1016/j.geomphys.2024.105337","DOIUrl":"10.1016/j.geomphys.2024.105337","url":null,"abstract":"<div><div>Let <span><math><mi>Q</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> be the queer Lie superalgebra. We determine conditions under which two 1-dimensional modules over the super-Yangian of <span><math><mi>Q</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> can be extended nontrivially. We describe the dual modules of the simple finite-dimensional modules over <span><math><mi>Y</mi><mi>Q</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span>. We use these results to describe blocks in the subcategory of finite-dimensional <span><math><mi>Y</mi><mi>Q</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span>-modules admitting the zero generalized central character.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142422306","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 : 2024-10-09DOI: 10.1016/j.geomphys.2024.105338
Andronikos Paliathanasis
We review the solution space for the field equations of Einstein's General Relativity for various static, spherically symmetric spacetimes. We consider the vacuum case, represented by the Schwarzschild black hole; the de Sitter-Schwarzschild geometry, which includes a cosmological constant; the Reissner-Nordström geometry, which accounts for the presence of charge. Additionally we consider the homogenenous and anisotropic locally rotational Bianchi II spacetime in the vacuum. Our analysis reveals that the field equations for these scenarios share a common three-dimensional group of point transformations, with the generators being the elements of the Lie algebra, known as the semidirect product of dilations and translations in the plane. Due to this algebraic property the field equations for the aforementioned gravitational models can be expressed in the equivalent form of the null geodesic equations for conformally flat geometries. Consequently, the solution space for the field equations is common, and it is the solution space for the free particle in a flat space. This approach open new directions on the construction of analytic solutions in gravitational physics and cosmology.
我们回顾了爱因斯坦广义相对论场方程在各种静态球对称空间的求解空间。我们考虑了以施瓦兹柴尔德黑洞为代表的真空情况；包含宇宙常数的德西特-施瓦兹柴尔德几何；考虑了电荷存在的赖斯纳-诺德斯特伦几何。此外，我们还考虑了真空中各向同性和各向异性的局部旋转比安奇 II 时空。我们的分析表明，这些场景的场方程共享一个共同的三维点变换群，其生成器是 D⊗sT2 Lie 代数的元素，即平面内扩张与平移的半直接乘积。由于这一代数特性，上述引力模型的场方程可以用保角平坦几何的空大地方程的等价形式表示。因此，场方程的解空间是共通的，它就是平面空间中自由粒子的解空间。这种方法为引力物理学和宇宙学中解析解的构建开辟了新的方向。
{"title":"The common solution space of general relativity","authors":"Andronikos Paliathanasis","doi":"10.1016/j.geomphys.2024.105338","DOIUrl":"10.1016/j.geomphys.2024.105338","url":null,"abstract":"<div><div>We review the solution space for the field equations of Einstein's General Relativity for various static, spherically symmetric spacetimes. We consider the vacuum case, represented by the Schwarzschild black hole; the de Sitter-Schwarzschild geometry, which includes a cosmological constant; the Reissner-Nordström geometry, which accounts for the presence of charge. Additionally we consider the homogenenous and anisotropic locally rotational Bianchi II spacetime in the vacuum. Our analysis reveals that the field equations for these scenarios share a common three-dimensional group of point transformations, with the generators being the elements of the <span><math><mi>D</mi><msub><mrow><mo>⊗</mo></mrow><mrow><mi>s</mi></mrow></msub><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> Lie algebra, known as the semidirect product of dilations and translations in the plane. Due to this algebraic property the field equations for the aforementioned gravitational models can be expressed in the equivalent form of the null geodesic equations for conformally flat geometries. Consequently, the solution space for the field equations is common, and it is the solution space for the free particle in a flat space. This approach open new directions on the construction of analytic solutions in gravitational physics and cosmology.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142422308","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 : 2024-10-09DOI: 10.1016/j.geomphys.2024.105340
A. Fotiadis, C. Daskaloyannis
We study locally harmonic maps between a Riemann surface or a Lorentz surface M and a Riemann or Lorentz surface N. All four cases are written using a unified formalism. Therefore properties and solutions to the harmonic map problem can be studied in a unified way.
It is known that harmonic maps between Riemannian surfaces are classified by the classification of the solutions of a sinh-Gordon equation. We extend this result to all the four cases of harmonic maps between Riemannian or pseudo-Riemannian surfaces. The calculation of the corresponding harmonic map can be calculated by the solutions of the corresponding Beltrami equations in all the cases.
We study the one-soliton solutions of this equation and we find the corresponding harmonic maps in a unified way.
Next, we discuss a Bäcklund transformation of the harmonic map equations that provides a connection between the solutions of two sine or sinh-Gordon type equations. Finally, we give an example of a harmonic map that is constructed by the use of a Bäcklund transformation.
{"title":"Harmonic maps between pseudo-Riemannian surfaces","authors":"A. Fotiadis, C. Daskaloyannis","doi":"10.1016/j.geomphys.2024.105340","DOIUrl":"10.1016/j.geomphys.2024.105340","url":null,"abstract":"<div><div>We study locally harmonic maps between a Riemann surface or a Lorentz surface <em>M</em> and a Riemann or Lorentz surface <em>N</em>. All four cases are written using a unified formalism. Therefore properties and solutions to the harmonic map problem can be studied in a unified way.</div><div>It is known that harmonic maps between Riemannian surfaces are classified by the classification of the solutions of a sinh-Gordon equation. We extend this result to all the four cases of harmonic maps between Riemannian or pseudo-Riemannian surfaces. The calculation of the corresponding harmonic map can be calculated by the solutions of the corresponding Beltrami equations in all the cases.</div><div>We study the one-soliton solutions of this equation and we find the corresponding harmonic maps in a unified way.</div><div>Next, we discuss a Bäcklund transformation of the harmonic map equations that provides a connection between the solutions of two sine or sinh-Gordon type equations. Finally, we give an example of a harmonic map that is constructed by the use of a Bäcklund transformation.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142438050","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 : 2024-10-05DOI: 10.1016/j.geomphys.2024.105335
Shanshan Liu , Liangyun Chen
In this paper, first, we give the notion of a compatible pre-Lie algebra and its representation. We study the relation between compatible Lie algebras and compatible pre-Lie algebras. We also construct a new bidifferential graded Lie algebra whose Maurer-Cartan elements are compatible pre-Lie structures. We give the bidifferential graded Lie algebra which controls deformations of a compatible pre-Lie algebra. Then, we introduce a cohomology of a compatible pre-Lie algebra with coefficients in itself. We study infinitesimal deformations of compatible pre-Lie algebras and show that equivalent infinitesimal deformations are in the same second cohomology group. We further give the notion of a Nijenhuis operator on a compatible pre-Lie algebra. We study formal deformations of compatible pre-Lie algebras. If the second cohomology group is trivial, then the compatible pre-Lie algebra is rigid. Finally, we give a cohomology of a compatible pre-Lie algebra with coefficients in arbitrary representation and study abelian extensions of compatible pre-Lie algebras using this cohomology. We show that abelian extensions are classified by the second cohomology group.
{"title":"Deformations and abelian extensions of compatible pre-Lie algebras","authors":"Shanshan Liu , Liangyun Chen","doi":"10.1016/j.geomphys.2024.105335","DOIUrl":"10.1016/j.geomphys.2024.105335","url":null,"abstract":"<div><div>In this paper, first, we give the notion of a compatible pre-Lie algebra and its representation. We study the relation between compatible Lie algebras and compatible pre-Lie algebras. We also construct a new bidifferential graded Lie algebra whose Maurer-Cartan elements are compatible pre-Lie structures. We give the bidifferential graded Lie algebra which controls deformations of a compatible pre-Lie algebra. Then, we introduce a cohomology of a compatible pre-Lie algebra with coefficients in itself. We study infinitesimal deformations of compatible pre-Lie algebras and show that equivalent infinitesimal deformations are in the same second cohomology group. We further give the notion of a Nijenhuis operator on a compatible pre-Lie algebra. We study formal deformations of compatible pre-Lie algebras. If the second cohomology group <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>g</mi><mo>;</mo><mi>g</mi><mo>)</mo></math></span> is trivial, then the compatible pre-Lie algebra is rigid. Finally, we give a cohomology of a compatible pre-Lie algebra with coefficients in arbitrary representation and study abelian extensions of compatible pre-Lie algebras using this cohomology. We show that abelian extensions are classified by the second cohomology group.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142421881","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 : 2024-10-05DOI: 10.1016/j.geomphys.2024.105339
Dmitrii Adler , Valery Gritsenko
We study rings of weak Jacobi forms invariant with respect to the action of the Weyl group for the root systems and , and provide an explicit construction of generators of such rings by using modular differential operators. The construction of generators in the form of a -tower () gives a simple proof that these graded rings of Jacobi forms are polynomial. We study in detail modular differential equations (MDEs), which are satisfied by generators of index 1. Interesting anomalies are noticed for the lattices , and . In particular, some generators for these lattices satisfy the Kaneko–Zagier type MDEs of order 2 or MDEs of order 1 similar to the differential equation of the elliptic genus of three-dimensional Calabi–Yau manifolds.
{"title":"Modular differential equations of W(Dn)-invariant Jacobi forms","authors":"Dmitrii Adler , Valery Gritsenko","doi":"10.1016/j.geomphys.2024.105339","DOIUrl":"10.1016/j.geomphys.2024.105339","url":null,"abstract":"<div><div>We study rings of weak Jacobi forms invariant with respect to the action of the Weyl group for the root systems <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>D</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, and provide an explicit construction of generators of such rings by using modular differential operators. The construction of generators in the form of a <span><math><msub><mrow><mi>D</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>-tower (<span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span>) gives a simple proof that these graded rings of Jacobi forms are polynomial. We study in detail modular differential equations (MDEs), which are satisfied by generators of index 1. Interesting anomalies are noticed for the lattices <span><math><msub><mrow><mi>D</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>, <span><math><msub><mrow><mi>D</mi></mrow><mrow><mn>8</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>D</mi></mrow><mrow><mn>12</mn></mrow></msub></math></span>. In particular, some generators for these lattices satisfy the Kaneko–Zagier type MDEs of order 2 or MDEs of order 1 similar to the differential equation of the elliptic genus of three-dimensional Calabi–Yau manifolds.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142422307","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 : 2024-10-03DOI: 10.1016/j.geomphys.2024.105334
Mohamed Boucetta , Hamza El Ouali , Hicham Lebzioui
A pseudo-Euclidean Novikov algebra is a Novikov algebra endowed with a non-degenerate symmetric bilinear form such that left multiplications are skew-symmetric. If is of signature then is called a Lorentzian Novikov algebra. In (H. Lebzioui, 2020 [11]), the author studied Lorentzian Novikov algebras and showed that a Lorentzian Novikov algebra is transitive. In this paper, we study pseudo-Euclidean Novikov algebras in the general case, where is of arbitrary signature. We show that a pseudo-Euclidean Novikov algebra of arbitrary signature must be transitive and the associated Lie algebra is two-solvable. This implies that a flat left-invariant pseudo-Riemannian metric on a corresponding Lie group is geodesically complete. We show that if is a pseudo-Euclidean Novikov algebra such that is non-degenerate then the underlying Lie algebra is a Milnor Lie algebra; that is , where is a sub-Lie algebra, is a sub-Lie ideal and is -skew symmetric for any . If is degenerate, then we show that we can obtain the pseudo-Euclidean Novikov algebra through a double extension process starting from a Milnor Lie algebra. Finally, as applications, we classify all pseudo-Euclidean Novikov algebras of dimension ≤5 such that is degenerate.
{"title":"Pseudo-Euclidean Novikov algebras of arbitrary signature","authors":"Mohamed Boucetta , Hamza El Ouali , Hicham Lebzioui","doi":"10.1016/j.geomphys.2024.105334","DOIUrl":"10.1016/j.geomphys.2024.105334","url":null,"abstract":"<div><div>A pseudo-Euclidean Novikov algebra <span><math><mo>(</mo><mi>g</mi><mo>,</mo><mo>•</mo><mo>,</mo><mo>〈</mo><mspace></mspace><mo>,</mo><mspace></mspace><mo>〉</mo><mo>)</mo></math></span> is a Novikov algebra <span><math><mo>(</mo><mi>g</mi><mo>,</mo><mo>•</mo><mo>)</mo></math></span> endowed with a non-degenerate symmetric bilinear form such that left multiplications are skew-symmetric. If <span><math><mo>〈</mo><mspace></mspace><mo>,</mo><mspace></mspace><mo>〉</mo></math></span> is of signature <span><math><mo>(</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span> then <span><math><mo>(</mo><mi>g</mi><mo>,</mo><mo>•</mo><mo>,</mo><mo>〈</mo><mspace></mspace><mo>,</mo><mspace></mspace><mo>〉</mo><mo>)</mo></math></span> is called a Lorentzian Novikov algebra. In (H. Lebzioui, 2020 <span><span>[11]</span></span>), the author studied Lorentzian Novikov algebras and showed that a Lorentzian Novikov algebra is transitive. In this paper, we study pseudo-Euclidean Novikov algebras in the general case, where <span><math><mo>〈</mo><mspace></mspace><mo>,</mo><mspace></mspace><mo>〉</mo></math></span> is of arbitrary signature. We show that a pseudo-Euclidean Novikov algebra of arbitrary signature must be transitive and the associated Lie algebra is two-solvable. This implies that a flat left-invariant pseudo-Riemannian metric on a corresponding Lie group is geodesically complete. We show that if <span><math><mo>(</mo><mi>g</mi><mo>,</mo><mo>•</mo><mo>,</mo><mo>〈</mo><mspace></mspace><mo>,</mo><mspace></mspace><mo>〉</mo><mo>)</mo></math></span> is a pseudo-Euclidean Novikov algebra such that <span><math><mi>g</mi><mo>•</mo><mi>g</mi></math></span> is non-degenerate then the underlying Lie algebra is a Milnor Lie algebra; that is <span><math><mi>g</mi><mo>=</mo><mi>b</mi><mo>⊕</mo><msup><mrow><mi>b</mi></mrow><mrow><mo>⊥</mo></mrow></msup></math></span>, where <span><math><mi>b</mi></math></span> is a sub-Lie algebra, <span><math><msup><mrow><mi>b</mi></mrow><mrow><mo>⊥</mo></mrow></msup></math></span> is a sub-Lie ideal and <span><math><msub><mrow><mi>ad</mi></mrow><mrow><mi>b</mi></mrow></msub></math></span> is <span><math><mo>〈</mo><mspace></mspace><mo>,</mo><mspace></mspace><mo>〉</mo></math></span>-skew symmetric for any <span><math><mi>b</mi><mo>∈</mo><mi>b</mi></math></span>. If <span><math><mi>g</mi><mo>•</mo><mi>g</mi></math></span> is degenerate, then we show that we can obtain the pseudo-Euclidean Novikov algebra through a double extension process starting from a Milnor Lie algebra. Finally, as applications, we classify all pseudo-Euclidean Novikov algebras of dimension ≤5 such that <span><math><mi>g</mi><mo>•</mo><mi>g</mi></math></span> is degenerate.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142422193","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 : 2024-10-03DOI: 10.1016/j.geomphys.2024.105336
Kazushi Kobayashi
Let be a mirror pair of an n-dimensional complex torus and its mirror partner . Then, by SYZ transform, we can construct a holomorphic line bundle with an integrable connection from each pair of a Lagrangian section of and a unitary local system along it, and those holomorphic line bundles with integrable connections form a dg-category . In this paper, we focus on a certain B-field transform of the generalized complex structure induced from the complex structure on , and interpret it as the deformation of by a flat gerbe . Moreover, we construct the deformation of associated to the deformation from to , and also discuss the homological mirror symmetry between and its mirror partner on the object level.
{"title":"On a B-field transform of generalized complex structures over complex tori","authors":"Kazushi Kobayashi","doi":"10.1016/j.geomphys.2024.105336","DOIUrl":"10.1016/j.geomphys.2024.105336","url":null,"abstract":"<div><div>Let <span><math><mo>(</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><msup><mrow><mover><mrow><mi>X</mi></mrow><mrow><mo>ˇ</mo></mrow></mover></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span> be a mirror pair of an <em>n</em>-dimensional complex torus <span><math><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> and its mirror partner <span><math><msup><mrow><mover><mrow><mi>X</mi></mrow><mrow><mo>ˇ</mo></mrow></mover></mrow><mrow><mi>n</mi></mrow></msup></math></span>. Then, by SYZ transform, we can construct a holomorphic line bundle with an integrable connection from each pair of a Lagrangian section of <span><math><msup><mrow><mover><mrow><mi>X</mi></mrow><mrow><mo>ˇ</mo></mrow></mover></mrow><mrow><mi>n</mi></mrow></msup><mo>→</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>/</mo><msup><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> and a unitary local system along it, and those holomorphic line bundles with integrable connections form a dg-category <span><math><mi>D</mi><msub><mrow><mi>G</mi></mrow><mrow><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span>. In this paper, we focus on a certain B-field transform of the generalized complex structure induced from the complex structure on <span><math><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, and interpret it as the deformation <span><math><msubsup><mrow><mi>X</mi></mrow><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> of <span><math><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> by a flat gerbe <span><math><mi>G</mi></math></span>. Moreover, we construct the deformation of <span><math><mi>D</mi><msub><mrow><mi>G</mi></mrow><mrow><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span> associated to the deformation from <span><math><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> to <span><math><msubsup><mrow><mi>X</mi></mrow><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span>, and also discuss the homological mirror symmetry between <span><math><msubsup><mrow><mi>X</mi></mrow><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> and its mirror partner on the object level.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142433957","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 : 2024-10-02DOI: 10.1016/j.geomphys.2024.105333
Ali Suri
Utilizing structure constants, we present a version of the Misiolek criterion for identifying conjugate points. We propose an approach that enables us to locate these points along solutions of the quasi-geostrophic equations on the sphere . We demonstrate that for any spherical harmonics with , except for and , conjugate points can be determined along the solution generated by the velocity field . Subsequently, we investigate the impact of the Coriolis force on the occurrence of conjugate points. Moreover, for any zonal flow generated by the velocity field , we demonstrate that proper rotation rate can lead to the appearance of conjugate points along the corresponding solution, where Additionally, we prove the existence of conjugate points along (complex) Rossby-Haurwitz waves and explore the effect of the Coriolis force on their stability.
{"title":"Conjugate points along spherical harmonics","authors":"Ali Suri","doi":"10.1016/j.geomphys.2024.105333","DOIUrl":"10.1016/j.geomphys.2024.105333","url":null,"abstract":"<div><div>Utilizing structure constants, we present a version of the Misiolek criterion for identifying conjugate points. We propose an approach that enables us to locate these points along solutions of the quasi-geostrophic equations on the sphere <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>. We demonstrate that for any spherical harmonics <span><math><msub><mrow><mi>Y</mi></mrow><mrow><mi>l</mi><mi>m</mi></mrow></msub></math></span> with <span><math><mn>1</mn><mo>≤</mo><mo>|</mo><mi>m</mi><mo>|</mo><mo>≤</mo><mi>l</mi></math></span>, except for <span><math><msub><mrow><mi>Y</mi></mrow><mrow><mn>1</mn><mo>±</mo><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>Y</mi></mrow><mrow><mn>2</mn><mo>±</mo><mn>1</mn></mrow></msub></math></span>, conjugate points can be determined along the solution generated by the velocity field <span><math><msub><mrow><mi>e</mi></mrow><mrow><mi>l</mi><mi>m</mi></mrow></msub><mo>=</mo><msup><mrow><mi>∇</mi></mrow><mrow><mo>⊥</mo></mrow></msup><msub><mrow><mi>Y</mi></mrow><mrow><mi>l</mi><mi>m</mi></mrow></msub></math></span>. Subsequently, we investigate the impact of the Coriolis force on the occurrence of conjugate points. Moreover, for any zonal flow generated by the velocity field <span><math><msup><mrow><mi>∇</mi></mrow><mrow><mo>⊥</mo></mrow></msup><msub><mrow><mi>Y</mi></mrow><mrow><msub><mrow><mi>l</mi></mrow><mrow><mn>1</mn></mrow></msub><mspace></mspace><mn>0</mn></mrow></msub></math></span>, we demonstrate that proper rotation rate can lead to the appearance of conjugate points along the corresponding solution, where <span><math><msub><mrow><mi>l</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn><mo>.</mo><mo>∈</mo><mi>N</mi></math></span> Additionally, we prove the existence of conjugate points along (complex) Rossby-Haurwitz waves and explore the effect of the Coriolis force on their stability.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142422195","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}
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