The classical Hochschild cohomology theory of rings is extended to abelian heaps with distributing multiplication or trusses. This cohomology is then employed to give necessary and sufficient conditions for a Nijenhuis product on a truss (defined by the extension of the Nijenhuis product on an associative ring introduced by Carinena, Grabowski and Marmo in [Internat. J. Modern Phys. A 15 (2000), 4797-4810, arXiv:math-ph/0610011]) to be associative. The definition of Nijenhuis product and operators on trusses is then linearised to the case of affine spaces with compatible associative multiplications or associative affgebras. It is shown that this construction leads to compatible Lie brackets on an affine space.
{"title":"Affine Nijenhuis Operators and Hochschild Cohomology of Trusses","authors":"Tomasz Brzezi'nski, J. Papworth","doi":"10.3842/SIGMA.2023.056","DOIUrl":"https://doi.org/10.3842/SIGMA.2023.056","url":null,"abstract":"The classical Hochschild cohomology theory of rings is extended to abelian heaps with distributing multiplication or trusses. This cohomology is then employed to give necessary and sufficient conditions for a Nijenhuis product on a truss (defined by the extension of the Nijenhuis product on an associative ring introduced by Carinena, Grabowski and Marmo in [Internat. J. Modern Phys. A 15 (2000), 4797-4810, arXiv:math-ph/0610011]) to be associative. The definition of Nijenhuis product and operators on trusses is then linearised to the case of affine spaces with compatible associative multiplications or associative affgebras. It is shown that this construction leads to compatible Lie brackets on an affine space.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48870075","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Green-hyperbolic operators - partial differential operators on globally hyperbolic spacetimes that (together with their formal duals) possess advanced and retarded Green operators - play an important role in many areas of mathematical physics. Here, we study modifications of Green-hyperbolic operators by the addition of a possibly nonlocal operator acting within a compact subset $K$ of spacetime, and seek corresponding '$K$-nonlocal' generalised Green operators. Assuming the modification depends holomorphically on a parameter, conditions are given under which $K$-nonlocal Green operators exist for all parameter values, with the possible exception of a discrete set. The exceptional points occur precisely where the modified operator admits nontrivial smooth homogeneous solutions that have past- or future-compact support. Fredholm theory is used to relate the dimensions of these spaces to those corresponding to the formal dual operator, switching the roles of future and past. The $K$-nonlocal Green operators are shown to depend holomorphically on the parameter in the topology of bounded convergence on maps between suitable Sobolev spaces, or between suitable spaces of smooth functions. An application to the LU factorisation of systems of equations is described.
{"title":"Modified Green-Hyperbolic Operators","authors":"C. Fewster","doi":"10.3842/SIGMA.2023.057","DOIUrl":"https://doi.org/10.3842/SIGMA.2023.057","url":null,"abstract":"Green-hyperbolic operators - partial differential operators on globally hyperbolic spacetimes that (together with their formal duals) possess advanced and retarded Green operators - play an important role in many areas of mathematical physics. Here, we study modifications of Green-hyperbolic operators by the addition of a possibly nonlocal operator acting within a compact subset $K$ of spacetime, and seek corresponding '$K$-nonlocal' generalised Green operators. Assuming the modification depends holomorphically on a parameter, conditions are given under which $K$-nonlocal Green operators exist for all parameter values, with the possible exception of a discrete set. The exceptional points occur precisely where the modified operator admits nontrivial smooth homogeneous solutions that have past- or future-compact support. Fredholm theory is used to relate the dimensions of these spaces to those corresponding to the formal dual operator, switching the roles of future and past. The $K$-nonlocal Green operators are shown to depend holomorphically on the parameter in the topology of bounded convergence on maps between suitable Sobolev spaces, or between suitable spaces of smooth functions. An application to the LU factorisation of systems of equations is described.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45864735","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We construct an asymptotic metric on the moduli space of two centred hyperbolic monopoles by working in the point particle approximation, that is treating well-separated monopoles as point particles with an electric, magnetic and scalar charge and re-interpreting the dynamics of the 2-particle system as geodesic motion with respect to some metric. The corresponding analysis in the Euclidean case famously yields the negative mass Taub-NUT metric, which asymptotically approximates the L2 metric on the moduli space of two Euclidean monopoles, the Atiyah-Hitchin metric. An important difference with the Euclidean case is that, due to the absence of Galilean symmetry, in the hyperbolic case it is not possible to factor out the centre of mass motion. Nevertheless we show that we can consistently restrict to a 3-dimensional configuration space by considering antipodal configurations. In complete parallel with the Euclidean case, the metric that we obtain is then the hyperbolic analogue of negative mass Taub-NUT. We also show how the metric obtained is related to the asymptotic form of a hyperbolic analogue of the Atiyah-Hitchin metric constructed by Hitchin.
{"title":"The Asymptotic Structure of the Centred Hyperbolic 2-Monopole Moduli Space","authors":"G. Franchetti, C. Ross","doi":"10.3842/SIGMA.2023.043","DOIUrl":"https://doi.org/10.3842/SIGMA.2023.043","url":null,"abstract":"We construct an asymptotic metric on the moduli space of two centred hyperbolic monopoles by working in the point particle approximation, that is treating well-separated monopoles as point particles with an electric, magnetic and scalar charge and re-interpreting the dynamics of the 2-particle system as geodesic motion with respect to some metric. The corresponding analysis in the Euclidean case famously yields the negative mass Taub-NUT metric, which asymptotically approximates the L2 metric on the moduli space of two Euclidean monopoles, the Atiyah-Hitchin metric. An important difference with the Euclidean case is that, due to the absence of Galilean symmetry, in the hyperbolic case it is not possible to factor out the centre of mass motion. Nevertheless we show that we can consistently restrict to a 3-dimensional configuration space by considering antipodal configurations. In complete parallel with the Euclidean case, the metric that we obtain is then the hyperbolic analogue of negative mass Taub-NUT. We also show how the metric obtained is related to the asymptotic form of a hyperbolic analogue of the Atiyah-Hitchin metric constructed by Hitchin.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43005447","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Yamabe invariant is a diffeomorphism invariant of smooth compact manifolds that arises from the normalized Einstein-Hilbert functional. This article highlights the manner in which one compelling open problem regarding the Yamabe invariant appears to be closely tied to static potentials and the first eigenvalue of the Laplacian.
{"title":"Yamabe Invariants, Homogeneous Spaces, and Rational Complex Surfaces","authors":"C. LeBrun","doi":"10.3842/SIGMA.2023.027","DOIUrl":"https://doi.org/10.3842/SIGMA.2023.027","url":null,"abstract":"The Yamabe invariant is a diffeomorphism invariant of smooth compact manifolds that arises from the normalized Einstein-Hilbert functional. This article highlights the manner in which one compelling open problem regarding the Yamabe invariant appears to be closely tied to static potentials and the first eigenvalue of the Laplacian.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42760070","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give explicit formulas for ten compatible Poisson brackets on $mathbb P^5$ found in arXiv:2007.12351.
我们给出了arXiv:2007.12351中$mathbb P^5$上十个兼容泊松括号的显式公式。
{"title":"Ten Compatible Poisson Brackets on $mathbb P^5$","authors":"Ville Nordstrom, A. Polishchuk","doi":"10.3842/SIGMA.2023.059","DOIUrl":"https://doi.org/10.3842/SIGMA.2023.059","url":null,"abstract":"We give explicit formulas for ten compatible Poisson brackets on $mathbb P^5$ found in arXiv:2007.12351.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47497627","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider soliton solutions of the U(1) gauged Skyrme model with the pion mass term. The domain of existence of gauged Skyrmions is restricted from above by the value of the pion mass. Concentrating on the solutions of topological degree one, we find that coupling to the electromagnetic field breaks the symmetry of the configurations, the Skyrmions carrying both an electric charge and a magnetic flux, with an induced dipole magnetic moment. The Skyrmions also possess an angular momentum, which is quantized in the units of the electric charge. The mass of the gauged Skyrmions monotonically decreases with increase of the gauge coupling.
{"title":"Solitons in the Gauged Skyrme-Maxwell Model","authors":"L. R. Livramento, E. Radu, Y. Shnir","doi":"10.3842/SIGMA.2023.042","DOIUrl":"https://doi.org/10.3842/SIGMA.2023.042","url":null,"abstract":"We consider soliton solutions of the U(1) gauged Skyrme model with the pion mass term. The domain of existence of gauged Skyrmions is restricted from above by the value of the pion mass. Concentrating on the solutions of topological degree one, we find that coupling to the electromagnetic field breaks the symmetry of the configurations, the Skyrmions carrying both an electric charge and a magnetic flux, with an induced dipole magnetic moment. The Skyrmions also possess an angular momentum, which is quantized in the units of the electric charge. The mass of the gauged Skyrmions monotonically decreases with increase of the gauge coupling.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47374385","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
W. Ballmann, Mayukh Mukherjee, Panagiotis Polymerakis
We show the absolute continuity of the spectrum and determine the spectrum as a set for two classes of Hadamard manifolds and for specific domains and quotients of one of the classes.
我们展示了谱的绝对连续性，并将谱确定为两类Hadamard流形以及其中一类的特定域和商的集合。
{"title":"On the Spectrum of Certain Hadamard Manifolds","authors":"W. Ballmann, Mayukh Mukherjee, Panagiotis Polymerakis","doi":"10.3842/SIGMA.2023.050","DOIUrl":"https://doi.org/10.3842/SIGMA.2023.050","url":null,"abstract":"We show the absolute continuity of the spectrum and determine the spectrum as a set for two classes of Hadamard manifolds and for specific domains and quotients of one of the classes.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44941314","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The fundamental invariants for vector ODEs of order $ge 3$ considered up to point transformations consist of generalized Wilczynski invariants and C-class invariants. An ODE of C-class is characterized by the vanishing of the former. For any fixed C-class invariant ${mathcal U}$, we give a local (point) classification for all submaximally symmetric ODEs of C-class with ${mathcal U} not equiv 0$ and all remaining C-class invariants vanishing identically. Our results yield generalizations of a well-known classical result for scalar ODEs due to Sophus Lie. Fundamental invariants correspond to the harmonic curvature of the associated Cartan geometry. A key new ingredient underlying our classification results is an advance concerning the harmonic theory associated with the structure of vector ODEs of C-class. Namely, for each irreducible C-class module, we provide an explicit identification of a lowest weight vector as a harmonic 2-cochain.
考虑到点变换的$ge 3$阶向量ode的基本不变量由广义Wilczynski不变量和c类不变量组成。c级ODE的特征是前者的消失。对于任意固定的c类不变量${mathcal U}$，我们给出了所有次极大对称的c类ode的局部(点)分类，其中${mathcal U} not equiv 0$和所有剩余的c类不变量完全消失。我们的结果推广了一个著名的经典结果，该结果是由索菲斯·李引起的标量ode。基本不变量对应于相关卡坦几何的调和曲率。我们的分类结果背后的一个关键新因素是关于c类矢量ode结构的谐波理论的进展。即，对于每个不可约的c类模，我们给出了一个最低权向量作为调和2-辅链的显式标识。
{"title":"On Uniqueness of Submaximally Symmetric Vector Ordinary Differential Equations of C-Class","authors":"Johnson Allen Kessy, D. The","doi":"10.3842/SIGMA.2023.058","DOIUrl":"https://doi.org/10.3842/SIGMA.2023.058","url":null,"abstract":"The fundamental invariants for vector ODEs of order $ge 3$ considered up to point transformations consist of generalized Wilczynski invariants and C-class invariants. An ODE of C-class is characterized by the vanishing of the former. For any fixed C-class invariant ${mathcal U}$, we give a local (point) classification for all submaximally symmetric ODEs of C-class with ${mathcal U} not equiv 0$ and all remaining C-class invariants vanishing identically. Our results yield generalizations of a well-known classical result for scalar ODEs due to Sophus Lie. Fundamental invariants correspond to the harmonic curvature of the associated Cartan geometry. A key new ingredient underlying our classification results is an advance concerning the harmonic theory associated with the structure of vector ODEs of C-class. Namely, for each irreducible C-class module, we provide an explicit identification of a lowest weight vector as a harmonic 2-cochain.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48633886","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The concept of a Spinh-manifold, which is a cousin of Spin- and Spinc-manifolds, has been at the center of much research in recent years. This article discusses some of the highlights of this story.
{"title":"Spinh Manifolds","authors":"H. B. Lawson","doi":"10.3842/SIGMA.2023.012","DOIUrl":"https://doi.org/10.3842/SIGMA.2023.012","url":null,"abstract":"The concept of a Spinh-manifold, which is a cousin of Spin- and Spinc-manifolds, has been at the center of much research in recent years. This article discusses some of the highlights of this story.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46481310","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In a first part, we give a new proof of Koenigs theorem and, in a second part, we determine the local form of all the superintegrable Riemannian Liouville metrics as well as their global geometries.
{"title":"Koenigs Theorem and Superintegrable Liouville Metrics","authors":"G. Valent","doi":"10.3842/SIGMA.2023.048","DOIUrl":"https://doi.org/10.3842/SIGMA.2023.048","url":null,"abstract":"In a first part, we give a new proof of Koenigs theorem and, in a second part, we determine the local form of all the superintegrable Riemannian Liouville metrics as well as their global geometries.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44584781","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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