Pub Date : 2024-11-05DOI: 10.1016/j.geomphys.2024.105354
Adara M. Blaga , Bang-Yen Chen
We provide conditions for a Riemannian manifold with a nontrivial closed affine conformal Killing vector field to be isometric to a Euclidean sphere or to the Euclidean space. Also, we formulate some triviality results for almost Ricci solitons with affine conformal Killing potential vector field.
{"title":"On conformal collineation and almost Ricci solitons","authors":"Adara M. Blaga , Bang-Yen Chen","doi":"10.1016/j.geomphys.2024.105354","DOIUrl":"10.1016/j.geomphys.2024.105354","url":null,"abstract":"<div><div>We provide conditions for a Riemannian manifold with a nontrivial closed affine conformal Killing vector field to be isometric to a Euclidean sphere or to the Euclidean space. Also, we formulate some triviality results for almost Ricci solitons with affine conformal Killing potential vector field.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142593032","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 : 2024-10-30DOI: 10.1016/j.geomphys.2024.105351
Shangshuai Li , Da-jun Zhang
The paper establishes a direct linearization scheme for the SU(2) anti-self-dual Yang-Mills (ASDYM) equation. The scheme starts from a set of linear integral equations with general measures and plane wave factors. After introducing infinite-dimensional matrices as master functions, we are able to investigate evolution relations and recurrence relations of these functions, which lead us to the unreduced ASDYM equation. It is then reduced to the ASDYM equation in the Euclidean space and two ultrahyperbolic spaces by reductions to meet the reality conditions and gauge conditions, respectively. Special solutions can be obtained by choosing suitable measures.
{"title":"Direct linearization of the SU(2) anti-self-dual Yang-Mills equation in various spaces","authors":"Shangshuai Li , Da-jun Zhang","doi":"10.1016/j.geomphys.2024.105351","DOIUrl":"10.1016/j.geomphys.2024.105351","url":null,"abstract":"<div><div>The paper establishes a direct linearization scheme for the SU(2) anti-self-dual Yang-Mills (ASDYM) equation. The scheme starts from a set of linear integral equations with general measures and plane wave factors. After introducing infinite-dimensional matrices as master functions, we are able to investigate evolution relations and recurrence relations of these functions, which lead us to the unreduced ASDYM equation. It is then reduced to the ASDYM equation in the Euclidean space and two ultrahyperbolic spaces by reductions to meet the reality conditions and gauge conditions, respectively. Special solutions can be obtained by choosing suitable measures.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142592951","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 : 2024-10-30DOI: 10.1016/j.geomphys.2024.105353
Pragya Belwal, Nishant Rathee, Mahender Singh
Relative Rota–Baxter groups are generalisations of Rota–Baxter groups and recently shown to be intimately related to skew left braces, which are well-known to yield bijective non-degenerate solutions to the Yang–Baxter equation. In this paper, we develop an extension theory of relative Rota–Baxter groups and introduce their low dimensional cohomology groups, which are distinct from the ones known in the context of Rota–Baxter operators on Lie groups. We establish an explicit bijection between the set of equivalence classes of extensions of relative Rota–Baxter groups and their second cohomology. Further, we delve into the connections between this cohomology and the cohomology of associated skew left braces. We prove that for bijective relative Rota–Baxter groups, the two cohomologies are isomorphic in dimension two.
{"title":"Cohomology and extensions of relative Rota–Baxter groups","authors":"Pragya Belwal, Nishant Rathee, Mahender Singh","doi":"10.1016/j.geomphys.2024.105353","DOIUrl":"10.1016/j.geomphys.2024.105353","url":null,"abstract":"<div><div>Relative Rota–Baxter groups are generalisations of Rota–Baxter groups and recently shown to be intimately related to skew left braces, which are well-known to yield bijective non-degenerate solutions to the Yang–Baxter equation. In this paper, we develop an extension theory of relative Rota–Baxter groups and introduce their low dimensional cohomology groups, which are distinct from the ones known in the context of Rota–Baxter operators on Lie groups. We establish an explicit bijection between the set of equivalence classes of extensions of relative Rota–Baxter groups and their second cohomology. Further, we delve into the connections between this cohomology and the cohomology of associated skew left braces. We prove that for bijective relative Rota–Baxter groups, the two cohomologies are isomorphic in dimension two.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142586337","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 : 2024-10-30DOI: 10.1016/j.geomphys.2024.105348
Eunjeong Lee , Kyeong-Dong Park
We classify fourfolds with trivial canonical bundle which are zero loci of general global sections of completely reducible equivariant vector bundles over exceptional homogeneous varieties of Picard number one. By computing their Hodge numbers, we see that there exist no hyperkähler fourfolds among them. This implies that a hyperkähler fourfold represented as the zero locus of a general global section of a completely reducible equivariant vector bundle over a rational homogeneous variety of Picard number one is one of the two cases described by Beauville–Donagi and Debarre–Voisin.
{"title":"Complete intersection hyperkähler fourfolds with respect to equivariant vector bundles over rational homogeneous varieties of Picard number one","authors":"Eunjeong Lee , Kyeong-Dong Park","doi":"10.1016/j.geomphys.2024.105348","DOIUrl":"10.1016/j.geomphys.2024.105348","url":null,"abstract":"<div><div>We classify fourfolds with trivial canonical bundle which are zero loci of general global sections of completely reducible equivariant vector bundles over exceptional homogeneous varieties of Picard number one. By computing their Hodge numbers, we see that there exist no hyperkähler fourfolds among them. This implies that a hyperkähler fourfold represented as the zero locus of a general global section of a completely reducible equivariant vector bundle over a rational homogeneous variety of Picard number one is one of the two cases described by Beauville–Donagi and Debarre–Voisin.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142592953","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 : 2024-10-29DOI: 10.1016/j.geomphys.2024.105350
David Alfaya , Indranil Biswas , Tomás L. Gómez , Swarnava Mukhopadhyay
Given any irreducible smooth complex projective curve X, of genus at least 2, consider the moduli stack of vector bundles on X of fixed rank and determinant. It is proved that the isomorphism class of the stack uniquely determines the isomorphism class of the curve X and the rank of the vector bundles. The case of trivial determinant, rank 2 and genus 2 is specially interesting: the curve can be recovered from the moduli stack, but not from the moduli space (since this moduli space is thus independently of the curve).
We also prove a Torelli theorem for moduli stacks of principal G-bundles on a curve of genus at least 3, where G is any non-abelian reductive group.
给定任何不可还原的光滑复射曲线 X（其属至少为 2），考虑 X 上具有固定秩和行列式的向量束的模堆栈。证明了堆栈的同构类唯一地决定了曲线 X 的同构类和向量束的秩。三阶行列式、秩 2 和属 2 的情况特别有趣：曲线可以从模数堆栈中恢复，但不能从模数空间中恢复（因为这个模数空间是 P3，因此与曲线无关）。我们还证明了关于至少属 3 的曲线上主 G 束的模数堆栈的托勒里定理，其中 G 是任何非阿贝尔还原群。
{"title":"Torelli theorem for moduli stacks of vector bundles and principal G-bundles","authors":"David Alfaya , Indranil Biswas , Tomás L. Gómez , Swarnava Mukhopadhyay","doi":"10.1016/j.geomphys.2024.105350","DOIUrl":"10.1016/j.geomphys.2024.105350","url":null,"abstract":"<div><div>Given any irreducible smooth complex projective curve <em>X</em>, of genus at least 2, consider the moduli stack of vector bundles on <em>X</em> of fixed rank and determinant. It is proved that the isomorphism class of the stack uniquely determines the isomorphism class of the curve <em>X</em> and the rank of the vector bundles. The case of trivial determinant, rank 2 and genus 2 is specially interesting: the curve can be recovered from the moduli stack, but not from the moduli space (since this moduli space is <span><math><msup><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> thus independently of the curve).</div><div>We also prove a Torelli theorem for moduli stacks of principal <em>G</em>-bundles on a curve of genus at least 3, where <em>G</em> is any non-abelian reductive group.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142592952","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 : 2024-10-29DOI: 10.1016/j.geomphys.2024.105349
Anastasia V. Vikulova
We describe MBM classes for irreducible holomorphic symplectic manifolds of K3 and Kummer types. These classes are the monodromy images of extremal rational curves which give the faces of the nef cone of some birational model. We study the connection between our results and A. Bayer and E. Macrì's theory. We apply the numerical method of description due to E. Amerik and M. Verbitsky in low dimensions to the K3 type and Kummer type cases.
我们描述了 K3 和 Kummer 型不可还原全形交映流形的 MBM 类。这些类是极值有理曲线的单旋转图像，它们给出了某个双向模型的内锥面。我们研究了我们的结果与 A. Bayer 和 E. Macrì 的理论之间的联系。我们将 E. Amerik 和 M. Verbitsky 提出的低维数值描述方法应用于 K3 型和 Kummer 型情况。
{"title":"An elementary description of nef cone for irreducible holomorphic symplectic manifolds","authors":"Anastasia V. Vikulova","doi":"10.1016/j.geomphys.2024.105349","DOIUrl":"10.1016/j.geomphys.2024.105349","url":null,"abstract":"<div><div>We describe MBM classes for irreducible holomorphic symplectic manifolds of K3 and Kummer types. These classes are the monodromy images of extremal rational curves which give the faces of the nef cone of some birational model. We study the connection between our results and A. Bayer and E. Macrì's theory. We apply the numerical method of description due to E. Amerik and M. Verbitsky in low dimensions to the K3 type and Kummer type cases.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142592950","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 : 2024-10-28DOI: 10.1016/j.geomphys.2024.105347
Yannick Mvondo-She
We discuss a partition-valued stochastic process in the logarithmic sector of critical cosmological topologically massive gravity. By applying results obtained in our previous works, we first show that the logarithmic sector can be modeled as an urn scheme, with a conceptual view of the random process occurring in the theory as an evolutionary process whose dynamical state space is the urn content. The urn process is then identified as the celebrated Hoppe urn model. We next show a one-to-one correspondence between Hoppe's urn model and the genus-zero Feynman diagram expansion of the log sector in terms of rooted trees. In this context, the balls in the urn model are represented by nodes in the random tree model, and the “special” ball in this Pólya-like urn construction finds a nice interpretation as the root in the recursive tree model. Furthermore, a partition-valued Markov process in which a sequence of partitions whose distribution is given by Hurwitz numbers is shown to be encoded in the log partition function. Given the bijection between the set of partitions of n and the conjugacy classes of the symmetric group , it is shown that the structure of the Markov chain consisting of a sample space that is also the set of permutations of n elements, leads to a further description of the Markov chain in terms of a random walk on the symmetric group. From this perspective, a probabilistic interpretation of the logarithmic sector of the theory as a two-dimensional gauge theory on the group manifold is given. We suggest that a possible holographic dual to cosmological topologically massive gravity at the critical point could be a logarithmic conformal field theory that takes into account non-equilibrium phenomena.
我们讨论临界宇宙拓扑大质量引力对数扇区中的分区值随机过程。通过应用我们之前工作中获得的结果，我们首先证明对数扇区可以建模为一个瓮方案，并从概念上将理论中发生的随机过程视为一个演化过程，其动力学状态空间就是瓮的内容。瓮过程就是著名的霍普瓮模型。接下来，我们展示了霍普瓮模型与对数扇形的零属费曼图展开之间的一一对应关系。在这种情况下，瓮模型中的球可以用随机树模型中的节点来表示，这种类似波利亚的瓮构造中的 "特殊 "球可以很好地解释为递归树模型中的根。此外，一个分区值马尔可夫过程的分区序列的分布是由赫维茨数给出的，这表明对数分区函数对该过程进行了编码。鉴于 n 的分区集与对称群 Sn 的共轭类之间的双射关系，证明了马尔可夫链的结构由样本空间（也是 n 元素的排列集）组成，从而进一步用对称群上的随机行走来描述马尔可夫链。从这个角度出发，我们给出了该理论的对数部门作为 Sn 群流形上的二维规理论的概率解释。我们认为，宇宙拓扑大质量引力在临界点的全息对偶可能是一种考虑到非平衡现象的对数共形场论。
{"title":"Urn models, Markov chains and random walks in cosmological topologically massive gravity at the critical point","authors":"Yannick Mvondo-She","doi":"10.1016/j.geomphys.2024.105347","DOIUrl":"10.1016/j.geomphys.2024.105347","url":null,"abstract":"<div><div>We discuss a partition-valued stochastic process in the logarithmic sector of critical cosmological topologically massive gravity. By applying results obtained in our previous works, we first show that the logarithmic sector can be modeled as an urn scheme, with a conceptual view of the random process occurring in the theory as an evolutionary process whose dynamical state space is the urn content. The urn process is then identified as the celebrated Hoppe urn model. We next show a one-to-one correspondence between Hoppe's urn model and the genus-zero Feynman diagram expansion of the log sector in terms of rooted trees. In this context, the balls in the urn model are represented by nodes in the random tree model, and the “special” ball in this Pólya-like urn construction finds a nice interpretation as the root in the recursive tree model. Furthermore, a partition-valued Markov process in which a sequence of partitions whose distribution is given by Hurwitz numbers is shown to be encoded in the log partition function. Given the bijection between the set of partitions of <em>n</em> and the conjugacy classes of the symmetric group <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, it is shown that the structure of the Markov chain consisting of a sample space that is also the set of permutations of <em>n</em> elements, leads to a further description of the Markov chain in terms of a random walk on the symmetric group. From this perspective, a probabilistic interpretation of the logarithmic sector of the theory as a two-dimensional gauge theory on the <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> group manifold is given. We suggest that a possible holographic dual to cosmological topologically massive gravity at the critical point could be a logarithmic conformal field theory that takes into account non-equilibrium phenomena.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142561462","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 : 2024-10-24DOI: 10.1016/j.geomphys.2024.105344
Xin Tang , Xingting Wang , James J. Zhang
We introduce a Poisson version of the graded twist of a graded associative algebra and prove that every graded Poisson structure on a connected graded polynomial ring is a graded twist of a unimodular Poisson structure on A, namely, if π is a graded Poisson structure on A, then π has a decomposition where E is the Euler derivation, is the unimodular graded Poisson structure on A corresponding to π, and m is the modular derivation of . This result is a generalization of the same one in the quadratic setting. The rigidity of graded twisting, -minimality, and H-ozoneness are studied. As an application, we compute the Poisson cohomologies of the quadratic Poisson structures on the polynomial ring of three variables when the potential is irreducible, but not necessarily having an isolated singularity.
我们引入了分级关联代数的分级捻度的泊松版本，并证明连通的分级多项式环 A 上的每一个分级泊松结构都是 A 上一个单模泊松结构的分级捻度：=k[x1,...,xn]上的每一个有级泊松结构都是 A 上的单模泊松结构的有级扭转，即如果 π 是 A 上的有级泊松结构，那么 π 有一个分解π=πunim+1∑i=1ndegxiE∧m，其中 E 是欧拉导数，πunim 是与 π 对应的 A 上的单模有级泊松结构，m 是 (A,π) 的模导数。这一结果是同一结果在二次方程中的推广。我们还研究了分级扭曲的刚性、PH1-最小性和 H-ozoneness 。作为应用，我们计算了三变量多项式环上二次泊松结构的泊松同调，当势能是不可还原的，但不一定具有孤立奇点时。
{"title":"Twists of graded Poisson algebras and related properties","authors":"Xin Tang , Xingting Wang , James J. Zhang","doi":"10.1016/j.geomphys.2024.105344","DOIUrl":"10.1016/j.geomphys.2024.105344","url":null,"abstract":"<div><div>We introduce a Poisson version of the graded twist of a graded associative algebra and prove that every graded Poisson structure on a connected graded polynomial ring <span><math><mi>A</mi><mo>:</mo><mo>=</mo><mi>k</mi><mo>[</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo></math></span> is a graded twist of a unimodular Poisson structure on <em>A</em>, namely, if <em>π</em> is a graded Poisson structure on <em>A</em>, then <em>π</em> has a decomposition<span><span><span><math><mi>π</mi><mspace></mspace><mo>=</mo><mspace></mspace><msub><mrow><mi>π</mi></mrow><mrow><mi>u</mi><mi>n</mi><mi>i</mi><mi>m</mi></mrow></msub><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup><mi>deg</mi><mo></mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac><mi>E</mi><mo>∧</mo><mi>m</mi></math></span></span></span> where <em>E</em> is the Euler derivation, <span><math><msub><mrow><mi>π</mi></mrow><mrow><mi>u</mi><mi>n</mi><mi>i</mi><mi>m</mi></mrow></msub></math></span> is the unimodular graded Poisson structure on <em>A</em> corresponding to <em>π</em>, and <strong>m</strong> is the modular derivation of <span><math><mo>(</mo><mi>A</mi><mo>,</mo><mi>π</mi><mo>)</mo></math></span>. This result is a generalization of the same one in the quadratic setting. The rigidity of graded twisting, <span><math><mi>P</mi><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-minimality, and <em>H</em>-ozoneness are studied. As an application, we compute the Poisson cohomologies of the quadratic Poisson structures on the polynomial ring of three variables when the potential is irreducible, but not necessarily having an isolated singularity.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142578359","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 : 2024-10-24DOI: 10.1016/j.geomphys.2024.105345
John W. Barrett, James Gaunt
Finite real spectral triples are defined to characterise the non-commutative geometry of a fuzzy torus. The geometries are the non-commutative analogues of flat tori with moduli determined by integer parameters. Each of these geometries has four different Dirac operators, corresponding to the four spin structures on a torus. The spectrum of the Dirac operator is calculated. It is given by replacing integers with their quantum integer analogues in the spectrum of the corresponding commutative torus.
{"title":"Finite spectral triples for the fuzzy torus","authors":"John W. Barrett, James Gaunt","doi":"10.1016/j.geomphys.2024.105345","DOIUrl":"10.1016/j.geomphys.2024.105345","url":null,"abstract":"<div><div>Finite real spectral triples are defined to characterise the non-commutative geometry of a fuzzy torus. The geometries are the non-commutative analogues of flat tori with moduli determined by integer parameters. Each of these geometries has four different Dirac operators, corresponding to the four spin structures on a torus. The spectrum of the Dirac operator is calculated. It is given by replacing integers with their quantum integer analogues in the spectrum of the corresponding commutative torus.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142578360","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 : 2024-10-22DOI: 10.1016/j.geomphys.2024.105343
Gaëtan Borot , Maksim Karev , Danilo Lewański
The general relation between Chekhov–Eynard–Orantin topological recursion and the intersection theory on the moduli space of curves, the deformation techniques in topological recursion, and the polynomiality properties with respect to deformation parameters can be combined to derive vanishing relations involving intersection indices of tautological classes. We apply this strategy to three different families of spectral curves and show they give vanishing relations for integrals involving Ω-classes. The first class of vanishing relations are genus-independent and generalises the vanishings found by Johnson–Pandharipande–Tseng [35] and by the authors jointly with Do and Moskovsky [8]. The two other classes of vanishing relations are of a different nature and depend on the genus.
{"title":"On ELSV-type formulae and relations between Ω-integrals via deformations of spectral curves","authors":"Gaëtan Borot , Maksim Karev , Danilo Lewański","doi":"10.1016/j.geomphys.2024.105343","DOIUrl":"10.1016/j.geomphys.2024.105343","url":null,"abstract":"<div><div>The general relation between Chekhov–Eynard–Orantin topological recursion and the intersection theory on the moduli space of curves, the deformation techniques in topological recursion, and the polynomiality properties with respect to deformation parameters can be combined to derive vanishing relations involving intersection indices of tautological classes. We apply this strategy to three different families of spectral curves and show they give vanishing relations for integrals involving Ω-classes. The first class of vanishing relations are genus-independent and generalises the vanishings found by Johnson–Pandharipande–Tseng <span><span>[35]</span></span> and by the authors jointly with Do and Moskovsky <span><span>[8]</span></span>. The two other classes of vanishing relations are of a different nature and depend on the genus.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142561523","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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