Pub Date : 2026-04-01Epub Date: 2026-01-09DOI: 10.1016/j.geomphys.2026.105760
Quo-Shin Chi , Zhenxiao Xie , Yan Xu
In this paper we classify sextic curves in the Fano 3-fold (the smooth quintic del Pezzo 3-fold) that admit rational Galois covers in the complex . We show that the moduli space of such sextic curves is of complex dimension 2 by studying the invariants of the respective Galois groups via explicit constructions. This raises the intriguing question of understanding the moduli space of sextic curves in through their Galois covers in .
{"title":"Classification of sextic curves in the Fano 3-fold V5 with rational Galois covers in P3","authors":"Quo-Shin Chi , Zhenxiao Xie , Yan Xu","doi":"10.1016/j.geomphys.2026.105760","DOIUrl":"10.1016/j.geomphys.2026.105760","url":null,"abstract":"<div><div>In this paper we classify sextic curves in the Fano 3-fold <span><math><msub><mrow><mi>V</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span> (the smooth quintic del Pezzo 3-fold) that admit rational Galois covers in the complex <span><math><msup><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>. We show that the moduli space of such sextic curves is of complex dimension 2 by studying the invariants of the respective Galois groups via explicit constructions. This raises the intriguing question of understanding the moduli space of sextic curves in <span><math><msub><mrow><mi>V</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span> through their Galois covers in <span><math><msup><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"222 ","pages":"Article 105760"},"PeriodicalIF":1.2,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145979755","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 : 2026-04-01Epub Date: 2026-01-08DOI: 10.1016/j.geomphys.2026.105758
Jie Fei , Jun Wang
In this paper, we investigate the classification problem for holomorphic two-spheres of constant curvature in the complex Grassmann manifold under additional geometric conditions. By considering the (1,0)-part of μ-th covariant differential about the second fundamental form denoted by , , its norm denoted by , we establish the following results: for unramified holomorphic two-spheres with constant curvature and constant squared norm of the second fundamental form, the quantity is necessarily constant. Moreover, under additional conditions that is positive and is identically zero, we obtain a complete classification of such holomorphic two-spheres.
{"title":"On classification of holomorphic two-spheres of constant curvature in the complex Grassmann manifold G(3,6)","authors":"Jie Fei , Jun Wang","doi":"10.1016/j.geomphys.2026.105758","DOIUrl":"10.1016/j.geomphys.2026.105758","url":null,"abstract":"<div><div>In this paper, we investigate the classification problem for holomorphic two-spheres of constant curvature in the complex Grassmann manifold <span><math><mi>G</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>6</mn><mo>)</mo></math></span> under additional geometric conditions. By considering the (1,0)-part of <em>μ</em>-th covariant differential about the second fundamental form denoted by <span><math><msub><mrow><mi>P</mi></mrow><mrow><mo>,</mo><mi>μ</mi></mrow></msub></math></span>, <span><math><mi>μ</mi><mo>≥</mo><mn>1</mn></math></span>, its norm denoted by <span><math><mo>|</mo><msub><mrow><mi>P</mi></mrow><mrow><mo>,</mo><mi>μ</mi></mrow></msub><mo>|</mo></math></span>, we establish the following results: for unramified holomorphic two-spheres with constant curvature and constant squared norm of the second fundamental form, the quantity <span><math><mo>|</mo><msub><mrow><mi>P</mi></mrow><mrow><mo>,</mo><mn>1</mn></mrow></msub><mo>|</mo></math></span> is necessarily constant. Moreover, under additional conditions that <span><math><mo>|</mo><msub><mrow><mi>P</mi></mrow><mrow><mo>,</mo><mn>1</mn></mrow></msub><mo>|</mo></math></span> is positive and <span><math><mo>|</mo><msub><mrow><mi>P</mi></mrow><mrow><mo>,</mo><mn>2</mn></mrow></msub><mo>|</mo></math></span> is identically zero, we obtain a complete classification of such holomorphic two-spheres.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"222 ","pages":"Article 105758"},"PeriodicalIF":1.2,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145980338","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 : 2026-04-01Epub Date: 2025-12-30DOI: 10.1016/j.geomphys.2025.105749
A.V. Smilga
It was recently found that a necessary and sufficient condition for a Kähler manifold to be hyperkähler reads(1) where is a complex metric, is a symplectic matrix and C is a positive constant.
In this note, we give a simple explicit proof of this fact.
{"title":"On the multidimensional heavenly equation","authors":"A.V. Smilga","doi":"10.1016/j.geomphys.2025.105749","DOIUrl":"10.1016/j.geomphys.2025.105749","url":null,"abstract":"<div><div>It was recently found that a necessary and sufficient condition for a Kähler manifold to be hyperkähler reads<span><span><span>(1)</span><span><math><mrow><msub><mrow><mi>h</mi></mrow><mrow><mi>i</mi><mover><mrow><mi>k</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow></msub><msub><mrow><mi>h</mi></mrow><mrow><mi>j</mi><mover><mrow><mi>l</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow></msub><msup><mrow><mi>Ω</mi></mrow><mrow><mover><mrow><mi>k</mi></mrow><mrow><mo>¯</mo></mrow></mover><mover><mrow><mi>l</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow></msup><mspace></mspace><mo>=</mo><mspace></mspace><mi>C</mi><msub><mrow><mi>Ω</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>,</mo></mrow></math></span></span></span> where <span><math><msub><mrow><mi>h</mi></mrow><mrow><mi>i</mi><mover><mrow><mi>k</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow></msub></math></span> is a complex metric, <span><math><msub><mrow><mi>Ω</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></math></span> is a symplectic matrix and <em>C</em> is a positive constant.</div><div>In this note, we give a simple explicit proof of this fact.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"222 ","pages":"Article 105749"},"PeriodicalIF":1.2,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145940683","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 : 2026-04-01Epub Date: 2026-01-21DOI: 10.1016/j.geomphys.2026.105770
Shigeki Matsutani
The real part of the focusing modified Korteweg-de Vries (MKdV) equation defined over the complex field is reduced to the focusing gauged MKdV (FGMKdV) equation. In this paper, we construct the real hyperelliptic solutions of FGMKdV equation in terms of data of the hyperelliptic curves of genus g and demonstrate the closed hyperelliptic plane curves of genus whose curvature obeys the FGMKdV equation by extending the previous results of genus three (Matsutani (2025) [29]). These are a generalization of Euler's elasticae.
{"title":"Closed real plane curves of hyperelliptic solutions of focusing gauged modified KdV equation of genus g","authors":"Shigeki Matsutani","doi":"10.1016/j.geomphys.2026.105770","DOIUrl":"10.1016/j.geomphys.2026.105770","url":null,"abstract":"<div><div>The real part of the focusing modified Korteweg-de Vries (MKdV) equation defined over the complex field <span><math><mi>C</mi></math></span> is reduced to the focusing gauged MKdV (FGMKdV) equation. In this paper, we construct the real hyperelliptic solutions of FGMKdV equation in terms of data of the hyperelliptic curves of genus <em>g</em> and demonstrate the closed hyperelliptic plane curves of genus <span><math><mi>g</mi><mo>=</mo><mn>5</mn></math></span> whose curvature obeys the FGMKdV equation by extending the previous results of genus three (Matsutani (2025) <span><span>[29]</span></span>). These are a generalization of Euler's elasticae.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"222 ","pages":"Article 105770"},"PeriodicalIF":1.2,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146090674","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 : 2026-04-01Epub Date: 2026-01-19DOI: 10.1016/j.geomphys.2026.105768
Zhengping Gui , Si Li , Xinxing Tang
This paper is devoted to study integrable deformations of chiral conformal field theories on elliptic curves from the viewpoint of contact algebra. We introduce the relevant integrable condition within the framework of conformal vertex algebra, and derive the contact term relations among certain local operators. We investigate three versions of genus one partition functions and derive the contact equations. This leads to a rigorous formulation of Dijkgraaf's master equation [6] for chiral deformations.
{"title":"Contact term algebras and Dijkgraaf's master equation","authors":"Zhengping Gui , Si Li , Xinxing Tang","doi":"10.1016/j.geomphys.2026.105768","DOIUrl":"10.1016/j.geomphys.2026.105768","url":null,"abstract":"<div><div>This paper is devoted to study integrable deformations of chiral conformal field theories on elliptic curves from the viewpoint of contact algebra. We introduce the relevant integrable condition within the framework of conformal vertex algebra, and derive the contact term relations among certain local operators. We investigate three versions of genus one partition functions and derive the contact equations. This leads to a rigorous formulation of Dijkgraaf's master equation <span><span>[6]</span></span> for chiral deformations.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"222 ","pages":"Article 105768"},"PeriodicalIF":1.2,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146038568","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 : 2026-04-01Epub Date: 2026-01-09DOI: 10.1016/j.geomphys.2026.105762
Jonathan Weitsman
We show that the number of lattice points in the boundary of a positive integer dilate of a Delzant integral polytope is a polynomial in the dilation parameter, analogous to the Ehrhart polynomial giving the number of lattice points in a lattice polytope. We give an explicit formula for this polynomial, analogous to the formula of Khovanskii-Pukhlikov for the Ehrhart polynomial. These counting polynomials satisfy a lacunarity principle, the vanishing of alternate coefficients, quite unlike the Ehrhart polynomial. We show that formal geometric quantization of singular Calabi Yau hypersurfaces in smooth toric varieties gives this polynomial, in analogy with the relation of the Khovanskii-Pukhlikov formula to the geometric quantization of toric varieties. The Atiyah-Singer theorem for the index of the Dirac operator gives a moral argument for the lacunarity of the counting polynomial. We conjecture that similar formulas should hold for arbitrary simple integral polytope boundaries.
{"title":"Lattice points in polytope boundaries and formal geometric quantization of singular Calabi Yau hypersurfaces in toric varieties","authors":"Jonathan Weitsman","doi":"10.1016/j.geomphys.2026.105762","DOIUrl":"10.1016/j.geomphys.2026.105762","url":null,"abstract":"<div><div>We show that the number of lattice points in the boundary of a positive integer dilate of a Delzant integral polytope is a polynomial in the dilation parameter, analogous to the Ehrhart polynomial giving the number of lattice points in a lattice polytope. We give an explicit formula for this polynomial, analogous to the formula of Khovanskii-Pukhlikov for the Ehrhart polynomial. These counting polynomials satisfy a lacunarity principle, the vanishing of alternate coefficients, quite unlike the Ehrhart polynomial. We show that formal geometric quantization of singular Calabi Yau hypersurfaces in smooth toric varieties gives this polynomial, in analogy with the relation of the Khovanskii-Pukhlikov formula to the geometric quantization of toric varieties. The Atiyah-Singer theorem for the index of the Dirac operator gives a moral argument for the lacunarity of the counting polynomial. We conjecture that similar formulas should hold for arbitrary simple integral polytope boundaries.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"222 ","pages":"Article 105762"},"PeriodicalIF":1.2,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145979757","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 : 2026-04-01Epub Date: 2026-01-21DOI: 10.1016/j.geomphys.2026.105771
Luisa Boateng , Matilde Marcolli
A construction that assigns a Boolean 1D TQFT with defects to a finite state automaton was recently developed by Gustafson, Im, Kaldawy, Khovanov, and Lihn. We show that the construction is functorial with respect to the category of finite state automata with transducers as morphisms. Certain classes of subregular languages correspond to additional cohomological structures on the associated TQFTs. We also show that the construction generalizes to context-free grammars through a categorical version of the Chomsky–Schützenberger representation theorem, due to Melliès and Zeilberger. The corresponding TQFTs are then described as morphisms of colored operads on an operad of cobordisms with defects.
{"title":"Formal languages and TQFTs with defects","authors":"Luisa Boateng , Matilde Marcolli","doi":"10.1016/j.geomphys.2026.105771","DOIUrl":"10.1016/j.geomphys.2026.105771","url":null,"abstract":"<div><div>A construction that assigns a Boolean 1D TQFT with defects to a finite state automaton was recently developed by Gustafson, Im, Kaldawy, Khovanov, and Lihn. We show that the construction is functorial with respect to the category of finite state automata with transducers as morphisms. Certain classes of subregular languages correspond to additional cohomological structures on the associated TQFTs. We also show that the construction generalizes to context-free grammars through a categorical version of the Chomsky–Schützenberger representation theorem, due to Melliès and Zeilberger. The corresponding TQFTs are then described as morphisms of colored operads on an operad of cobordisms with defects.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"222 ","pages":"Article 105771"},"PeriodicalIF":1.2,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146038566","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 : 2026-04-01Epub Date: 2026-01-23DOI: 10.1016/j.geomphys.2026.105772
Wen-Xiu Ma , Chaudry Masood Khalique
Pairs of group reductions or similarity transformations involving off-diagonal block matrices are proposed and analyzed for a specific type of Ablowitz-Kaup-Newell-Segur (AKNS) matrix spectral problem. The corresponding reduced integrable hierarchies of AKNS matrix integrable models are presented, complementing the standard AKNS matrix integrable hierarchies. The Lax formulation plays a key role in generating these reduced matrix integrable models.
针对一类特殊类型的ablowitz - kap - newwell - segur （AKNS）矩阵谱问题，提出并分析了涉及非对角块矩阵的群约简或相似变换对。给出了相应的AKNS矩阵可积模型的约简可积层次，补充了标准AKNS矩阵可积层次。Lax公式在生成这些约简矩阵可积模型中起着关键作用。
{"title":"Matrix integrable models associated with reduced AKNS Lax pairs","authors":"Wen-Xiu Ma , Chaudry Masood Khalique","doi":"10.1016/j.geomphys.2026.105772","DOIUrl":"10.1016/j.geomphys.2026.105772","url":null,"abstract":"<div><div>Pairs of group reductions or similarity transformations involving off-diagonal block matrices are proposed and analyzed for a specific type of Ablowitz-Kaup-Newell-Segur (AKNS) matrix spectral problem. The corresponding reduced integrable hierarchies of AKNS matrix integrable models are presented, complementing the standard AKNS matrix integrable hierarchies. The Lax formulation plays a key role in generating these reduced matrix integrable models.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"222 ","pages":"Article 105772"},"PeriodicalIF":1.2,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146090671","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 : 2026-03-01Epub Date: 2025-12-17DOI: 10.1016/j.geomphys.2025.105740
Tung Tran
We present twistor BV actions that encompass many classically consistent bosonic holomorphic twistorial higher-spin theories with vanishing cosmological constant. Upon quantization, these actions are shown to be quantum consistent, i.e. no gauge anomaly, for some subclasses of twistorial higher-spin theories. Anomaly-free twistorial theories can be identified through an index theorem, which is a higher-spin extension of the Hirzebruch-Riemann-Roch index theorem. We also discuss the anomaly cancellation mechanisms on twistor space to render anomalous theories quantum consistent at one loop.
{"title":"Anomaly-free twistorial higher-spin theories","authors":"Tung Tran","doi":"10.1016/j.geomphys.2025.105740","DOIUrl":"10.1016/j.geomphys.2025.105740","url":null,"abstract":"<div><div>We present twistor BV actions that encompass many classically consistent bosonic holomorphic twistorial higher-spin theories with vanishing cosmological constant. Upon quantization, these actions are shown to be quantum consistent, i.e. no gauge anomaly, for some subclasses of twistorial higher-spin theories. Anomaly-free twistorial theories can be identified through an index theorem, which is a higher-spin extension of the Hirzebruch-Riemann-Roch index theorem. We also discuss the anomaly cancellation mechanisms on twistor space to render anomalous theories quantum consistent at one loop.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"221 ","pages":"Article 105740"},"PeriodicalIF":1.2,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145841320","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 : 2026-03-01Epub Date: 2025-12-03DOI: 10.1016/j.geomphys.2025.105723
Andrey Losev , Dmitrii Sheptunov , Xin Geng
One of the approaches to quantum gravity is to formulate it in terms of de Rham algebra, choose a triangulation of space-time, and replace differential forms by cochains (that form a finite dimensional vector space). The key issue in general relativity is the action of diffeomorphisms of space-time on fields. In this paper, we induce the action of diffeomorphisms on cochains by homotopy transfer (or, equivalently, BV integral) that leads to an action. We explicitly compute this action for the space-time, being an interval and a circle.
{"title":"On induced L∞ action of diffeomorphisms on cochains","authors":"Andrey Losev , Dmitrii Sheptunov , Xin Geng","doi":"10.1016/j.geomphys.2025.105723","DOIUrl":"10.1016/j.geomphys.2025.105723","url":null,"abstract":"<div><div>One of the approaches to quantum gravity is to formulate it in terms of de Rham algebra, choose a triangulation of space-time, and replace differential forms by cochains (that form a finite dimensional vector space). The key issue in general relativity is the action of diffeomorphisms of space-time on fields. In this paper, we induce the action of diffeomorphisms on cochains by homotopy transfer (or, equivalently, BV integral) that leads to an <span><math><msub><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span> action. We explicitly compute this action for the space-time, being an interval and a circle.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"221 ","pages":"Article 105723"},"PeriodicalIF":1.2,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145697786","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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