Pub Date : 2026-01-16DOI: 10.1016/j.geomphys.2026.105767
Deng-Yun Yang , Hai-Ping Fu
Let M be a compact Willmore surface in the unit sphere. Denote by the component of the traceless second fundamental form of M. We prove that if , where and is the second largest eigenvalue of matrix , then M is either totally umbilic, , or the Veronese surface. We also give an estimate for the first eigenvalue of the Schrödinger operator .
{"title":"New rigidity of compact Willmore surfaces in S2+m","authors":"Deng-Yun Yang , Hai-Ping Fu","doi":"10.1016/j.geomphys.2026.105767","DOIUrl":"10.1016/j.geomphys.2026.105767","url":null,"abstract":"<div><div>Let <em>M</em> be a compact Willmore surface in the unit sphere. Denote by <span><math><msubsup><mrow><mover><mrow><mi>h</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>α</mi></mrow></msubsup></math></span> the component of the traceless second fundamental form of <em>M</em>. We prove that if <span><math><mn>0</mn><mo>≤</mo><msup><mrow><mi>ρ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msub><mrow><mover><mrow><mi>λ</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mn>2</mn></mrow></msub><mo>≤</mo><mi>n</mi></math></span>, where <span><math><msup><mrow><mi>ρ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><munder><mo>∑</mo><mrow><mi>α</mi><mo>,</mo><mi>i</mi><mo>,</mo><mi>j</mi></mrow></munder><msup><mrow><mo>(</mo><msubsup><mrow><mover><mrow><mi>h</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>α</mi></mrow></msubsup><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></math></span> and <span><math><msub><mrow><mover><mrow><mi>λ</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mn>2</mn></mrow></msub></math></span> is the second largest eigenvalue of matrix <span><math><mo>(</mo><munder><mo>∑</mo><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></munder><msubsup><mrow><mover><mrow><mi>h</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>α</mi></mrow></msubsup><msubsup><mrow><mover><mrow><mi>h</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mi>β</mi></mrow></msubsup><mo>)</mo></math></span>, then <em>M</em> is either totally umbilic, <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>(</mo><msqrt><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msqrt><mo>)</mo><mo>×</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>(</mo><msqrt><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msqrt><mo>)</mo></math></span>, or the Veronese surface. We also give an estimate for the first eigenvalue of the Schrödinger operator <span><math><mi>L</mi><mo>=</mo><mo>−</mo><mi>Δ</mi><mo>−</mo><msup><mrow><mi>ρ</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"222 ","pages":"Article 105767"},"PeriodicalIF":1.2,"publicationDate":"2026-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146038565","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 : 2026-01-12DOI: 10.1016/j.geomphys.2026.105765
Thomas Chen, Patrícia Muñoz Ewald
We prove that the standard gradient flow in parameter space that underlies many training algorithms in deep learning can be continuously deformed into an adapted gradient flow which yields (constrained) Euclidean gradient flow in output space. Moreover, for the loss, if the Jacobian of the outputs with respect to the parameters is full rank (for fixed training data), then the time variable can be reparametrized so that the resulting flow is simply linear interpolation, and a global minimum can be achieved. For the cross-entropy loss, under the same rank condition and assuming the labels have positive components, we derive an explicit formula for the unique global minimum.
{"title":"Gradient flow in parameter space is equivalent to linear interpolation in output space","authors":"Thomas Chen, Patrícia Muñoz Ewald","doi":"10.1016/j.geomphys.2026.105765","DOIUrl":"10.1016/j.geomphys.2026.105765","url":null,"abstract":"<div><div>We prove that the standard gradient flow in parameter space that underlies many training algorithms in deep learning can be continuously deformed into an adapted gradient flow which yields (constrained) Euclidean gradient flow in output space. Moreover, for the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> loss, if the Jacobian of the outputs with respect to the parameters is full rank (for fixed training data), then the time variable can be reparametrized so that the resulting flow is simply linear interpolation, and a global minimum can be achieved. For the cross-entropy loss, under the same rank condition and assuming the labels have positive components, we derive an explicit formula for the unique global minimum.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"222 ","pages":"Article 105765"},"PeriodicalIF":1.2,"publicationDate":"2026-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145979754","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 : 2026-01-12DOI: 10.1016/j.geomphys.2026.105766
Chunguang Xia , Tianyu Ma , Wei Wang , Mingjing Zhang
We construct and study non-finitely graded Lie algebras related to Heisenberg-Virasoro type Lie algebras, where are complex numbers, and . Using combinatorial techniques, we completely classify the free -modules of rank one over . It turns out that these modules are more varied and complex than those over non-finitely graded Virasoro algebras, and in particular admit infinitely many free parameters if and . Meanwhile, we also determine the simplicity and isomorphism classes of these modules.
{"title":"Representations of non-finitely graded Heisenberg-Virasoro type Lie algebras","authors":"Chunguang Xia , Tianyu Ma , Wei Wang , Mingjing Zhang","doi":"10.1016/j.geomphys.2026.105766","DOIUrl":"10.1016/j.geomphys.2026.105766","url":null,"abstract":"<div><div>We construct and study non-finitely graded Lie algebras <span><math><mrow><mi>HV</mi></mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>;</mo><mi>ϵ</mi><mo>)</mo></math></span> related to Heisenberg-Virasoro type Lie algebras, where <span><math><mi>a</mi><mo>,</mo><mi>b</mi></math></span> are complex numbers, and <span><math><mi>ϵ</mi><mo>=</mo><mo>±</mo><mn>1</mn></math></span>. Using combinatorial techniques, we completely classify the free <span><math><mi>U</mi><mo>(</mo><mi>h</mi><mo>)</mo></math></span>-modules of rank one over <span><math><mrow><mi>HV</mi></mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>;</mo><mi>ϵ</mi><mo>)</mo></math></span>. It turns out that these modules are more varied and complex than those over non-finitely graded Virasoro algebras, and in particular admit infinitely many free parameters if <span><math><mi>b</mi><mo>=</mo><mn>1</mn></math></span> and <span><math><mi>ϵ</mi><mo>=</mo><mo>−</mo><mn>1</mn></math></span>. Meanwhile, we also determine the simplicity and isomorphism classes of these modules.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"222 ","pages":"Article 105766"},"PeriodicalIF":1.2,"publicationDate":"2026-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145979758","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 : 2026-01-12DOI: 10.1016/j.geomphys.2026.105759
Chris Elliott, Owen Gwilliam, Matteo Lotito
We take first steps toward a theory of “conformal twists” for superconformal field theories in dimension 3 to 6, extending the well-known analysis of twists for supersymmetric theories. A conformal twist is a square-zero odd element in the superconformal Lie algebra, and we classify all twists and describe their orbits under the adjoint action of the superconformal group. We work mostly with the complexified superconformal algebras, unless explicitly stated otherwise; real forms of the superconformal algebra may have important physical implications, but we only discuss these subtleties in a few special cases. Conformal twists can give rise to interesting subalgebras and protected sectors of operators in a superconformal field theory, with the Donaldson–Witten topological field theory and the vertex operator algebras of 4-dimensional SCFTs being prominent examples. To obtain mathematical precision, we explain how to extract vertex algebras and algebras from a twisted superconformal field theory using factorization algebras.
{"title":"Twists of superconformal algebras","authors":"Chris Elliott, Owen Gwilliam, Matteo Lotito","doi":"10.1016/j.geomphys.2026.105759","DOIUrl":"10.1016/j.geomphys.2026.105759","url":null,"abstract":"<div><div>We take first steps toward a theory of “conformal twists” for superconformal field theories in dimension 3 to 6, extending the well-known analysis of twists for supersymmetric theories. A conformal twist is a square-zero odd element in the superconformal Lie algebra, and we classify all twists and describe their orbits under the adjoint action of the superconformal group. We work mostly with the complexified superconformal algebras, unless explicitly stated otherwise; real forms of the superconformal algebra may have important physical implications, but we only discuss these subtleties in a few special cases. Conformal twists can give rise to interesting subalgebras and protected sectors of operators in a superconformal field theory, with the Donaldson–Witten topological field theory and the vertex operator algebras of 4-dimensional <span><math><mi>N</mi><mo>=</mo><mn>2</mn></math></span> SCFTs being prominent examples. To obtain mathematical precision, we explain how to extract vertex algebras and <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> algebras from a twisted superconformal field theory using factorization algebras.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"222 ","pages":"Article 105759"},"PeriodicalIF":1.2,"publicationDate":"2026-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145979756","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 : 2026-01-12DOI: 10.1016/j.geomphys.2026.105764
Joakim Arnlind, Victor Hildebrandsson
We study the existence of Levi-Civita connections, i.e. torsion free connections compatible with a hermitian form, in the setting of derivation based noncommutative differential calculi over ⁎-algebras. We prove a necessary and sufficient condition for the existence of Levi-Civita connections in terms of the image of an operator derived from the hermitian form. Moreover, we identify a necessary symmetry condition on the hermitian form that extends the classical notion of metric symmetry in Riemannian geometry. The theory is illustrated with explicit computations for free modules of rank three, including noncommutative 3-tori. We note that our approach is algebraic and does not rely on analytic tools such as -algebra norms.
{"title":"On the existence of noncommutative Levi-Civita connections in derivation based calculi","authors":"Joakim Arnlind, Victor Hildebrandsson","doi":"10.1016/j.geomphys.2026.105764","DOIUrl":"10.1016/j.geomphys.2026.105764","url":null,"abstract":"<div><div>We study the existence of Levi-Civita connections, i.e. torsion free connections compatible with a hermitian form, in the setting of derivation based noncommutative differential calculi over ⁎-algebras. We prove a necessary and sufficient condition for the existence of Levi-Civita connections in terms of the image of an operator derived from the hermitian form. Moreover, we identify a necessary symmetry condition on the hermitian form that extends the classical notion of metric symmetry in Riemannian geometry. The theory is illustrated with explicit computations for free modules of rank three, including noncommutative 3-tori. We note that our approach is algebraic and does not rely on analytic tools such as <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-algebra norms.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"222 ","pages":"Article 105764"},"PeriodicalIF":1.2,"publicationDate":"2026-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146038567","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 : 2026-01-09DOI: 10.1016/j.geomphys.2026.105761
Rafael Azuaje , Xuefeng Zhao
This paper is focused on the development of the notions of canonical and canonoid transformations within the framework of Hamiltonian mechanics on locally conformal symplectic manifolds. Both, time-independent and time-dependent dynamics are considered. Noether-like theorems relating one-parameter groups of transformations with canonical and non-canonical symmetries, are formulated, proved as well as illustrated with elementary examples.
{"title":"Canonical and canonoid transformations for Hamiltonian systems on locally conformal symplectic manifolds","authors":"Rafael Azuaje , Xuefeng Zhao","doi":"10.1016/j.geomphys.2026.105761","DOIUrl":"10.1016/j.geomphys.2026.105761","url":null,"abstract":"<div><div>This paper is focused on the development of the notions of canonical and canonoid transformations within the framework of Hamiltonian mechanics on locally conformal symplectic manifolds. Both, time-independent and time-dependent dynamics are considered. Noether-like theorems relating one-parameter groups of transformations with canonical and non-canonical symmetries, are formulated, proved as well as illustrated with elementary examples.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"222 ","pages":"Article 105761"},"PeriodicalIF":1.2,"publicationDate":"2026-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145979759","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 : 2026-01-09DOI: 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":"2026-01-09","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 : 2026-01-09DOI: 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":"2026-01-09","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 : 2026-01-08DOI: 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":"2026-01-08","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 : 2026-01-07DOI: 10.1016/j.geomphys.2025.105750
Paul Norbury
In this paper we relate volumes of moduli spaces of super Riemann surfaces to integrals over the moduli space of stable Riemann surfaces . This allows us to prove via algebraic geometry a recursion between the volumes of moduli spaces of super hyperbolic surfaces previously proven via super geometry techniques by Stanford and Witten. The recursion between the volumes of moduli spaces of super hyperbolic surfaces is proven to be equivalent to the property that a generating function for the intersection numbers of a natural collection of cohomology classes with tautological classes on is a KdV tau function. This is analogous to Mirzakhani's proof of the Kontsevich-Witten theorem, which relates a generating function for the intersection numbers of tautological classes on to KdV, using volumes of moduli spaces of hyperbolic surfaces.
{"title":"Enumerative geometry via the moduli space of super Riemann surfaces","authors":"Paul Norbury","doi":"10.1016/j.geomphys.2025.105750","DOIUrl":"10.1016/j.geomphys.2025.105750","url":null,"abstract":"<div><div>In this paper we relate volumes of moduli spaces of super Riemann surfaces to integrals over the moduli space of stable Riemann surfaces <span><math><msub><mrow><mover><mrow><mi>M</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>g</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span>. This allows us to prove via algebraic geometry a recursion between the volumes of moduli spaces of super hyperbolic surfaces previously proven via super geometry techniques by Stanford and Witten. The recursion between the volumes of moduli spaces of super hyperbolic surfaces is proven to be equivalent to the property that a generating function for the intersection numbers of a natural collection of cohomology classes <span><math><msub><mrow><mi>Θ</mi></mrow><mrow><mi>g</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span> with tautological classes on <span><math><msub><mrow><mover><mrow><mi>M</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>g</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span> is a KdV tau function. This is analogous to Mirzakhani's proof of the Kontsevich-Witten theorem, which relates a generating function for the intersection numbers of tautological classes on <span><math><msub><mrow><mover><mrow><mi>M</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>g</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span> to KdV, using volumes of moduli spaces of hyperbolic surfaces.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"222 ","pages":"Article 105750"},"PeriodicalIF":1.2,"publicationDate":"2026-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146038569","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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