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The hidden M-group 隐藏的m群
IF 1.2 3区 数学
Journal of Geometry and Physics Pub Date : 2026-04-01 Epub Date: 2025-12-30 DOI: 10.1016/j.geomphys.2025.105743
Grigorios Giotopoulos , Hisham Sati , Urs Schreiber
{"title":"The hidden M-group","authors":"Grigorios Giotopoulos ,&nbsp;Hisham Sati ,&nbsp;Urs Schreiber","doi":"10.1016/j.geomphys.2025.105743","DOIUrl":"10.1016/j.geomphys.2025.105743","url":null,"abstract":"<div><div>We give a modernized and streamlined review, aimed at mathematical physicists, of the origin and nature of the super Lie-algebra known as the (“hidden”) <em>M-algebra</em>, which arises somewhat subtly in analysis of 11D supergravity. Following arguments that this (hidden) M-algebra serves in fact as the maximal super-exceptional tangent space for 11D supergravity, we particularly make explicit here its integration to a (super-Lie) <em>group</em>. This is equipped with a left-invariant extension of the “decomposed” M-theory 3-form, such that it constitutes the Kleinian space on which super-exceptional spacetimes are to be locally modeled as Cartan geometries.</div><div>As a simple but consequential application, we highlight how to describe lattice subgroups <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mi>k</mi><mo>≤</mo><mn>528</mn></mrow></msup></math></span> of the hidden M-group that allow to toroidially compactify also the “hidden” dimensions of a super-exceptional spacetime, akin to the familiar situation in topological T-duality.</div><div>In order to deal with subtleties in these constructions, we (i) provide a computer-checked re-derivation of the “decomposed” M-theory 3-form, and (ii) present a streamlined conception of super-Lie groups, that is both rigorous while still close to physics intuition and practice.</div><div>Thereby this article highlights modernized super-Lie theory along the example of the hidden M-algebra, with an eye towards laying foundations for super-exceptional geometry. Among new observations which we touch on along the way is the dimensional reduction of the hidden M-algebra to a “hidden IIA-algebra” which in <span><span>[45]</span></span> we have explained as the exceptional extension of the T-duality doubled super-spacetime.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"222 ","pages":"Article 105743"},"PeriodicalIF":1.2,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145898029","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}
引用次数: 0
Gradient flow in parameter space is equivalent to linear interpolation in output space 参数空间中的梯度流等价于输出空间中的线性插值
IF 1.2 3区 数学
Journal of Geometry and Physics Pub Date : 2026-04-01 Epub Date: 2026-01-12 DOI: 10.1016/j.geomphys.2026.105765
Thomas Chen, Patrícia Muñoz Ewald
{"title":"Gradient flow in parameter space is equivalent to linear interpolation in output space","authors":"Thomas Chen,&nbsp;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":"2026-04-01","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}
引用次数: 0
Positive scalar curvature on foliations and the Euler class 叶上的正标量曲率和欧拉类
IF 1.2 3区 数学
Journal of Geometry and Physics Pub Date : 2026-04-01 Epub Date: 2025-12-30 DOI: 10.1016/j.geomphys.2025.105746
Guolin An , Guangxiang Su
{"title":"Positive scalar curvature on foliations and the Euler class","authors":"Guolin An ,&nbsp;Guangxiang Su","doi":"10.1016/j.geomphys.2025.105746","DOIUrl":"10.1016/j.geomphys.2025.105746","url":null,"abstract":"<div><div>Let <span><math><mo>(</mo><mi>M</mi><mo>,</mo><msup><mrow><mi>g</mi></mrow><mrow><mi>T</mi><mi>M</mi></mrow></msup><mo>)</mo></math></span> be a closed Riemannian manifold of dimension <em>n</em>, and let <em>F</em> be an integrable subbundle of <em>TM</em>. Let <span><math><msup><mrow><mi>k</mi></mrow><mrow><mi>F</mi></mrow></msup></math></span> be the leafwise scalar curvature associated to <span><math><msup><mrow><mi>g</mi></mrow><mrow><mi>F</mi></mrow></msup><mo>=</mo><msup><mrow><mi>g</mi></mrow><mrow><mi>T</mi><mi>M</mi></mrow></msup><msub><mrow><mo>|</mo></mrow><mrow><mi>F</mi></mrow></msub></math></span>. Let <em>E</em> be an oriented flat vector bundle. We show that if either <em>TM</em> or <em>F</em> is spin, and <em>TM</em> carries a metric <span><math><msup><mrow><mi>g</mi></mrow><mrow><mi>T</mi><mi>M</mi></mrow></msup></math></span> satisfying that <span><math><msup><mrow><mi>k</mi></mrow><mrow><mi>F</mi></mrow></msup></math></span>, the leafwise scalar curvature along <em>F</em>, is positive everywhere, then <span><math><mo>〈</mo><mover><mrow><mi>A</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>(</mo><mi>T</mi><mi>M</mi><mo>)</mo><mi>e</mi><mo>(</mo><mi>E</mi><mo>)</mo><mo>,</mo><mo>[</mo><mi>M</mi><mo>]</mo><mo>〉</mo><mo>=</mo><mn>0</mn></math></span>, where <span><math><mover><mrow><mi>A</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>(</mo><mi>T</mi><mi>M</mi><mo>)</mo></math></span> is the Hirzebruch <span><math><mover><mrow><mi>A</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span>-class of <em>TM</em> and <span><math><mi>e</mi><mo>(</mo><mi>E</mi><mo>)</mo></math></span> is the Euler class of <em>E</em>. This extends the generalization of the Lichnerowicz vanishing theorem concerning the Euler class proved by Yu and Zhang to the case of foliations.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"222 ","pages":"Article 105746"},"PeriodicalIF":1.2,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145898028","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}
引用次数: 0
Loop general BMS-Kac-Moody Lie conformal algebra 环一般BMS-Kac-Moody Lie共形代数
IF 1.2 3区 数学
Journal of Geometry and Physics Pub Date : 2026-04-01 Epub Date: 2026-01-27 DOI: 10.1016/j.geomphys.2026.105775
Fu Liu
{"title":"Loop general BMS-Kac-Moody Lie conformal algebra","authors":"Fu Liu","doi":"10.1016/j.geomphys.2026.105775","DOIUrl":"10.1016/j.geomphys.2026.105775","url":null,"abstract":"<div><div>In this paper, we construct two kinds of Lie conformal algebras <span><math><mi>cgm</mi></math></span> and <span><math><mi>cm</mi></math></span>, associated with the loop general BMS-Kac-Moody algebra <span><math><mi>gm</mi></math></span> and the loop BMS-Kac-Moody algebra <span><math><mi>m</mi></math></span>, respectively. The second cohomology groups of these two conformal algebras are completely determined. And nontrivial free conformal modules of rank one and <span><math><mi>Z</mi></math></span>-graded free intermediate series modules over these two conformal algebras are also classified.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"222 ","pages":"Article 105775"},"PeriodicalIF":1.2,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146090673","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}
引用次数: 0
Representations of non-finitely graded Heisenberg-Virasoro type Lie algebras 非有限梯度Heisenberg-Virasoro型李代数的表示
IF 1.2 3区 数学
Journal of Geometry and Physics Pub Date : 2026-04-01 Epub Date: 2026-01-12 DOI: 10.1016/j.geomphys.2026.105766
Chunguang Xia , Tianyu Ma , Wei Wang , Mingjing Zhang
{"title":"Representations of non-finitely graded Heisenberg-Virasoro type Lie algebras","authors":"Chunguang Xia ,&nbsp;Tianyu Ma ,&nbsp;Wei Wang ,&nbsp;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":"2026-04-01","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}
引用次数: 0
Integral-integral affine geometry, geometric quantization, and Riemann–Roch 积分-积分仿射几何，几何量化，和黎曼-洛克
IF 1.2 3区 数学
Journal of Geometry and Physics Pub Date : 2026-04-01 Epub Date: 2025-12-29 DOI: 10.1016/j.geomphys.2025.105745
Mark D. Hamilton , Yael Karshon , Takahiko Yoshida
{"title":"Integral-integral affine geometry, geometric quantization, and Riemann–Roch","authors":"Mark D. Hamilton ,&nbsp;Yael Karshon ,&nbsp;Takahiko Yoshida","doi":"10.1016/j.geomphys.2025.105745","DOIUrl":"10.1016/j.geomphys.2025.105745","url":null,"abstract":"<div><div>We give a simple proof that, for a pre-quantized compact symplectic manifold with a Lagrangian torus fibration, its Riemann–Roch number coincides with its number of Bohr–Sommerfeld fibres. This can be viewed as an instance of the “independence of polarization” phenomenon of geometric quantization. The base space for such a fibration acquires a so-called integral-integral affine structure. The proof uses the following simple fact, whose proof is trickier than we expected: on a compact integral-integral affine manifold, the total volume is equal to the number of integer points.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"222 ","pages":"Article 105745"},"PeriodicalIF":1.2,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145940686","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}
引用次数: 0
Twists of superconformal algebras 超共形代数的扭转
IF 1.2 3区 数学
Journal of Geometry and Physics Pub Date : 2026-04-01 Epub Date: 2026-01-12 DOI: 10.1016/j.geomphys.2026.105759
Chris Elliott, Owen Gwilliam, Matteo Lotito
{"title":"Twists of superconformal algebras","authors":"Chris Elliott,&nbsp;Owen Gwilliam,&nbsp;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":"2026-04-01","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}
引用次数: 0
Canonical and canonoid transformations for Hamiltonian systems on locally conformal symplectic manifolds 局部共形辛流形上哈密顿系统的正则和仿正则变换
IF 1.2 3区 数学
Journal of Geometry and Physics Pub Date : 2026-04-01 Epub Date: 2026-01-09 DOI: 10.1016/j.geomphys.2026.105761
Rafael Azuaje , Xuefeng Zhao
{"title":"Canonical and canonoid transformations for Hamiltonian systems on locally conformal symplectic manifolds","authors":"Rafael Azuaje ,&nbsp;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":"2026-04-01","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}
引用次数: 0
On the existence of noncommutative Levi-Civita connections in derivation based calculi 基于导数的微积分中非交换Levi-Civita连接的存在性
IF 1.2 3区 数学
Journal of Geometry and Physics Pub Date : 2026-04-01 Epub Date: 2026-01-12 DOI: 10.1016/j.geomphys.2026.105764
Joakim Arnlind, Victor Hildebrandsson
{"title":"On the existence of noncommutative Levi-Civita connections in derivation based calculi","authors":"Joakim Arnlind,&nbsp;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":"2026-04-01","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}
引用次数: 0
Corrigendum to ‘A reconstruction theorem for Connes–Landi deformations of commutative spectral triples’ [J. Geom. Phys. 98 (2015) 82–109] 对交换谱三元组的cones - landi变形的重构定理的更正[J]。几何学。物理学报，98 (2015)82-109]
IF 1.2 3区 数学
Journal of Geometry and Physics Pub Date : 2026-04-01 Epub Date: 2026-01-26 DOI: 10.1016/j.geomphys.2026.105769
Branimir Ćaćić
{"title":"Corrigendum to ‘A reconstruction theorem for Connes–Landi deformations of commutative spectral triples’ [J. Geom. Phys. 98 (2015) 82–109]","authors":"Branimir Ćaćić","doi":"10.1016/j.geomphys.2026.105769","DOIUrl":"10.1016/j.geomphys.2026.105769","url":null,"abstract":"<div><div>We strengthen the orientability condition in our definition of <em>θ</em>-commutative spectral triple to resolve an issue with the proof of our main theorem. In particular, we show that this corrected condition is still satisfied in the relevant commutative case.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"222 ","pages":"Article 105769"},"PeriodicalIF":1.2,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146090672","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}
引用次数: 0
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