Journal of Geometry and Physics最新文献

英文中文

The hidden M-group 隐藏的m群

IF 1.2 3区数学 Q1 MATHEMATICS

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

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”) M-algebra, 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) group. 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.

As a simple but consequential application, we highlight how to describe lattice subgroups

Z^{k \leq 528}

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.

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.

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 [45] we have explained as the exceptional extension of the T-duality doubled super-spacetime.

我们以数学物理学家为对象，对被称为（“隐藏的”）m -代数的超级李代数的起源和性质进行了现代化和精简的回顾，m -代数在分析11D超重力时有些微妙。在论证了这个（隐藏的）m代数实际上是11D超引力的最大超例外切空间之后，我们特别在这里明确了它对一个（超李）群的积分。它配备了“分解”m理论3-形式的左不变扩展，这样它就构成了克莱因空间，在克莱因空间上，超例外时空将被局部建模为卡尔坦几何。作为一个简单但重要的应用，我们强调了如何描述隐藏m群的晶格子群Zk≤528，这些子群允许超例外时空的“隐藏”维度也环向紧化，类似于拓扑t二象性中熟悉的情况。为了处理这些结构中的微妙之处，我们(i)提供了“分解”m理论3-形式的计算机检查的重新推导，并且（ii）提出了超李群的流线概念，这既严格又接近物理直觉和实践。因此，本文以隐m代数为例，重点介绍了现代超李理论，并着眼于为超例外几何奠定基础。在我们讨论的新观测中，隐藏的m -代数降维为“隐藏的iia -代数”，在b[45]中，我们将其解释为t -对偶性加倍的超时空的特殊扩展。

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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 Q1 MATHEMATICS

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

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

L^{2}

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.

我们证明了作为深度学习中许多训练算法基础的参数空间中的标准梯度流可以连续地变形为自适应梯度流，从而在输出空间中产生（约束）欧几里得梯度流。此外，对于L2损失，如果输出相对于参数的雅可比矩阵是满秩的（对于固定的训练数据），那么时间变量可以被重新参数化，这样得到的流就是简单的线性插值，并且可以实现全局最小值。对于交叉熵损失，在相同秩条件下，假设标签有正分量，我们导出了唯一全局最小值的显式公式。

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 Q1 MATHEMATICS

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

Let

(M, g^{T M})

be a closed Riemannian manifold of dimension n, and let F be an integrable subbundle of TM. Let

k^{F}

be the leafwise scalar curvature associated to

g^{F} = g^{T M} |_{F}

. Let E be an oriented flat vector bundle. We show that if either TM or F is spin, and TM carries a metric

g^{T M}

satisfying that

k^{F}

, the leafwise scalar curvature along F, is positive everywhere, then

〈 \hat{A} (T M) e (E), [M] 〉 = 0

, where

\hat{A} (T M)

is the Hirzebruch

\hat{A}

-class of TM and

e (E)

is the Euler class of E. This extends the generalization of the Lichnerowicz vanishing theorem concerning the Euler class proved by Yu and Zhang to the case of foliations.

设（M,gTM）为n维的封闭黎曼流形，设F为TM的可积子束。设kF为与gF=gTM|F相关的叶向标量曲率。设E是一个有方向的平面向量束。我们证明了如果TM或F是自旋，并且TM携带一个度量gTM，满足沿F的叶向标量曲率kF处处为正，则< a - (TM)e(e),[M] > =0，其中a - (TM)是TM的Hirzebruch a -类，e(e)是e的欧拉类。这将Yu和Zhang证明的关于欧拉类的Lichnerowicz消失定理推广到叶分的情况。

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 Q1 MATHEMATICS

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

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.

我们给出了一个简单的证明，对于具有拉格朗日环面振动的预量子化紧辛流形，其黎曼-罗赫数与其玻尔-索默菲尔德纤维数重合。这可以看作是几何量子化的“极化独立性”现象的一个实例。这种振动的基空间获得所谓的积分-积分仿射结构。这个证明使用了下面这个简单的事实，它的证明比我们想象的要棘手：在紧致的积分-积分仿射流形上，总体积等于整数点的个数。

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 Q1 MATHEMATICS

Journal of Geometry and Physics

Pub Date : 2026-04-01 Epub Date: 2026-01-27 DOI: 10.1016/j.geomphys.2026.105775

Fu Liu

In this paper, we construct two kinds of Lie conformal algebras

cgm

and

cm

, associated with the loop general BMS-Kac-Moody algebra

gm

and the loop BMS-Kac-Moody algebra

m

, respectively. The second cohomology groups of these two conformal algebras are completely determined. And nontrivial free conformal modules of rank one and

Z

-graded free intermediate series modules over these two conformal algebras are also classified.

本文构造了两类李共形代数cgm和cm，分别与环一般BMS-Kac-Moody代数gm和环BMS-Kac-Moody代数m相关联。这两个共形代数的第二上同调群是完全确定的。并对这两个共形代数上的1阶非平凡自由共形模和z阶自由中间级数模进行了分类。

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 Q1 MATHEMATICS

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

We construct and study non-finitely graded Lie algebras

HV (a, b; ϵ)

related to Heisenberg-Virasoro type Lie algebras, where

a, b

are complex numbers, and

ϵ = \pm 1

. Using combinatorial techniques, we completely classify the free

U (h)

-modules of rank one over

HV (a, b; ϵ)

. 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

b = 1

and

ϵ = - 1

. Meanwhile, we also determine the simplicity and isomorphism classes of these modules.

我们构建并研究了与Heisenberg-Virasoro型李代数相关的非有限梯度李代数HV(a,b; λ)，其中a，b为复数，且λ =±1。使用组合技术，我们完全分类了秩1 / HV（a,b; λ）的自由U(h)-模块。结果表明，这些模比非有限梯度Virasoro代数上的模更多样、更复杂，特别是当b=1和λ =−1时，它们允许无限多个自由参数。同时，我们还确定了这些模块的简单性和同构类。

引用次数: 0

Twists of superconformal algebras 超共形代数的扭转

IF 1.2 3区数学 Q1 MATHEMATICS

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

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

N = 2

SCFTs being prominent examples. To obtain mathematical precision, we explain how to extract vertex algebras and

E_{n}

algebras from a twisted superconformal field theory using factorization algebras.

我们为3至6维超共形场理论的“共形扭转”理论迈出了第一步，扩展了众所周知的超对称理论的扭转分析。在超共形李代数中，共形扭转是一个零平方奇元，我们对所有扭转进行了分类，并描述了它们在超共形群的伴随作用下的轨道。我们主要研究复形超共形代数，除非另有明确说明；超共形代数的实形式可能具有重要的物理含义，但我们只在少数特殊情况下讨论这些微妙之处。共形扭转可以在超共形场论中产生有趣的子代数和算子的保护扇区，Donaldson-Witten拓扑场论和四维N=2 SCFTs的顶点算子代数是突出的例子。为了获得数学精度，我们解释了如何利用分解代数从扭曲超共形场论中提取顶点代数和En代数。

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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 Q1 MATHEMATICS

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

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.

在局部共形辛流形的哈密顿力学框架下，研究了正则变换和仿正则变换的概念。同时考虑了时间无关动力学和时间相关动力学。本文给出了关于正则对称和非正则对称变换的单参数群的类诺瑟定理，并给出了证明，并用初等例子进行了说明。

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 Q1 MATHEMATICS

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

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

C^{⁎}

-algebra norms.

研究了在基于微分的非交换代数上的非交换微分微积分集合中，与厄米特形式相容的无扭利维-奇维塔连接的存在性。我们用厄米特形式的算子的象证明了列维-奇维塔连接存在的一个充分必要条件。此外，我们确定了厄米形式的一个必要对称条件，扩展了黎曼几何中度量对称的经典概念。用包含非交换3环面的3阶自由模的显式计算说明了该理论。我们注意到我们的方法是代数的，不依赖于C -代数范数等分析工具。

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 Q1 MATHEMATICS

Journal of Geometry and Physics

Pub Date : 2026-04-01 Epub Date: 2026-01-26 DOI: 10.1016/j.geomphys.2026.105769

Branimir Ćaćić

We strengthen the orientability condition in our definition of θ-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.

我们在θ-可交换谱三重体的定义中加强了可定向条件，从而解决了证明主要定理的一个问题。特别地，我们证明了在相应的交换情况下，这个修正条件仍然是满足的。

引用次数: 0

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