MDPI - Publisher of Open Access Journals

21 pages, 402 KB

Open AccessArticle

Frobenius Pullback Map on Moduli of Principal G-Bundles

by Xiaoyu Su and Yumin Zhong

Mathematics 2026, 14(12), 2076; https://doi.org/10.3390/math14122076 - 10 Jun 2026

Viewed by 131

Let G be a connected reductive group over K with

char K = p > 0

. In this paper, we prove via deformation-theoretic techniques that the Frobenius pullback of a general stable G-bundle on a curve X remains stable in the [...] Read more.

Let G be a connected reductive group over K with

char K = p > 0

. In this paper, we prove via deformation-theoretic techniques that the Frobenius pullback of a general stable G-bundle on a curve X remains stable in the moduli stack of G-bundles with fixed topological type. We then present some applications of the main theorem. Full article

(This article belongs to the Special Issue Advanced Researches in Algebraic Geometry)

19 pages, 350 KB

Open AccessFeature PaperArticle

The Moduli Space of Octonionic Bundles as a Subvariety of Orthogonal Bundles

by Álvaro Antón-Sancho

Mathematics 2026, 14(8), 1330; https://doi.org/10.3390/math14081330 - 15 Apr 2026

Viewed by 307

Abstract

Let X be a compact Riemann surface of genus

g \geq 2

. An octonionic bundle over X is a fiber bundle whose fiber is the non-associative algebra of complex octonions, equivalently a principal

G_{2} (C)

-bundle, where [...] Read more.

Let X be a compact Riemann surface of genus

g \geq 2

. An octonionic bundle over X is a fiber bundle whose fiber is the non-associative algebra of complex octonions, equivalently a principal

G_{2} (C)

-bundle, where

G_{2} (C)

is the exceptional Lie group of automorphisms of the octonions. We prove that the natural inclusion

G_{2} (C) ↪ SO (7, C)

induces a closed embedding of the moduli space

M_{Oct} (X)

into the moduli space

M_{SO (7, C)} (X)

of

SO (7, C)

-bundles. We further analyze the normal bundle to this embedding, computing its rank as

7 (g - 1)

and providing an explicit cohomological description of its fibers, which enables explicit computations of tangent spaces and provides a foundation for deformation theory. As applications of the embedding, we prove that the image is a closed irreducible subvariety not contained in the singular locus of the ambient space, and we derive the Whitney formula

c (T_{amb}) = c (T) \cdot c (N)

relating the Chern classes of the tangent bundle of

M_{Oct} (X)

, the pullback of the ambient tangent bundle, and the normal bundle over the smooth locus. Full article

(This article belongs to the Special Issue Advances in Design Theory, Combinatorial Algebraic Geometry and Applications)

► Show Figures

Figure 1

25 pages, 371 KB

Open AccessArticle

Involutions of the Moduli Space of Principal E₆-Bundles over a Compact Riemann Surface

by Álvaro Antón-Sancho

Axioms 2025, 14(6), 423; https://doi.org/10.3390/axioms14060423 - 29 May 2025

Cited by 1 | Viewed by 1202

Abstract

In this paper, the fixed points of involutions on the moduli space of principal

E_{6}

-bundles over a compact Riemann surface X are investigated. In particular, it is proved that the combined action of a representative

σ

of the outer involution of [...] Read more.

In this paper, the fixed points of involutions on the moduli space of principal

E_{6}

-bundles over a compact Riemann surface X are investigated. In particular, it is proved that the combined action of a representative

σ

of the outer involution of

E_{6}

with the pull-back action of a surface involution

τ

admits fixed points if and only if a specific topological obstruction in

H^{2} (X / τ, π_{0} (E_{6}^{σ}))

vanishes. For an involution

τ

with

2 k

fixed points, it is proved that the fixed point set is isomorphic to the moduli space of principal H-bundles over the quotient curve

X / τ

, where H is either

F_{4}

or

PSp (8, C)

and it consists of

2^{g - k + 1}

components. The complex dimensions of these components are computed, and their singular loci are determined as corresponding to H-bundles admitting non-trivial automorphisms. Furthermore, it is checked that the stability of fixed

E_{6}

-bundles implies the stability of their corresponding H-bundles over

X / τ

, and the behavior of characteristic classes is discussed under this correspondence. Finally, as an application of the above results, it is proved that the fixed points correspond to octonionic structures on

X / τ

, and an explicit construction of these octonionic structures is provided. Full article

(This article belongs to the Special Issue Trends in Differential Geometry and Algebraic Topology)

► Show Figures

Figure 1

20 pages, 351 KB

Open AccessFeature PaperArticle

The Moduli Space of Principal G₂-Bundles and Automorphisms

by Álvaro Antón-Sancho

Mathematics 2025, 13(7), 1086; https://doi.org/10.3390/math13071086 - 26 Mar 2025

Viewed by 1634

Abstract

Let X be a compact Riemann surface of genus

g \geq 2

and

M (G_{2})

be the moduli space of polystable principal bundles over X, the structure group of which is the simple complex Lie group of exceptional type [...] Read more.

Let X be a compact Riemann surface of genus

g \geq 2

and

M (G_{2})

be the moduli space of polystable principal bundles over X, the structure group of which is the simple complex Lie group of exceptional type

G_{2}

. In this work, it is proved that the only automorphisms that

M (G_{2})

admits are those defined as the pull-back action of an automorphism of the base curve X. The strategy followed uses specific techniques that arise from the geometry of the gauge group

G_{2}

. In particular, some new results that provide relations between the stability, simplicity, and irreducibility of

G_{2}

-bundles over X have been proved in the paper. The inclusion of groups

G_{2} ↪ Spin (8, C)

where

G_{2}

is viewed as the fixed point subgroup of an order of 3 automorphisms of

Spin (8, C)

that lifts the triality automorphism is also considered. Specifically, this inclusion induces the forgetful map of moduli spaces of principal bundles

M (G_{2}) \to M (Spin (8, C))

. In the paper, it is also proved that the forgetful map is an embedding. Finally, some consequences are drawn from the results above on the geometry of

M (G_{2})

in relation to

M (Spin (8, C))

. Full article

16 pages, 429 KB

Open AccessArticle

Pullback Bundles and the Geometry of Learning

by Stéphane Puechmorel

Entropy 2023, 25(10), 1450; https://doi.org/10.3390/e25101450 - 15 Oct 2023

Cited by 2 | Viewed by 2980

Abstract

Explainable Artificial Intelligence (XAI) and acceptable artificial intelligence are active topics of research in machine learning. For critical applications, being able to prove or at least to ensure with a high probability the correctness of algorithms is of utmost importance. In practice, however, [...] Read more.

Explainable Artificial Intelligence (XAI) and acceptable artificial intelligence are active topics of research in machine learning. For critical applications, being able to prove or at least to ensure with a high probability the correctness of algorithms is of utmost importance. In practice, however, few theoretical tools are known that can be used for this purpose. Using the Fisher Information Metric (FIM) on the output space yields interesting indicators in both the input and parameter spaces, but the underlying geometry is not yet fully understood. In this work, an approach based on the pullback bundle, a well-known trick for describing bundle morphisms, is introduced and applied to the encoder–decoder block. With constant rank hypothesis on the derivative of the network with respect to its inputs, a description of its behavior is obtained. Further generalization is gained through the introduction of the pullback generalized bundle that takes into account the sensitivity with respect to weights. Full article

(This article belongs to the Special Issue Information Geometry for Data Analysis)

► Show Figures

Figure 1

16 pages, 323 KB

Open AccessReview

R-Symmetries and Curvature Constraints in A-Twisted Heterotic Landau–Ginzburg Models

by Richard S. Garavuso

Particles 2023, 6(3), 746-761; https://doi.org/10.3390/particles6030047 - 7 Aug 2023

Viewed by 1792

Abstract

In this paper, we discuss various aspects of a class of A-twisted heterotic Landau–Ginzburg models on a Kähler variety X. We provide a classification of the R-symmetries in these models which allow the A-twist to be implemented, focusing on the case in [...] Read more.

In this paper, we discuss various aspects of a class of A-twisted heterotic Landau–Ginzburg models on a Kähler variety X. We provide a classification of the R-symmetries in these models which allow the A-twist to be implemented, focusing on the case in which the gauge bundle is either a deformation of the tangent bundle of X or a deformation of a sub-bundle of the tangent bundle of X. Some anomaly-free examples are provided. The curvature constraint imposed by supersymmetry in these models when the superpotential is not holomorphic is reviewed. Constraints of this nature have been used to establish properties of analogues of pullbacks of Mathai–Quillen forms which arise in the correlation functions of the corresponding A-twisted or B-twisted heterotic Landau–Ginzburg models. The analogue most relevant to this paper is a deformation of the pullback of a Mathai–Quillen form. We discuss how this deformation may arise in the class of models studied in this paper. We then comment on how analogues of pullbacks of Mathai–Quillen forms not discussed in previous work may be obtained. Standard Mathai–Quillen formalism is reviewed in an appendix. We also include an appendix which discusses the deformation of the pullback of a Mathai–Quillen form. Full article

(This article belongs to the Collection High Energy Physics)

56 pages, 563 KB

Open AccessFeature PaperArticle

Riemannian Calculus of Variations Using Strongly Typed Tensor Calculus

by Victor Dods

Mathematics 2022, 10(18), 3231; https://doi.org/10.3390/math10183231 - 6 Sep 2022

Viewed by 2789

Abstract

In this paper, the notion of strongly typed language will be borrowed from the field of computer programming to introduce a calculational framework for linear algebra and tensor calculus for the purpose of detecting errors resulting from inherent misuse of objects and for [...] Read more.

In this paper, the notion of strongly typed language will be borrowed from the field of computer programming to introduce a calculational framework for linear algebra and tensor calculus for the purpose of detecting errors resulting from inherent misuse of objects and for finding natural formulations of various objects. A tensor bundle formalism, crucially relying on the notion of pullback bundle, will be used to create a rich type system with which to distinguish objects. The type system and relevant notation is designed to “telescope” to accommodate a level of detail appropriate to a set of calculations. Various techniques using this formalism will be developed and demonstrated with the goal of providing a relatively complete and uniform method of coordinate-free computation. The calculus of variations pertaining to maps between Riemannian manifolds will be formulated using the strongly typed tensor formalism and associated techniques. Energy functionals defined in terms of first-order Lagrangians are the focus of the second half of this paper, in which the first variation, the Euler–Lagrange equations, and the second variation of such functionals will be derived. Full article

(This article belongs to the Special Issue Variational Methods on Riemannian Manifolds: Theory and Applications)

► Show Figures

Figure 1

19 pages, 313 KB

Open AccessArticle

Minimum Time Problem Controlled by Affine Connection

by Constantin Udriste and Ionel Tevy

Symmetry 2021, 13(8), 1391; https://doi.org/10.3390/sym13081391 - 31 Jul 2021

Cited by 1 | Viewed by 1864

Abstract

Geometrically, the affine connection is the main ingredient that underlies the covariant derivative, the parallel transport, the auto-parallel curves, the torsion tensor field, and the curvature tensor field on a finite-dimensional differentiable manifold. In this paper, we come up with a new idea [...] Read more.

Geometrically, the affine connection is the main ingredient that underlies the covariant derivative, the parallel transport, the auto-parallel curves, the torsion tensor field, and the curvature tensor field on a finite-dimensional differentiable manifold. In this paper, we come up with a new idea of controllability and observability of states by using auto-parallel curves, and the minimum time problem controlled by the affine connection. The main contributions refer to the following: (i) auto-parallel curves controlled by a connection, (ii) reachability and controllability on the tangent bundle of a manifold, (iii) examples of equiaffine connections, (iv) minimum time problem controlled by a connection, (v) connectivity by stochastic perturbations of auto-parallel curves, and (vi) computing the optimal time and the optimal striking time. The connections with bounded pull-backs result in bang–bang optimal controls. Some significative examples on bi-dimensional manifolds clarify the intention of our paper and suggest possible applications. At the end, an example of minimum striking time with simulation results is presented. Full article

9 pages, 250 KB

Open AccessArticle

General Relativity with a Positive Cosmological Constant Λ as a Gauge Theory

by Marta Dudek and Janusz Garecki

Axioms 2019, 8(1), 24; https://doi.org/10.3390/axioms8010024 - 21 Feb 2019

Cited by 2 | Viewed by 3581

Abstract

In this paper, we show that the general relativity action (and Lagrangian) in recent Einstein–Palatini formulation is equivalent in four dimensions to the action (and Langrangian) of a gauge field. First, we briefly showcase the Einstein–Palatini (EP) action, and then we present how [...] Read more.

In this paper, we show that the general relativity action (and Lagrangian) in recent Einstein–Palatini formulation is equivalent in four dimensions to the action (and Langrangian) of a gauge field. First, we briefly showcase the Einstein–Palatini (EP) action, and then we present how Einstein fields equations can be derived from it. In the next section, we study Einstein–Palatini action integral for general relativity with a positive cosmological constant

Λ

in terms of the corrected curvature

Ω_{c o r}

. We see that in terms of

Ω_{c o r}

this action takes the form typical for a gauge field. Finally, we give a geometrical interpretation of the corrected curvature

Ω_{c o r}

. Full article

(This article belongs to the Special Issue Applications of Differential Geometry)

Search Results (9)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

Saved Queries

Search Filter Reset All

Years

Feature Papers

Subjects

Journals

Article Types

Countries / Regions

Search Results (9)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI