Alexander I Nesterov
A nonassociative generalization of the principal fiber bundles with a smooth loop mapping on the fiber is presented. Our approach allows us to construct a new kind of gauge theories that involve higher ''nonassociative'' symmetries.
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We construct manifestly superconformal field theories in six dimensions which contain a non-Abelian tensor multiplet. In particular, we show how principal 3-bundles over a suitable twistor space encode solutions to these self-dual tensor field theories via a Penrose-Ward transform. The resulting higher or categorified gauge theories significantly generalise those obtained previously from principal 2-bundles in that the so-called Peiffer identity is relaxed in a systematic fashion. This transform also exposes various unexplored structures of higher gauge theories modelled on principal 3-bundles…
In this paper, we initiate the study of a parametrised version of Rieffel's strict deformation quantization. We apply it to give a classification of noncommutative principal torus bundles, in terms of parametrised strict deformation quantization of ordinary principal torus bundles. The paper also contains a putative definition of noncommutative non-principal torus bundles.
Let $π:P\to M^n$ be a principal G-bundle, and let ${\mathcal{L}}: J^1P \toΛ^n(M)$ be a G-invariant Lagrangian density. We obtain the Euler-Poincare equations for the reduced Lagrangian l defined on ${\mathcal C}(P)$, the bundle of connections on P.
On n-tuple principal bundles
该研究从李群主作用的相容性条件出发，定义并发展了二重乃至更一般的n重主丛的数学概念。
We define double principal bundles (DPBs), for which the frame bundle of a double vector bundle, double Lie groups and double homogeneous spaces are basic examples. It is shown that a double vector bundle can be realized as the associated bundle of its frame bundle. Also dual structures, gauge transformations and connections in DPBs are investigated.
Let $P$ be a parabolic subgroup of a connected simply connected complex semisimple Lie group $G$. Given a compact Kähler manifold $X$, the dimensional reduction of $G$-equivariant holomorphic vector bundles over $X\times G/P$ was carried out by the first and third authors. This raises the question of dimensional reduction of holomorphic principal bundles over $X\times G/P$. The method used for equivariant vector bundles does not generalize to principal bundles. In this paper, we adapt to equivariant principal bundles the Tannakian approach of Nori, to describe the dimensional reduction of $G$-…
We introduce the concepts of a multisymplectic structure and a polysymplectic structure on a general fiber bundle over a general base manifold, define the concept of the symbol of a multisymplectic form, which is a polysymplectic form representing its leading order contribution, and prove Darboux theorems for the existence of canonical local coordinates.
In this paper we introduce a notion of parallel transport for principal bundles with connections over differentiable stacks. We show that principal bundles with connections over stacks can be recovered from their parallel transport thereby extending the results of Barrett, Caetano and Picken, and Schreiber and Waldof from manifolds to stacks. In the process of proving our main result we simplify Schreiber and Waldorf's definition of a transport functor for principal bundles with connections over manifolds and provide a more direct proof of the correspondence between principal bundles with conn…
This work revisits, from a geometric perspective, the notion of discrete connection on a principal bundle, introduced by M. Leok, J. Marsden and A. Weinstein. It provides precise definitions of discrete connection, discrete connection form and discrete horizontal lift and studies some of their basic properties and relationships. An existence result for discrete connections on principal bundles equipped with appropriate Riemannian metrics is proved.
A constructive approach to generalized principal connections
本文通过构造性方法研究广义主丛与广义主联络，证明李群纤维丛是广义主丛，给出其联络的刻画，复现相关已知结果，为广义规范理论研究提供基础。
On GKM fiber bundles and realizability with full flag fibers
本文研究GKM流形纤维丛诱导GKM图纤维丛的条件，给出反例，并针对全旗流形纤维的带符号GKM纤维丛，分类扭转自同构以判定可实现性。
We study the Teichmüller space of negatively curved metrics on a high dimensional manifold, with applications to bundles with negatively curved fibers.
We define contact fiber bundles and investigate conditions for the existence of contact structures on the total space of such a bundle. The results are analogous to minimal coupling in symplectic geometry. The two applications are construction of K-contact manifolds generalizing Yamazaki's fiber join construction and a cross-section theorem for contact moment maps
We prove that all smooth sphere bundles that admit fiberwise 1/4-pinched metrics are induced bundles of vector bundles, so their structure groups reduce from the diffeomorphism group of the sphere to the orthogonal group. This result implies the existence of many smooth n-sphere bundles over a k-sphere that do not support strictly 1/4-pinched positively curved Riemannian metrics on their fibers.
The moduli space of holomorphic fiber bundles ${\cal M}_n(\Si)$ over a compact Riemann surface $\Si$ is considered. A formula for the regularised determinant and an other for the symplectic form at trivial bundle are proposed.
Let $P(M,G)$ be a principal fiber bundle, let $ω$ be a connection form on $P(M,G)$, and consider a projectable connection $\nabla^{P}$ on $P(M,G)$. The aim of this work is to determine the $\nabla^{P}$-martingales in $P(M,G)$. Our results allow establishing new characterizations of harmonic maps from Riemannian manifolds to principal fiber bundles.
We give sufficient conditions for the existence of a Dirac structure on the total space of a Poisson fiber bundle endowed with a compatible connection. We also show that Cartan and Cartan-Hannay-Berry connections give rise to coupling Dirac structures.
We provide very general symmetrization theorems in arbitrary dimension and codimension, in products, warped products, and certain fiber bundles such as lens spaces, including Steiner, Schwarz, and spherical symmetrization and admitting density.
The equivalence of principal bundles with transitive Lie groupoids due to Ehresmann is a well known result. A remarkable generalisation of this equivalence, due to Mackenzie, is the equivalence of principal bundle extensions with those transitive Lie groupoids over the total space of a principal bundle, which also admit an action of the structure group by automorphisms. This paper proves the existence of suitably equivariant transition functions for such groupoids, generalising consequently the classification of principal bundles by means of their transition functions, to extensions of princip…
In our previous papers [Far East Journal of Mathematical Sciences, 35 (2009), 211-223] and [International Journal of Pure and Applied Mathematics, 60 (2010), 15-24] we have developed the theory of Weil prolongation, Weil exponentiability and microlinearity for Frolicher spaces. In this paper we will relativize it so as to obtain the theory of fiber bundles for Frölicher spaces. It is shown that any Weil functor naturally gives rise to a fiber bundle. We will see that the category of fiber bundles over a fixed Frölicher space M and their smooth mappings over M is cartesian closed. We will see a…
Let $G$ be a connected complex Lie group and $Γ\subset G$ a cocompact lattice. Let $H$ be a complex Lie group. We prove that a holomorphic principal $H$-bundle $E_H$ over $G/Γ$ admits a holomorphic connection if and only if $E_H$ is invariant. If $G$ is simply connected, we show that a holomorphic principal $H$-bundle $E_H$ over $G/Γ$ admits a flat holomorphic connection if and only if $E_H$ is homogeneous.
In this article we consider the continuity of the eigenvalues of the connection Laplacian of $G$-connections on vector bundles over Riemannian manifolds. To show it, we introduce the notion of the asymptotically $G$-equivariant measured Gromov-Hausdorff topology on the space of metric measure spaces with isometric $G$-actions, and apply it to the total spaces of principal $G$-bundles equipped with $G$-connections over Riemannian manifolds.
Relative trisections of fiber bundles over the circle
本文研究定向四维圆上纤维丛的三剖分构造，提出从纤维的缝合 Heegaard 分裂出发，通过算法显式构造相对三剖分图，并利用图的粘合恢复闭纤维丛和旋流形的三剖分，同时给出四维开书分解的三剖分。
We obtain an explicit formula for the symplectic form over the double quotient with help of the Green function of a Riemann surface.
In this work, generalized principal bundles modelled by Lie group bundle actions are investigated. In particular, the definition of equivariant connections in these bundles, associated to Lie group bundle connections, is provided, together with the analysis of their existence and their main properties. The final part gives some examples. In particular, since this research was initially originated by some problems on geometric reduction of gauge field theories, we revisit the classical Utiyama Theorem from the perspective investigated in the article.
Bundles with even-dimensional spherical space form as fibers and fiberwise quarter pinched Riemannian metrics
本文证明：若光滑丛的每个纤维（实射影空间）带有连续变化的正曲率逐点强1/4夹挤度量，则丛的结构群可约化为标准圆度量的等距群。
In this paper, we introduce the concept of principal bundles on statistical manifolds. After necessary preliminaries on information geometry and principal bundles on manifolds, we study the $α$-structure of frame bundles over statistical manifolds with respect to $α$-connections, by giving geometric structures. The manifold of one-dimensional normal distributions appears in the end as an application and a concrete example.
We prove a stability result for volume forms on fiber bundles with compact base and noncompact fibers. This generalizes the classical results of Moser and Greene--Shiohama, and recent work by the authors.
In this paper, we give concrete descriptions of leafwise cohomology groups and show the regularized determinant expression of the dynamical zeta function for fiber bundles over $S^{1}$. As applications, we show a functional equation and some formulas for special values of the dynamical zeta function.
This work is devoted to new constructions of symplectically fat fiber bundles. The latter are constructed in two ways: using the Kirwan map and expressing the fatness condition in terms of the isotropy representation related to the G-structure over some homogeneous spaces.
The aim of this article is to proof a necessary and sufficient condition for the existence of a Cartan connection on a principal bundle. After collecting the essentially well known facts to fix the terminology, soldering forms and geometrizable principal bundles are defined to finally prove the existence criterion.
A principal Higgs bundle $(P,ϕ)$ over a singular curve $X$ is a pair consisting of a principal bundle $P$ and a morphism $ϕ:X\to\text{Ad}P \otimes Ω^1_X$. We construct the moduli space of principal Higgs G-bundles over an irreducible singular curve $X$ using the theory of decorated vector bundles. More precisely, given a faithful representation $ρ:G\to Sl(V)$ of $G$, we consider principal Higgs bundles as triples $(E,q,ϕ)$ where $E$ is a vector bundle with $\rk{E}=\dim V$ over the normalization $\xtilde$ of $X$, $q$ is a parabolic structure on $E$ and $ϕ:E\ab{}\to L$ is a morphism of bundles, …
A new construction of a universal connection was given in \cite{BHS}. The main aim here is to explain this construction. A theorem of Atiyah and Weil says that a holomorphic vector bundle $E$ over a compact Riemann surface admits a holomorphic connection if and only if the degree of every direct summand of $E$ is degree. In \cite{AB}, this criterion was generalized to principal bundles on compact Riemann surfaces. This criterion for principal bundles is also explained.
Diffeological Principal Bundles and Principal Infinity Bundles
本文研究微分学空间的∞-层上同调，对比现有Čech上同调概念，并证明微分学主G-丛与主∞-丛范畴神经弱同伦等价，沟通微分学丛理论与高阶拓扑斯理论。
For a Riemann surface $X$ and the moduli of regularly stable $G$-bundles $M$, there is a naturally occuring "$adjoint$" vector bundle over $X \times M$. One can take the determinant of this vector bundle with respect to the projection map onto $M$. Our aim here is to study the curvature of the determinant bundle as the conformal structure on $X$ varies over the Teichmüller space.
In this paper, we prove that principal circle bundles over the complex projective space equipped with the standard Sasakian structures are volume rigid among all $K$-contact manifolds satisfying positivity conditions of tensors involing the Tanaka-Webster curvature.
We construct new examples of manifolds with cyclic-parallel Ricci tensor, so called A-manifolds, on a r-torus bundle over a product of almost Hodge A-manifolds.
Let $X$ be a compact connected Kähler manifold equipped with an anti-holomorphic involution which is compatible with the Kähler structure. Let $G$ be a connected complex reductive affine algebraic group equipped with a real form $σ_G$. We define pseudo-real principal $G$--bundles on $X$; these are generalizations of real algebraic principal $G$--bundles over a real algebraic variety. Next we define stable, semistable and polystable pseudo-real principal $G$--bundles. Their relationships with the usual stable, semistable and polystable principal $G$--bundles are investigated. We then prove that…
We give a complete characterization of invariant integrable complex structures on principal bundles defined over hermitian symmetric spaces, using the Jordan algebraic approach for the curvature computations. In view of possible generalizations, the general setup of invariant holomorphic principal fibre bundles is described in a systematic way.
We construct and study general connections on Lie groupoids and differentiable stacks as well as on principal bundles over them using Atiyah sequences associated to transversal tangential distributions.
Let $H$ be a semisimple algebraic group. We prove the semistable reduction theorem for $μ$--semistable principal $H$--bundles over a {\it smooth projective variety $X$} defined over the field $\bc$. When $X$ is a {\it smooth projective surface} and $H$ is simple, we construct the algebro--geometric Donaldson--Uhlenbeck compactification of the moduli space of $μ$--semistable principal $H$--bundles with fixed characteristic classes and describe its points. For large characteristic classes we show that the moduli space of $μ$--stable principal $H$--bundles is non--empty.
Frames of group-sets and their application in bundle theory
本文研究纤维为自由G-空间而非群的半主丛，通过定义G-集的基并利用圈积对称群构造标架丛，以函子形式将半主丛收缩为主丛，并保持平行移动性质。
In this note we show that the holonomies for higher local systems provided by the higher Riemann-Hilbert correspondence coincide with those associated to principal $2$-bundles where both formalisms meet.
Using a model for the bundle $\hat{\mathcal F}^2M$ of semi-holonomic second order frames of a manifold $M$ as an extension of the bundle ${\mathcal F}^2M$ of holonomic second order frames of $M$, we introduce in $\hat{\mathcal F}^2M$ a principal bundle structure over ${\mathcal F}^2M$, the structure group being the additive group $A_2(n)$ of skew-symmetric bilinear maps from ${\Bbb R}^2 \times {\Bbb R}^n$ into ${\Bbb R}^n$. The composition of the projection of that structure with the existing projection of the bundle $\tilde{\mathcal F}^2M$ of non-holonomic second order frames of $M$ over $\ha…
Fiber Bundle Networks: A Geometric Machine Learning Paradigm
纤维束网络通过几何优化提升机器学习的可解释性与效率，结合黎曼度量与变分原型优化实现分类任务的清晰几何解释。
Generalized Killing spinors as Lagrangian sections of principal spin bundles
本文建立了三维伪黎曼旋量流形上ε-单位广义Killing旋量与其主旋量丛上Lagrangian截面的一一对应关系。
Intermediate Pseudoconvexity of Fiber Bundles
研究亏格≥2的紧黎曼曲面上的局部平凡全纯球丛的拟凸性及其在相伴射影空间丛中的补集的中拟凸性，证明球丛是1-凸的而补集是n-凸的。
We show a relationship between Chern-Simons 1- and 3-forms and harmonic forms on a principal bundle. Doing so requires one to consider an adiabatic limit. For the 3-form case, assume that G is simple and the corresponding Chern-Weil 4-form is exact. Then, the Chern-Simons 3-form on the princpal bundle G-bundle, minus a canonical term from the base, is harmonic in the adiabatic limit.
Let $P\to M$ be a principal bundle. Consider a sequence of metrics on $P$ obtained by re-scaling the fibers to points. The Gromov-Hausdorff limit of the tangent bundles over these principal bundles with their Sasaki metric is seen herein to be a locally trivial fiber bundle containing the tangent space to the base as a subbundle in a natural way. Berger 3-spheres provide an example where the limit fibers are still of dimension 3. The fibers are shown to be entirely determined by the riemannian holonomy of the chosen bi-invariant metric.
We consider groupoids in the category of principal bundles, which we call principal bundles (PB) groupoids. Inspired by work by Th. Nikolaus and K. Waldorf, we generalise bundle gerbes over manifolds to bundle gerbes over groupoids and discuss a functorial correspondence between PB groupoids and bundle gerbes over groupoids. From a PB groupoid over a fibre product groupoid, we build a bundle gerbe over another fibre product groupoid. Conversely, from a bundle gerbe over a Lie groupoid, we build a PB groupoid. It has a trivial base and from any PB groupoid with trivial base, we build a bundle g…
Let K be a a Lie group, modeled on a locally convex space, and M a finite-dimensional paracompact manifold with corners. We show that each continuous principal K-bundle over M is continuously equivalent to a smooth one and that two smooth principal K-bundles over M which are continuously equivalent are also smoothly equivalent. In the concluding section, we relate our results to neighboring topics.
We construct a new example of an A-manifold, i.e. a Riemannian manifold with a cyclic-parallel Ricci tensor, which can be viewed as a generalization of the Einstein condition. The underlying manifold for our construction is a principal torus bundle over Kähler-Einstein manifold a with fibre a torus of arbitrary dimension.
We prove that the Teichmüller space $\mathcal{T}^{<0}(M)$ of negatively curved metrics on a hyperbolic manifold $M$ has nontrivial $i$-th rational homotopy groups for some $i> \dim M$. Moreover, some elements of infinite order in $π_i B\mbox{Diff}(M)$ can be represented by bundles over $S^i$ with fiberwise negatively curved metrics.
We present a derivation-based Atiyah sequence for noncommutative principal bundles. Along the way we treat the problem of deciding when a given $^*$-automorphism on the quantum base space lifts to a $^*$-automorphism on the quantum total space that commutes with the underlying structure group.
Holomorphic principal G-bundles over a complex manifold M can be studied using non-abelian cohomology groups H^1(M,G). On the other hand, if M=Σis a closed Riemann surface, there is a correspondence between holomorphic principal G-bundles over Σand coadjoint orbits in the dual of a central extension of the Lie algebra C^\infty(Σ, \g). We review these results and provide the details of an integrability condition for almost complex structures on smoothly trivial bundles. This article is a shortened version of the author's Diplom thesis.
For elliptic principal bundles $π:X\ra B$ over Kähler manifolds it was shown by Blanchard that $X$ has a Kähler metric if and only both Chern classes (with real coefficients) of $π$ vanish. For some elliptic principal bundles, when the span of these Chern classes is 1-dimensional, it was shown by Vaisman that $X$ carry locally conformally Kähler (LCK, for short) metrics. We show that in the case when the Chern classes are linearly independent, $X$ carries no LCK metric.
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