Principal Bundle Research Articles

A note on holomorphic families of Abelian varieties

Let π : X → Δ \pi :X\to \Delta be a smooth family of complex manifolds. Suppose that X X is Kähler and the fibers X t X_t are biholomorphic to S S for all t ∈ Δ ∗ ≔ Δ ∖ 0 t\in \Delta ^*≔\Delta \setminus 0 , where S S is a fixed complex manifold. We prove that the central fiber X 0 X_0 is biholomorphic to S S when S S is an Abelian variety or a holomorphic principal bundle of Abelian varieties over a smooth curve T T with genus g ( T ) > 1 g(T)>1 .

A Unified View on Geometric Phases and Exceptional Points in Adiabatic Quantum Mechanics

We present a formal geometric framework for the study of adiabatic quantum mechanics for arbitrary finite-dimensional non-degenerate Hamiltonians. This framework generalizes earlier holonomy interpretations of the geometric phase to non-cyclic states appearing for non-Hermitian Hamiltonians. We start with an investigation of the space of non-degenerate operators on a finite-dimensional state space. We then show how the energy bands of a Hamiltonian family form a covering space. Likewise, we show that the eigenrays form a bundle, a generalization of a principal bundle, which admits a natural connection yielding the (generalized) geometric phase. This bundle provides in addition a natural generalization of the quantum geometric tensor and derived tensors, and we show how it can incorporate the non-geometric dynamical phase as well. We finish by demonstrating how the bundle can be recast as a principal bundle, so that both the geometric phases and the permutations of eigenstates can be expressed simultaneously by means of standard holonomy theory.

Einstein gravity as a gauge theory for the conformal group

General relativity in dimension n = p + q can be formulated as a gauge theory for the conformal group , along with an additional field reducing the structure group down to the Poincaré group . In this paper, we propose a new variational principle for Einstein geometry which realizes this fact. Importantly, as opposed to previous treatments in the literature, our action functional gives first order field equations and does not require supplementary constraints on gauge fields, such as torsion-freeness. Our approach is based on the ‘first order formulation’ of conformal tractor geometry. Accordingly, it provides a straightforward variational derivation of the tractor version of the Einstein equation. To achieve this, we review the standard theory of tractor geometry with a gauge theory perspective, defining the tractor bundle a priori in terms of an abstract principal bundle and providing an identification with the standard conformal tractor bundle via a dynamical soldering form. This can also be seen as a generalization of the so called Cartan–Palatini formulation of general relativity in which the ‘internal’ orthogonal group is extended to an appropriate parabolic subgroup of the conformal group.

An 8-periodic exact sequence of Witt groups of Azumaya algebras with involution

Given an Azumaya algebra with involution $$(A,\sigma )$$ over a commutative ring R and some auxiliary data, we construct an 8-periodic chain complex involving the Witt groups of $$(A,\sigma )$$ and other algebras with involution, and prove it is exact when R is semilocal. When R is a field, this recovers an 8-periodic exact sequence of Witt groups of Grenier-Boley and Mahmoudi, which in turn generalizes exact sequences of Parimala–Sridharan–Suresh and Lewis. We apply this result in several ways: We establish the Grothendieck–Serre conjecture on principal homogeneous bundles and the local purity conjecture for certain outer forms of $${\mathbf {GL}}_n$$ and $${\mathbf {Sp}}_{2n}$$ , provided some assumptions on R. We show that a 1-hermitian form over a quadratic étale or quaternion Azumaya algebra over a semilocal ring R is isotropic if and only if its trace (a quadratic form over R) is isotropic, generalizing a result of Jacobson. We also apply it to characterize the kernel of the restriction map $$W(R)\rightarrow W(S)$$ when R is a (non-semilocal) 2-dimensional regular domain and S is a quadratic étale R-algebra, generalizing a theorem of Pfister. In the process, we establish many fundamental results concerning Azumaya algebras with involution and hermitian forms over them.

Principal bundles on two-dimensional CW-complexes with disconnected structure group

AbstractGiven any topological group G, the topological classification of principal G-bundles over a finite CW-complex X is long known to be given by the set of free homotopy classes of maps from X to the corresponding classifying space BG. This classical result has been long-used to provide such classification in terms of explicit characteristic classes. However, even when X has dimension 2, there is a case in which such explicit classification has not been explicitly considered. This is the case where G is a Lie group, whose group of components acts nontrivially on its fundamental group $\pi_1G$ . Here, we deal with this case and obtain the classification, in terms of characteristic classes, of principal G-bundles over a finite CW-complex of dimension 2, with G is a Lie group such that $\pi_0G$ is abelian.

Cardinal-indexed classifying spaces for families of subgroups of any topological group

For G a topological group, existence theorems by Milnor (1956), Gelfand–Fuks (1968), and Segal (1975) of classifying spaces for principal G-bundles are generalized to G-spaces with torsion. Namely, any G-space approximately covered by tubes (a generalization of local trivialization) is the pullback of a universal space indexed by the orbit types of tubes and cardinality of the cover. For G a Lie group, via a metric model we generalize the corresponding uniqueness theorem by Palais (1960) and Bredon (1972) for compact G. Namely, the G-homeomorphism types of proper G-spaces over a metric space correspond to stratified-homotopy classes of orbit classifying maps.The former existence result is enabled by Segal's clever but esoteric use of non-Hausdorff spaces. The latter uniqueness result is enabled by our own development of equivariant ANR theory for noncompact Lie G. Applications include the existence part of classification for unstructured fiber bundles with locally compact Hausdorff fiber and with locally connected base or fiber, as well as for equivariant principal bundles which in certain cases via other models is due to Lashof–May (1986) and to Lück–Uribe (2014). From a categorical perspective, our general model EFκG is a final object inspired by the formulation of the Baum–Connes conjecture (1994).

Lifting spectral triples to noncommutative principal bundles

Given a free action of a compact Lie group G on a unital C⁎-algebra A and a spectral triple on the corresponding fixed point algebra AG, we present a systematic and in-depth construction of a spectral triple on A that is build upon the geometry of AG and G. We compare our construction with a selection of established examples.

A New Bundle Picture for the Drone

A new <inline-formula><tex-math notation="LaTeX">$\mathrm{S}^{1}$</tex-math></inline-formula> principal bundle over a vector space with the origin removed is constructed for the drone with the yaw dynamics evolving on the <inline-formula><tex-math notation="LaTeX">$\mathrm{S}^{1}$</tex-math></inline-formula> fiber and the remaining part of drone dynamics on the base space. Two local trivializations are constructed to cover the whole bundle such that on each trivialization the drone dynamics is transformed via dynamic feedback to a linear controllable system with the yaw dynamics decoupled from the rest of the drone dynamics to make easy the design of drone trajectories and controls. The manifold <inline-formula><tex-math notation="LaTeX">$\mathrm{S}^{2}$</tex-math></inline-formula> often used for thrust directions is made obsolete by the new bundle.

Deformation theory of holomorphic Cartan geometries, II

Abstract In this continuation of [4], we investigate the deformations of holomorphic Cartan geometries where the underlying complex manifold is allowed to move. The space of infinitesimal deformations of a flat holomorphic Cartan geometry is computed. We show that the natural forgetful map, from the infinitesimal deformations of a flat holomorphic Cartan geometry to the infinitesimal deformations of the underlying flat principal bundle on the topological manifold, is an isomorphism.

Geometric Structure of the Set of Pairs of Matrices under Simultaneous Similarity

In this paper pairs of matrices under similarity are considered because of their scientific applications, especially pairs of matrices being simultaneously diagonalizable. For example, a problem in quantum mechanics is the position and momentum operators, because they do not have a shared base representing the system's states. They do not commute, and that is why switching operators form a crucial element in quantum physics. A study of the set of linear operators consisting of pairs of simultaneously diagonalizable matrices is done using geometric constructions such as the principal bundles. The main goal of this work is to construct connections that allow us to establish a relationship between the local geometry around a point with the local geometry around another point. The connections give us a way to help distinguish bundle sections along tangent vectors.

Bigerbes

The bigerbes introduced here give a refinement of the notion of 2-gerbes, representing degree four integral cohomology classes of a space. Defined in terms of bisimplicial line bundles, bigerbes have a symmetry with respect to which they form 'bundle 2-gerbes' in two ways; this structure replaces higher associativity conditions. We provide natural examples, including a Brylinski-McLaughlin bigerbe associated to a principal G-bundle for a simply connected simple Lie group. This represents the first Pontryagin class of the bundle, and is the obstruction to the lifting problem on the associated principal bundle over the loop space to the structure group consisting of a central extension of the loop group; in particular, trivializations of this bigerbe for a spin manifold are in bijection with string structures on the original manifold. Other natural examples represent 'decomposable' 4-classes arising as cup products, a universal bigerbe on K(Z,4) involving its based double loop space, and the representation of any 4-class on a space by a bigerbe involving its free double loop space. The generalization to 'multigerbes' of arbitrary degree is also described.

A note on induced connections

In this note, we will exploit the classical bijective correspondence between sections of an associated vector bundle and equivariant functions on the underlying principal bundle to revisit a global formula for induced connections on associated vector bundles. Consequently, we give the expression of the curvature in terms of the curvature 2-form of a connection on a principal bundle.

Non-unimodular transversely homogeneous foliations

We give sufficient conditions for the tautness of a transversely homogenous foliation defined on a compact manifold, by computing its base-like cohomology. As an application, we prove that if the foliation is non-unimodular then either the ambient manifold, the closure of the leaves or the total space of an associated principal bundle fiber over $S^1$.

Numerical flatness and principal bundles on Fujiki manifolds

Let M be a compact connected Fujiki manifold, G a semisimple affine algebraic group over C with one simple factor and P a fixed proper parabolic subgroup of G. For a holomorphic principal G–bundle EG over M, let EP be the holomorphic principal P–bundle EG⟶EG/P given by the quotient map. We prove that the following three statements are equivalent: (1) ad(EG) is numerically flat, (2) the holomorphic line bundle ⋀topad(EP)⁎ is nef, and (3) for every reduced irreducible compact complex analytic space Z with a Kähler form ω, holomorphic map γ:Z⟶M, and holomorphic reduction of structure group EP⊂γ⁎EG to P, the inequality degree(ad(EP))≤0 holds.

On uniform flag bundles on Fano manifolds

As a natural extension of the theory of uniform vector bundles on Fano manifolds, we consider uniform principal bundles, and study them by means of the associated flag bundles, as their natural projective geometric realizations. In this paper, we develop the necessary background and prove some theorems that are flag bundle counterparts of some of the central results in the theory of uniform vector bundles.

Discrete connections on principal bundles: The discrete Atiyah sequence

In this work we study discrete analogues of an exact sequence of vector bundles introduced by M. Atiyah in 1957, associated to any smooth principal G-bundle π:Q→Q/G. In the original setting, the splittings of the exact sequence correspond to connections on the principal bundle π. The discrete analogues that we consider here can be studied in two different categories: the category of fiber bundles with a (chosen) section, Fbs, and the category of local Lie groupoids, lLGpd. In Fbs we find a correspondence between a) (semi-local) splittings of the discrete Atiyah sequence (DAS) of π, b) discrete connections on the same bundle π, and c) isomorphisms of the DAS with certain fiber product extensions in Fbs. We see that the right splittings of the DAS (in Fbs) are not necessarily right splittings in lLGpd: we use this obstruction to define the discrete curvature of a discrete connection. Then, there is a correspondence between the right splittings of the DAS in lLGpd and discrete connections with trivial discrete curvature, that is, flat discrete connections. We also introduce a semidirect product between (some) local Lie groupoids and prove that there is a correspondence between semidirect product extensions and right splittings of the DAS in lLGpd.

Smooth loops and loop bundles

A loop is a rather general algebraic structure that has an identity element and division, but is not necessarily associative. Smooth loops are a direct generalization of Lie groups. A key example of a non-Lie smooth loop is the loop of unit octonions. In this paper, we study properties of smooth loops and their associated tangent algebras, including a loop analog of the Maurer-Cartan equation. Then, given a manifold, we introduce a loop bundle as an associated bundle to a particular principal bundle. Given a connection on the principal bundle, we define the torsion of a loop bundle structure and show how it relates to the curvature, and also develop aspects of a non-associative gauge theory. Throughout, we see how some of the known properties of G2-structures can be seen from this more general setting.

The Picard group of the universal moduli stack of principal bundles on pointed smooth curves II

The Picard group of the universal moduli stack of principal bundles on pointed smooth curves II

Gauge groups and bialgebroids

We study the Ehresmann–Schauenburg bialgebroid of a noncommutative principal bundle as a quantization of the gauge groupoid of a classical principal bundle. We show that the gauge group of the noncommutative bundle is isomorphic to the group of bisections of the bialgebroid, and we give a crossed module structure for the bisections and the automorphisms of the bialgebroid. Examples include: Galois objects of Taft algebras, a monopole bundle over a quantum sphere and a not faithfully flat Hopf–Galois extension of commutative algebras. For each of the latter two examples, there is in fact a suitable invertible antipode for the bialgebroid making it a Hopf algebroid.

Gauge Theory on Noncommutative Riemannian Principal Bundles

We present a new, general approach to gauge theory on principal $G$-spectral triples, where $G$ is a compact connected Lie group. We introduce a notion of vertical Riemannian geometry for $G$-$C^\ast$-algebras and prove that the resulting noncommutative orbitwise family of Kostant's cubic Dirac operators defines a natural unbounded $KK^G$-cycle in the case of a principal $G$-action. Then, we introduce a notion of principal $G$-spectral triple and prove, in particular, that any such spectral triple admits a canonical factorisation in unbounded $KK^G$-theory with respect to such a cycle: up to a remainder, the total geometry is the twisting of the basic geometry by a noncommutative superconnection encoding the vertical geometry and underlying principal connection. Using these notions, we formulate an approach to gauge theory that explicitly generalises the classical case up to a groupoid cocycle and is compatible in general with this factorisation; in the unital case, it correctly yields a real affine space of noncommutative principal connections with affine gauge action. Our definitions cover all locally compact classical principal $G$-bundles and are compatible with $\theta$-deformation; in particular, they cover the $\theta$-deformed quaternionic Hopf fibration $C^\infty(S^7_\theta) \hookleftarrow C^\infty(S^4_\theta)$ as a noncommutative principal $\operatorname{SU}(2)$-bundle.