# Questions tagged [principal-bundles]

A principal $G$-bundle, where $G$ denotes any topological group, is a fiber bundle $\pi :P → X$ together with a continuous right action $P × G → P$ such that $G$ preserves the fibers of $P$ and acts freely and transitively on them.

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### Is every covering space associated to a "principal" covering space? Can they be classified?

Every fiber bundle is associated to a principal bundle with the same structure group. A covering space is just a fiber bundle with a discrete fiber, and hence a structure group $G$ that is a (subgroup ...

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### I'm looking for the NLab page on particle species

This is just a reference request.
I came across an NLab page on particle species described as certain vector bundles. But I can't seem to find it again when I searched recently.
If someone can point ...

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### Principal circle bundles over punctured $3$-sphere

Let $M$ be $S^3$ with $k$ disjoint open balls $D^3$ removed.
Can we classify all principal circle bundles over $M$ such that the total space is simply-connected?

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### How is the $k$-times iterative frame bundle $FF\cdots FM$ associated to the higher order frame bundle $F^k M$?

$\DeclareMathOperator\Gl{Gl}$As I understand it a natural bundle is one for which a diffeomorphism on the base space lifts to an automorphism on the total space of the bundle. It is my understanding ...

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### Connection of principal fiber bundles — history

I wonder who was the first to discover the notion of principal fiber bundle and its connection (gauge field in the physical language). Wikipedia cites the book by Steenrod (1951). But was he the ...

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### Classification of bundles with fixed total space

I am aware of classification theorems for principal bundles, vector bundles, and covering spaces $\pi:E\to B$ over a fixed base space $B$. Principal and vector bundles over $B$ are classified by ...

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### Local triviality of torsors for relative reductive groups

Let $X \to S$ be a relative (smooth proper) curve, and $G \to X$ a reductive group scheme. The following two results are well-known:
(Drinfeld-Simpson) For arbitrary $S$, if $G$ is defined over $S$, ...

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### Are two homotopic principal bundles isomorphic?

Let $E_1 \to B$ and $E_2 \to B$ be two principal $G$-bundles, where $E_1$ and $E_2$ are two simply-connected manifolds and $G$ is a compact Lie group.
Suppose there exists a $G$-equivariant continuous ...

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### Are there geometric $\mathbb{G}_a$-quotients with trivial stabilizers, not being principal bundles?

Consider algebraic $\mathbb{C}$-schemes. The group scheme $\mathbb{G}_a$ is the scheme $\mathbb{A}^1$ with the addition. This is not a reductive group. Here I want to know some examples of $\mathbb{G}...

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### Nonholonomic version of bijection between $r$th order connection on $TM$ and principal connection on $r$th order frame bundle $P^r M$?

Given a smooth manifold $M$, Kolář - On the torsion of linear higher order connections showed that there is an equivalence between a linear, $r$-th order connection the tangent bundle $TM$ of $M$, and ...

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### Direct image of associated line bundle (on an associated fiber bundle) is associated vector bundle

Let $k$ be an algebraically closed field of characteristic $0$, $G/k$ be a linear algebraic group and $C/k$ be a curve. Let $F$ on $C$ be a $G$-bundle, $X/k$ be a projective $G$-variety and $L$ on $X$ ...

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### Orthogonal bundles with values in a line bundle vs. reductions of structure group to $O(n)$

I have already posted this question in math.stackexchange here, but didn't get any response, so I'm posting my question here as well.
Let $X$ be a smooth projective variety over $\mathbb{C}$, and $\pi:...

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### Lifting action of torus to torus bundle

Preamble: Let $X$ be a simply connected smooth manifold and $P \to X$ be a principal $T^\ell$ bundle on it.
Let $\phi$ be a smooth action of $T^k$ on $X$.
The paper "Lifting compact group actions ...

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### Homogeneous metric connections on 3-dimensional Lie groups

Let $G$ be a 3-dimensional unimodular Lie group equipped with a left-invariant metric $q$. Call $P_{SO}$ its oriented orthonormal frame bundle.
Considering the moduli space of connections $\mathscr{B}$...

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### Bianchi's identity in a principal bundle

Let us consider a principal bundle $P$, with a Lie-algebra-valued connection one-form $\omega\in\mathfrak{g}\otimes\Omega^1(P)$ and a Lie-algebra-valued curvature two-form $\Omega\in\mathfrak{g}\...

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### G-local systems via the classifying stack BG

First, let $BG$ be the classifying stack of a Lie group $G$ in either Top or Diff (compactly generated topological spaces or differentiable manifolds). A map $f: X \to BG$ determines a principal $G$-...

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### Invariants associated to a principal bundle whose total space is a symplectic manifold acted symplectically by group structure

The following question - proposal came to my mind about 4 years ago but I did not find any solution to this question and did not find any answer via e-personal comunication with some ...

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### Spin connection vs. Cartan connection

I am studying the tetradic Palatini formalism of general relativity. In this formalism, one usually considers a manifold $M$, which is either non-compact or compact with Euler-characteristic $\chi(M)=...

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### Non-invertible map between principal bundles only locally trivial in a supercanonical Grothendieck topology?

It is a truth universally acknowledged that a map between principal bundles must be in want of an inverse.
However, the construction of said inverse in the context of a more general site $(S,J)$ ...

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### Learning roadmap for holonomy theory

During my Master's thesis I encountered the theory of holonomy for the first time. Unluckily it was only tangentially related to the topic of my thesis, so I couldn't dive into it.
The book I was ...

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### What does homotopy invariance mean for twisted K-theory?

In ordinary K-theory, homotopy invariance means that if $f,g \colon X \to Y$ are homotopic maps then their induced maps on K-theory are equal: $f^* = g^* \colon K(Y) \to K(X)$.
My question is how to ...

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### Frame bundle of $\mathbb{C}P^n$ as homogeneous space

I am reading "Dirac Operator in Riemannian Geometry" by T. Friedrich, where he writes that (the total space of) the frame bundle $R$ of the tangent space of $\mathbb{C}P^n$ is:
$$ R = SU(n+1)...

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### Confusion about Turaev's description of G-bundles on the cylinder and pairs of pants

In Homotopy Field Theory in dimension 2 and group algebras, section 4.6, page 24, Turaev considers an annulus $C = S^1 \times [0,1]$ (thought of as a cobordism from $C_0 = S^1 \times \{ 0\}$ to $C_1 = ...

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### Holonomy as integration of curvature for principal $G$-bundles?

Holonomy and curvature may seem to be slightly advanced topics in
geometry. However, their origins are easily imaginable. Namely,
picture the surface of earth $S$, and pick an arbitrary
contractible ...

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### Reference request: Weil's uniformization theorem

The Weil uniformization theorem says that if $k$ is an algebraically closed field, $G$ a reductive group, and $C$ a curve, we have an isomorphism of stacks $Bun_G(C)\cong G(F_C)\backslash G(\mathbb{A}...

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### Existence (or non existence) of principal bundle charts compatible with an $f$-reduction

I asked this question on math stack exchange here, but I wonder if it would be better received here.
Let $\pi:P\rightarrow M$ and $\pi':P'\rightarrow M$ be principal $G$ and $H$ bundles respectively, ...

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### References on principal $\mathbf{C}$-bundles, where $\mathbf{C}$ is a category?

Is there any treatment on principal "categorical" bundles - principal $\mathbf{C}$-bundles, where $\mathbf{C}$ is some (topological) category?
I know that one can define "categorical ...

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### Construct a 'nice' trivializing cover of universal principal $G$-bundle $EG \to BG$

Let G be a discrete or say for sake of simplicity a finite group. In Hatcher's book Algebraic Topology on p 89 the construction of universal bundle $EG$ carries structure of a $\Delta$-complex whose $...

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### Physical intuition for curvature on higher order frame bundles?

$\DeclareMathOperator\SO{SO}$A priori: I apologize if this isn't up to Mathoverflow standards, I've had very little luck getting questions on this subject answered elsewhere.
I'm looking for a physics ...

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### Completion of the classifying stack $BG$ at a point

With the classifying stack $BG$ I have come across "the formal completion of $BG$ at point", which is denoted $\widehat{BG}$, for instance on page 7 of https://arxiv.org/pdf/1703.08578.pdf, ...

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### Given a Lie $2$-group $G$ does every principal $G$ $2$-bundle admit a $2$-connection?

The statement is true for Lie groups and principal bundles, with every principal bundle admitting a connection and I see no reason for the analogue result not to hold in the Lie $2$-group case but I ...

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### references for holomorphic principal bundles (over complex manifolds)

principal bundles in differential geometry is a classical notion and there are so many references that discuss these notion (even in text books). But, when it comes to its version in complex geometry, ...

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### Concrete descriptions of $S^1$-bundles over smooth manifold $Y$ underying a K3 surface

Let $Y$ be the smooth manifold underlying a K3 surface. As a manifold, $Y$ is diffeomorphic to $\{[x_0:x_1:x_2:x_3]\in\mathbb{C}P^3\colon X_0^4+x_1^4+X_2^4+X_3^4=1\}$. It is well known that $H^2(Y,\...

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### Fiber bundle orientability vs manifold orientability

This question seems like a pretty straightforward generalization of a result from vector bundles but its been on MSE for over a week with no answers so I'm reposting https://math.stackexchange.com/...

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### Moduli space of flat connection over homology 3-sphere

I'm trying to understand the space of flat connections of the trivial $\mathrm{SU}(2)$-bundle over a closed, oriented homology three-sphere (for the purpose of understanding the instanton Floer ...

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### Different definitions of "charged spinors": "bundle splicing" vs. "twisted spinor bundles"

Currently I study the mathematical formulation of the (classical) standard model of particle physics using the language of gauge theory and spin geometry. One of the central objects in the standard ...

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### On the "integrality condition" of the bilinear form in the Chern-Simons action

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ and let $\pi:P\to\mathcal{M}$ be a principal $G$-bundle over a smooth orientable manifold $\mathcal{M}$. Furthermore, let $\langle\cdot,\cdot\...

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### Non existence of preferred Horizontal subspace on a bundle [closed]

If I choose a principal bundle, let us say $G\rightarrow P \rightarrow B$, with $G=U(1)$, $P=S^1 \times S^1$ and $B=S^1$. Can I follow the identity element of the group over a curve at the base. How ...

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### Do classifying spaces determine categories of principal bundles?

If $X$ is a topological space, $G$ a topological group and $E G \to BG$ a universal bundle, isomorphism classes of numerable principal $G$-bundles over $X$ are in one-to-one correspondence with ...

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### The contravariant mapping space represented by a homotopical classifying space (e.g. BG)

In classical homotopy theory, there are a number of spaces which are important because they represent an interesting functor on $\operatorname{Ho(Top)}$; for example, $K(G,n)$ represents singular ...

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### Principal bundles with no trivializable extensions

Let $Q \to M$ be a principal $G$-bundle. Given a homomorphism $\phi: G \to H$, we can ‘extend the structure group’ of $Q$ to $H$, by defining an associated principal $H$-bundle: $Q_{H} := (Q \times H)/...

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### Classifying space of bundles over bundles

Consider a sufficiently nice topological space $X$ as well as topological groups $G$ and $H$. Consider the functor $F$ that associates to $X$ the set of all isomorphism classes of all principal $H$-...

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### Principal bundles from a fibration of homogeneous spaces

Let $G$ be a compact (Lie) group, and $H \subseteq H'$ two compact (Lie) subgroups. It is clear that we have an obvious surjective map of homogeneous spaces
$$
G/H \twoheadrightarrow G/H'.
$$
Will it ...

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### Change of two normal coordinates based on two nearby points?

Let $M$ be a manifold and $L(M)$ be the tangent frame bundle on $M$. Let $\Gamma$ be a linear connection on $L(M)$ which induces a covariant derivative $\nabla$ on $TM$.
Let $p, q$ be two ...

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### Internal principal $G$-bundles

Let $(C, J)$ be a small site and let $\mathsf{Sh}_{(2, 1)}(C, J)$ be the $(2, 1)$-sheaf topos of sheaves of (small) groupoids on $(C, J)$. Let $G$ be a sheaf of groups on $(C, J)$, and let $\mathbf{...

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### Cartan geometry: jet space perspective on the tractor bundle

Let $G$ a Lie group and $H\subset G$ a Lie subgroup. For simplicity we assume that the adjoint action of $H$ on $\mathfrak g/\mathfrak h$ is faithful.
Let $M$ a differentiable manifold of the same ...

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### Finite groups principal bundles

I am studying principal bundles from the point of view of algebraic geometry and I have come up with the following question. For the sake of clarity, a principal $G$-bundle over a scheme $X$ is just a ...

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### Definition of trace in topological BF-theories

I very important example of topological field theories are "BF-theories", which are usually defined as follows: Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ and let $\pi:P\to\...

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### Torsors are geometric quotients

I have been reading lately a lot about torsors in algebraic geometry and some authors say that a torsor over a scheme $X$, which is defined as a faithfully flat map of finite type $f:P\rightarrow X$ ...

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### Relationship between two bundles approaches of spontaneous symmetry breaking

I am trying understand if there is a relation between two formulations of the spontaneous symmetry breaking.
The first is provide by Derdzinski in his book "Geometry of the standard model of ...