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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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 ...
3 votes
1 answer
152 views

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?
2 votes
1 answer
317 views

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}\...
12 votes
1 answer
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Reference request: Gauge natural bundles, and calculus of variation via the equivariant bundle approach

Let $P\rightarrow M$ be a principal fibre bundle with structure group $G$, $F$ a manifold and $\alpha: G\times F\rightarrow F$ a smooth left action. There is an associated fibre bundle $E\rightarrow ...
2 votes
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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 ...
1 vote
1 answer
789 views

Characterisation of (integrable) connections on (trivial) principal bundle

Let $M$ be a manifold. Let $G$ be a Lie group and $\mathfrak{g}$ be its Lie algebra. Let $P(M,G)$ be a principal bundle. Recall that, a connection on $P(M,G)$ is a distribution $\mathcal{H}\subseteq ...
7 votes
1 answer
177 views

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 ...
2 votes
0 answers
158 views

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 ...
11 votes
4 answers
936 views

$G$-bundles on affine spaces

Let $G$ be a connected algebraic group. Is it true that every $G$-bundle on ${\mathbb A}^n$ is trivial? What is the reference? I am actually only interested in the case $n=2$.
5 votes
2 answers
1k views

Some questions about Hitchin's self-duality paper

I am reading this paper (The self-duality equations on a Riemann surface by N. Hitchin; DOI: 10.1112/plms/s3-55.1.59), and I don't understand a few things in page 67. In proof of Theorem 2.1 after ...
5 votes
1 answer
246 views

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 $S$, ...
3 votes
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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}...
14 votes
1 answer
547 views

Mednykh's formula for 2d manifold with boundaries? How to count principal G-bundles with prescribed holonomies?

Let S be a closed, orientable 2d manifold and G a finite group. Since a principal G-bundle over S is specified by maps $\phi : \pi_1(S) \rightarrow G$ modulo the adjoint action by G, the way to count ...
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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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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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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 $X$ ...
2 votes
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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:...
1 vote
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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 ...
7 votes
1 answer
164 views

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 $\mathscr{B}$...
2 votes
1 answer
284 views

Extrinsic horizontal path lifting

As a follow up question to my previous question about the orthonormal frame bundle, I would like to understand a simple example explicitly. Let $\mathbb{S}^2$ be written extrinsically as $$\mathbb{S}...
4 votes
1 answer
222 views

Integrability of certain distribution associated to a connection form on the total space of a principal bundle (Principal Frobenius condition)

Let $P\to M$ be a $G$-principal bundle where $P,M$ are smooth manifolds and $G$ is a Lie group with Lie algebra $\mathfrak{g}$, whose center is denoted by $C(\mathfrak{g})$. Let $\omega$ be the ...
8 votes
2 answers
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What does reduction of structure group of principal bundle say?

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SO{SO}$Let $G$ be a Lie group and $\pi:P\rightarrow M$ be a principal $G$ bundle. The notion of reduction of structure group is standard but I will ...
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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 $G$-...
1 vote
1 answer
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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)=...
3 votes
4 answers
3k views

References on principal G bundle and connections

I am trying to understand about principal G bundle given a Lie group $G$. For that, I started with the action of Lie groups on manifold $M$ and convinced myself that if the action is smooth, proper, ...
6 votes
0 answers
228 views

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 $(S,J)$ ...
2 votes
1 answer
246 views

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 ...
3 votes
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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 ...
3 votes
1 answer
283 views

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 = ...
6 votes
2 answers
533 views

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 ...
1 vote
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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}...
3 votes
1 answer
317 views

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, ...
5 votes
2 answers
230 views

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 ...
9 votes
6 answers
3k views

Motivation for construction of associated fiber bundle from a principal bundle

Given a principal $G$ bundle $P(M,G)$ and a manifold $F$ with an action of $G$ on it from left, we construct a fiber bundle over $M$ with fiber $F$ and call this the associated fiber bundle for $P(M,G)...
5 votes
2 answers
319 views

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 $...
4 votes
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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 ...
6 votes
1 answer
247 views

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 ...
6 votes
1 answer
427 views

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, ...
8 votes
2 answers
326 views

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 ...
0 votes
1 answer
138 views

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 ...
4 votes
0 answers
410 views

Symplectic principal bundles

A symplectic principal bundle is a principal bundle $(X,B, G)$ with projection map $q:X\to B$ such that $X$ and $B$ are symplectic manifolds and the right action of $G$ preserves the symplectic ...
1 vote
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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, ...
2 votes
1 answer
179 views

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,\...
1 vote
0 answers
456 views

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/...
3 votes
1 answer
234 views

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 ...
2 votes
0 answers
84 views

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\...
2 votes
1 answer
302 views

Scheme of Higgs reductions

I'm reading the Bruzzo and Graña Otero's paper Semistable and Numerically Effective Principal (Higgs) Bundles; here: $X$ is a smooth, complex, projective variety; $G$ is a connected, complex, ...
5 votes
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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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