# 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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### 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 ...

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### Universal principal bundle on stack

I am studying the notes of Sorger concerning the moduli problem of principal bundles over curve https://inis.iaea.org/collection/NCLCollectionStore/_Public/38/005/38005695.pdf and there is something I ...

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### Extending $G$-torsors on open subsets of affine space

Let $k$ be a characteristic zero field, $V \subset \mathbb{A}^n_k$ an open subscheme, $G$ a split reductive group over $k$ and $T$ a $G$-torsor over $V$ (in the etale, equivalently fppf topology). ...

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### Picard group of moduli of principal bundles

I am looking for the Picard group of the moduli space of principal $G$-bundles for a connected reductive complex algebraic group $G$.
Is it isomorphic to $\mathbb{Z}$? If not, what can we say when $G=\...

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### Does lifting correspondence hold for principal bundles too?

Let $P$ be a (nontrivial) principal bundle over the base space $\mathbb{R}^4$ and fibers diffeomorphic to $SU(3)$.
Also assume that $P$ is equipped with an Ehresmann connection.
Then, for for any two ...

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### The Yang-Mills Higgs Lagrangian

Let's say we have a principal bundle $(P,B,\pi;G)$ and associated bundle $E=P \times_{(G,\rho)}V$and $Ad(P)=P\times_{(G,Ad)} \mathfrak{g}$ the adjoint bundle. The Yang-Mills-Higgs action (without ...

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### Spin connection in the tetradic Palatini-formalism of general relativity

$\DeclareMathOperator\SO{SO}$I am trying to understand the tetradic Palatini-formalism of general relativity from a mathematical point of view. I am graduate student and quite new to mathematical ...

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### Tensor product of associated vector bundles

Let $(P, X, \pi, G)$ and $(P', X, \pi', G')$ be two principal bundles (with Lie groups $G$, $G'$ respectively). Given a vector space $V$ and representations $\rho, \rho'$ of the Lie groups in this ...

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### Curvature of principal bundle

Let $(P,M,G)$ be a principal bundle with connection 1-form $\omega$. In all books I have seen so far, the curvature is defined by
\begin{equation}
F:=D_{\omega}\omega \in \Omega({P,\mathfrak{g}})
\end{...

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### Equivalence of $Spin^C$-Structures

I'm trying to understand the equivalence of $Spin^C(n)$-structures in the book "Dirac Operators in Riemannian Geometry" by Thomas Friedrich, p. 47 ff, but I got somehow stuck because I'm not ...

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### Is there a smooth Weyl equivariant map from this quotient space into $G_2/T^2$?

It is known that $G_2$ acts transitively on $S^6$ with fibers $SU(3)$. Let us consider the following set $P$ of complex unitary $7 \times7$ matrices $A$, where
$$ A = (v_0, \, v_1, \, v_2, \, v_3, \, ...

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### Local coordinates of one form on a principal bundle

I am reading "Natural and Gauge Natural Formalism for Classical Field Theory" by Lorenzo Fatibene and I am really confused by his definition of a connection in local coordinates.
Let's say ...

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### Reference for mathematical Palatini formalism of general relativity

I know that this is maybe not a research level question, but since the topic is quite special, I thought that the chance to get some reference is higher in this community.
I am looking for a reference ...

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### Principal bundle over associated bundle

Let $P$ be a principal $G$ bundle.
Let $S$ be a space with left action of $G$, and let $Q$ be a principal $H$ bundle over $S$ with the property that the action of $G$ can be lifted to $Q$.
Then
$$
P \...

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### $1$-cocycle associated to universal $G$-bundle $EG \to BG$

Let $G$ be a (topological) group whose identity element $e_G$ is
a nondegenerated basepoint (e.g. if $G$ is a Lie group). Then that's a
known fact that there is for every 'nice' enough topological ...

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### About irreducible connection

The irreducible connection is a connection whose holonomy group is just $G$ (let us just assume the base space $X$ is just connected). Otherwise, it is called reducible if the holonomy group $H_A$ ...

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### The notion of a "relatively" flat connection

Suppose that $X$ is a connected smooth manifold and $\Gamma$ is a group acting smoothly, freely, properly and discretely on $X$, so that $Y=X/\Gamma$ is another smooth manifold endowed with a covering ...

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### For what topological groups $G$ can we take $EG \rightarrow BG$ to be of the form $S^{\infty} \rightarrow BG$?

Title. For what topological groups $G$ can we take $EG \rightarrow BG$ to be of the form $S^{\infty} \rightarrow BG$?
If $G$ is a subgroup of either $S^0,S^1,S^3$ or $S^7$ this induces a free action ...

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### Principal G-bundles over the circle

To edify my understanding of fiber bundles with structure groups, I was currently trying to reconcile two classifications (in a particular case). For simplicity, I'm taking the base to be $S^1$ and ...

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### Dot product of functions on cosets

Some time ago I asked this same question at Math Stackexchange, because I thought that the question is nearly elementary.
To my surprise, it was never answered. So I am elevating it to MathOverflow.
I ...