# 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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### Torsor descriptions of $Bun_G$

The moduli stack $Bun_{SL_n}X$ of $SL_n$-principal bundles over a projective curve $X$ is a $\mathbb{G}_m$-torsor over the sublocus of $Bun_{GL_n}X$ corresponding to vector bundles with zero ...

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### Morphism of Lie groups $\theta:G\rightarrow H$ giving an equivalence of categories $BG\rightarrow BH$?

Given a morphism of Lie groups $ \theta:G\rightarrow H$ and a principal $G$ bundle $ \pi:P\rightarrow M$ there are (at least) two ways to assign a principal $ H$ bundle.
See that the morphism of Lie ...

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### Different definitions of a structure on principal bundles

Let $P\to B$ be a principal $G$-bundle and $\psi:H\to G$ a homomorphism of topological groups. A $\psi$-structure for $P$ can be defined in two different ways. I am trying to prove their equivalence.
...

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### On the existence and classification of prequantization spaces over a closed symplectic manifold

Let $(M,\omega)$ be a closed symplectic manifold. If the cohomology class $[\omega]$ is rational, that is if it lies in the image of the natural homomorphism $H^2(M,\mathbb{Z}) \to H^2(M,\mathbb{R})$, ...

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### Invariant integration on principal bundles

Let $G$ be a sufficiently nice topological or Lie group (e.g. compact), and let $H$ be a closed subgroup. This data determines a principal $H$ bundle $G \rightarrow G/H$ defined by the projection $g \...

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### Is there any Lie algebra structure on the sheaf of sections of adjoint bundle

Let $X$ be an irreducible smooth projective variety over $\mathbb{C}$.
Let $G$ be an affine algebraic group over $\mathbb{C}$.
Let $p : E_G \longrightarrow X$ be a holomorphic principal $G$-bundle ...

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### Characteristic classes of the bundle of trace free, skew adjoint endomorphisms

In "Floer Homology groups in Yang-Mills theory", Donaldson says that if we take an $U(2)$-vector bundle $E$ and we construct the bundle $\mathfrak{g}_E$ of trace-free, skew adjoint automorphisms of $...

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### Principal bundle analogue for Hodge bundle

Let $X$ be a connected smooth complex projective variety.
A holomorphic Higgs bundle is a pair $(E, \theta)$ consists of a holomorphic vector bundle $E$ on $X$ together with a Higgs field $\theta \...

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### Group scheme with isomorphic fibers

Let $X$ be a smooth irreducible algebraic curve over $\mathbb C$. Let $\mathcal G\rightarrow X$ be a smooth affine group scheme over $X$ such that for any closed points $p\in X$, we have $\mathcal G_p\...

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### When is G->G/H a trivial bundle

Suppose that $H\subseteq G$ are connected lie groups. Then $G\mapsto G/H$ is a principal $H$-bundle. I would like to know if there is some non-tautological characterization of when this is a trivial ...

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### A question about affine Grassmannian

I am reading Sorger's Lectures on the Moduli of $G$-bundles, and I am confused about a detail in the proof of proposition 5.3.2., where he proves that the $G$-bundle description of affine Grassmannian ...

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### Connecting torsors by a rational curve

Assume that $k$ is an infinite field. Let $G$ be a finite (constant) group, let $V$ be a faithful $G$-representation over $k$, $U$ a non-empty open subset of $V$ where the action is free. The map $\pi:...

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### 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 fibre bundle over $M$ with fiber $F$ and call this the associated fiber bundle for $P(M,G)...

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### Alternative (easier) Proof of Ambrose Singer Holonomy theorem

Let $P(M,G)$ be a principal bundle. Giving a connection on $P(M,G)$ means two equivalent things. One as an assignment of subspace of $T_pP$ for each $p\in P$ and another as a $\mathfrak{g}$ valued $1$ ...

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### Classification of Principal $G$ bundles and vector bundles in smooth sense

Suppose $G$ is a Topological group then classification theorem of Principal $G$ bundles says that
there is a Principal $G$ bundle $EG\rightarrow BG$ such that any principal $G$ bundle over a ...

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### What does reduction of structure group of principal bundle say?

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 recall here in case some one needs it.
Let $f:P(M,G)\...

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### When is the Chern class of a principal $G$-bundle same to the Chern class of the associated complex vector bundle?

Let $P\rightarrow M$ be a principal bundle, with structure group $G$. If $G$ has a representation $\rho:G\rightarrow GL(n,\mathbb{C})$, then we can define its associated vector bundle $E=P\times_{\rho}...

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### Slice theorem for proper groupoids

Let $G$ be a locally compact Hausdorff (second countable) groupoid with Hausdorff (second countable) unit space $X$.
Assume $G$ is étale, i.e., the source and range maps of $G$ are local ...

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### In what sense bibundles are called as generalized morphisms

Definition : Let $\mathcal{G}$ and $\mathcal{H}$ be Lie groupoids. A bibundle from $\mathcal{G}$ to $\mathcal{H}$ is a manifold $P$ together with two maps $a_L:P\rightarrow \mathcal{G}_0,a_R:P\...

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### Connection 1-form of the frame bundle associated to a vector bundle with a connection

Let $\lambda = (P,\pi,M;G)$ be a smooth principal $G$-bundle (projection $\pi : P \to M)$, $V$ a finite dimensional vector space, and $\rho : G \to GL(V)$ a smooth representation of $G$ in $V$.
We ...

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### $SU(2)\times SU(2)$ invariant $SU(3)$-structure on $\{t\} \times M^6$

I am reading Jason Lotay and Goncalo Oliveira's paper -$SU(2)^2$ invariant $G_2$-instantons, and have few questions from the same.
If we consider the space $M = S^3 \times S^3$. Then the cone metric ...

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### Exercise in the book “Lectures on Kähler geometry”

I am currently studying the book "Lectures on Kähler geometry" by Andrei Moroianu and am looking for help concerning Exercise 5.8 (3) which is to prove the following Lemma 5.11
Let $f: M \...

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### Principal $G$-bundles on affine toric varieties

Let $X_\sigma$ be an affine toric variety for an action of a torus $T$ and let $\mathcal{P}$ be a toric principal $G$-bundle over $X_\sigma$ where $G$ is an affine algebraic group (here base field $k$ ...

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### What are the applications of the Atiyah-Bott Yang Mills paper?

I recently finished a seminar going through Atiyah and Bott's paper ''The Yang-Mills Equations over Riemann surfaces''. The ideas going into the proof were surprising and very beautiful to me.
...

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### Global trivialization of a Principal G bundle

Let $X$ be a variety and $E$ be a principal $G$ bundles, where $G$ is a semisimple group. Is there a variety $f: \tilde{X}\rightarrow X$ such that $f^*E$ is trivial $G$ bundle?

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### The bundle of symmetric affine connections as quotient of the second-order frame bundle

This post is not about finding an answer to a certain problem - because the answer already exists - but rather about finding the simplest possible answer.
The problem is: how to define the bundle $C(...

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### Notion of Torsors

I am trying to read this paper by Lawrence Breen.
It starts with the definition of a torsor.
Let $G$ be a bundle of groups on a space $X$. The following definition of a principal space is standard,...

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### Extend a gauge transformation

Suppose $M$ is a smooth manifold and $P$ is a principal bundle on $M$. Let $U^\prime\Subset U\Subset M$ be strictly contained precompact open subsets. Let $g\in C^\infty(U, \hbox{Ad}P|_U)$ be a ...

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### Classification of principal bundles

I'm trying to reconcile two results on the classification of principal bundles. First, we have $\mathrm{Prin}_G(X)$ (the equivalence classes of $G$-bundles on $X$) is isomorphic to $H^1(X;G)$ (the ...

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### Is there an easy example of group action where the slice theorem produces a non-trivial principal bundle?

Let $\rho$ be a group action by a compact group $G$
\begin{equation}
\rho:G\times M \rightarrow M \\
\rho:(g,p) \rightarrow \rho_g(p)
\end{equation}
Denote the orbit of $p\in M$ by $\...

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### How does one introduce characteristic classes [closed]

How does one introduce, or how were you introduced to characteristic classes?
You can assume that the student is comfortable with principal bundles and connections on principal bundles.
I am not ...

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### The topology of subgroups of gauge groups

I am reading Atiyah and Bott's classic paper "Yang-Mills Equations over Riemann surfaces" and struggling with proposition 2.16 (p. 542)
Let $P$ be a principal $U(n)$-bundle over a compact Riemann ...

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### Universal covariant derivative decomposition in a Cartan geometry

I'm reading the book "Differential Geometry: Cartan's Generalisation of Klein's Erlangen Program" from Sharpe. Given a reductive model geometry, where $\mathfrak g=\mathfrak h\oplus \mathfrak p$ with $...

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### 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 action of Lie groups on manifold $M$ and convinced my self that if the action is smooth, proper, free ...

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### Classifying spaces and Brown's representability theorem

Let $G\text{-}PF(X)$ be the set of isomorphism classes of principal topological fibrations over the space $X$ with structural group $G$, and $G\text{-}PF_{cw} : hCW \to Set$ the contravariant functor $...

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### Compute characteristic classes of principal bundle over closed surfaces

Let $G$ be a connected Lie group and $\Sigma$ a closed oriented surface. We know that principal $G$-bundles $P$ can be topologically classified by a characteristic class $c(P)\in H^2(\Sigma,\pi_1G)\...

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### 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 ...

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### A bounded operator associated with a principal bundle

Assume that $(X, B, G)$ is a $G$- principal bundle where $G$ is a compact topological or Lie group.
The normalized Haar measure of (each fiber of ) $X$ is denoted by $\mu$.
The space of continuos ...

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### Is there a book on differential geometry that doesn't mention the notion of charts?

What are some books/texts that use chart free coordinate free language for things otherwise written in a coordinate based formulation? I would like to learn about covariant differentiation, curvature, ...

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### Does an oriented $S^3$ fiber bundle admit the structure of a principal $SU(2)$-bundle?

Let $S \to X$ be an $S^3$-fiber bundle over a smooth manifold $X$. If $S$ is an oriented manifold does this fiber bundle admit the structure of an $SU(2)$-principal bundle?
There is a similar theorem ...

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### Compatibility of Definitions of Universal G-bundles

In the book Fibre Bundles by Husemoller, universal G-bundles are introduced as bundles over a homotopy type $BG$, for which the cofunctor $[-,BG]\rightarrow k_G(-)$ is a natural isomorphism.
...

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### introduction textbook to the Laplacian on a circle bundle

I am looking for an introduction to spectral theory of $\Delta$ on a circle bundle over a compact M. Is there an analog of Selberg trace formula?

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### Automorphisms of semistable $G$-bundles

Let $X$ be a projective smooth curve over an algebraically closed field $\mathbb k$ of any characteristic. Let also $G$ be a reductive group over $\mathbb k$. (Probably) following Ramanathan, a $G$-...

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### Identify the sphere bundle of a complex line bundle $BD_{2n}\to BU(1)$

I'd like to know whether it is possible to identify the sphere bundle arising as follow:
Let $\xi \colon BD_{2n}\to BU(1)$ the complex line bundle corresponding to the element $y^2 \in H^2(D_{2n};\...

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### Contracted product of torsors

Given a group $G$ and a left $G$-set $X$, then we can make $X$ a right $G$-set defining the action as $xg:=g^{-1}x$ , or if you prefer we are considering the opposite group $G^{op}$ to make the left ...

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### Largest elementary neighbourhood

The real projective space $\mathbb{R}P^n$ can be defined as the quotient space of $\mathbb{S}^n$by the equivalence relation that identifies antipodal points. The largest open set of $\mathbb{S}^n$ ...

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### Connected components of gauge groups of principal bundles over generalized flag manifolds

Let $G$ be a compact connected Lie group and $P$ a principal $G$-bundle over a finite CW complex $X$. The gauge group $\mathcal{G}(P)$ is defined to be the group of principal bundle automorphisms of $...

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### Transition functions and twisting functions

A principle bundle over a topological space can be given by transitions functions. On the other hand, in the simplicial world principle bundles can be given by twisting functions. Is there an explicit ...

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### Principal bundle: A criterion

I am referring to page 300 of Okonek et al. book "Vector Bundles on Complex Projective Spaces" (1988.)
Let $G=GL_n(\mathbb{C})$ act holomorphically and freely on a complex manifold $X$. Define the ...

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### from 2-cocycle to classifying map

Let $A,E,G:\mathrm{Set}_*\to\mathrm{Grp}_*$ be functors from pointed sets to (discrete) groups ($*=1$) together with natural transformations $i:A\to E, \ p: E\to G$ such that for any set $X$
\begin{...