In mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of the Cartesian product $X\times G$ of a space $X$ with a group $G$.

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259 views

Group actions and associated bundles

Let $P$ be a principal $G$-bundle over $B$, and let $G$ act on some space $F$ (feel free to work in your favorite category of spaces, if this helps). Then $\text{Aut}{P}$ (aka the group of gauge ...
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135 views

Exact sequence of tangent spaces of principal $G$-bundles

Let $P$ be a smooth manifold, $G$ a Lie group, $\alpha:P\times G\to P$ a smooth action and $p:P\to P/G$ a smooth principal $G$-bundle. Then, we have the sequence $$ G \xrightarrow{\alpha_a} P \...
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107 views

Formal adjoint of curvature (Yang Mills)

Currently reading a paper on finding solutions to the Yang Mills equation $D^*\Omega=0$, where $\Omega$ is the curvature and $D^*$ is the formal adjoint of the exterior covariant derivative $D$. ...
5
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97 views

Does a left group action on a principal bundle induce an action on associated vector bundles?

Let $G\hookrightarrow P\xrightarrow{\pi}M$ be a principal $G$-bundle with right action $\cdot $ and suppose we are also given a left action $\rho: U\times P\rightarrow P$ of some group $U$ on $P$. ...
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64 views

Quotient space of equivariant vector bundle.

Let $p: P \to B$ be a principal $G$-bundle, and $\pi : E \to P$ a vector bundle with action of $G$ on $E$ such that $G$ acts by vector bundle isomorphisms and $\pi$ is equivariant. Is it always the ...
5
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56 views

Cartan geometry on manifolds with boundary

I was reading Sharpe's text on Cartan geometry, and I started to wonder: Does the theory change in any significant way if the base manifold for the Cartan principal bundle is allowed to be a smooth ...
4
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60 views

Classifying vector bundles with a reduction of its structure group

Let $Bun(X)$ denote the set of equivalence classes of complex rank 2 vector bundles with a reduction of its structure group to $\mathbb{H}^*$. How can I proof that there is a bijection between $Bun(X)$...
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88 views

Does naturality for characteristic classes imply the classifying space is universal for them?

Let $G$ be a Lie group, $\mathfrak g$ its Lie algebra, $K$ its maximal compact subgroup. To every flat $G$-bundle $P$ over a smooth manifold $M$ I can associate a homomorphism $w_P: H^*(\mathfrak g, ...
4
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39 views

Isomorphism of Principal Bundles with structure groupoid

Let $\mathcal{G}\rightrightarrows M$ be a Lie groupoid. Suppose that $\pi:P\rightarrow B$ is a $\mathcal{G}$-principal bundle and let $(h_s,g_s):(P,B)\rightarrow (P,B); s\in [0,1]$, be a homotopy of $\...
4
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154 views

Are all quotients of a weakly contractible space via a free group action classifying spaces of the group?

First of all, I don't want to restrict to any kind of "nice spaces" since I am interested in the most general situation. Especially I do not work in any "convenient" category of spaces. Wikipedia ...
4
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73 views

Heisenberg manifold

I am interested in the Heisenberg manifold, which is the quotient of the real Heisenberg group by the discrete Heisenberg (sub)group. It is a 3 -manifold which may be viewed as a circle bundle over ...
3
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25 views

Two ways to define the fundamental vector field on a principal bundle?

Let $\pi:P\longrightarrow M$ be a $G$-principal bundle. Let $\mathfrak{g}$ be the Lie algebra of left-invariant vector fields on $G$ and $\mathfrak{V}(P)$ be the space of vertical vector fields on $P$,...
3
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59 views

Rigorously what is this integral?

I've been studying some gauge theories approach to problems in mechanics in order to get a better understanding of the ideas from gauge theories and to see some applications of fibre bundle theory. ...
3
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70 views

How are the components of a connection on a homogenous space related to the Mauer-Cartan form?

I am finding it hard to understand in what way the Mauer-Cartan form $\omega_G$ of a Lie group $G$ can be used to define a connection on a bundle $G \to G/H$ in the same way that parallel transport of ...
3
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161 views

What is the commutator of a horizontal and vector field for a connection on a Fiber bundle?

I would be tempted to rephrase my question as : why do people seem to care only about the curvature of a connection on fiber bundles ? Indeed, the curvature gives the vertical part of the commutator ...
2
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23 views

Lifting the Einstein-Hilbert action into the frame bundle

If we have a four dimensional real spacetime $(M,g)$, with $g$ being a $(-+++)$ signature Lorentz-metric, and $\{\theta^0,\theta^1,\theta^2,\theta^3\}$ is a local orthornormal coframe defined in some ...
2
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34 views

Spin^c structure induced by a spin structure

I wondered how it works exactly to induce a $\mathrm{spin^c}$-structure if a spin structure is given. I wanted to use the following definitions as used in Friedrich`s "Dirac operators in Riemannian ...
2
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34 views

Reduction to the special orthogonal group

It is well known that an $SL_n$-bundle $E$ on an algebraic curve $X$ is self dual (i.e $E\cong E^*$) iff it is an $SO_n$-bundle However, I can't see why, because the isomorphism $E\cong E^*$ means ...
2
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53 views

How to Induce a Metric on Homogeneous Space $G/H$ by the Metric from Bundle G

I am having a question on how to induce a metric $g$ on homogeneous space $G/H$, if one is given a ${\rm Ad}_H$-invariant metric $\bar{g}$ on G. More specifically and simply, consider principal ...
2
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76 views

Conjugate group homomorphisms induce homotopic maps on classifying spaces

Let $G$ be a group and let $\phi: G \to G$ be the inner automorphism given by conjugation by an element $g' \in G$, i.e., $\phi(g) = g'^{-1} g g'$. I want to show that the induced map on classifying ...
2
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42 views

Identity of curvature in a principal bundle

I saw in wikipedia (http://en.m.wikipedia.org/wiki/Curvature_form) the following identity for the curvature 2-form of a principal bundle $$2\Omega(X,Y)=h[X,Y]-[hX,hY]$$ where $X,Y\in T_uP$, $P$ being ...
2
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63 views

What is the principal bundle structure of $O(n)$?

Consider the map $\pi:O(n)\rightarrow G(k,n)$ which maps $A\in O(n)$ to the subspace of $\mathtt{R}^n$ spanned by the first $k$ columns of $A$. Here $G(k,n)$ is the Grassmannian manifold. My question ...
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43 views

Contraction of second exterior covariant derivative with metric

Let $G \hookrightarrow P \to M$ be a principal $G$ bundle, $P \times_\rho V$ be a vector bundle associated to representation $\rho$ of $G$ on $V$. If $\omega$ is a connection $1$-form on $P$ then we ...
2
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52 views

Invariant characterization of vector bundles associated to a principal bundle?

I have two related questions. Suppose I have a principal $G$-bundle $P\xrightarrow{\pi} M$. The usual construction of an associated vector bundle goes as follows. Fix some representation $\rho : G\...
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66 views

How to decompose connections on the complexified orthonormal frame bundle?

Let $E\rightarrow M$ be an orientable vector bundle of rank n equipped with some Riemannian metric, $P:=F_{SO(n)}(E)$ the orthonormal frame bundle. I say that $P^{c}:=F_{SO(n)}(E)\times_{SO(n)} SO(n,\...
2
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69 views

Clarification on some notation and “assumptions” in page 143-144 of the book “Quantum Fields and Strings: A Course for Mathematicians, Volume 1”

I was trying to read the chapter $1$ (at page $143$) of this book Quantum Fields and Strings: A Course for Mathematicians, Volume 1 that is supposed to be an introduction to modern quantum field ...
2
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182 views

Local trivializations for orthonormal frame bundle

Let $(E,\pi, M)$ be a real vector bundle of Rank $N$. Then one can define its frame bundle $GL(E)$ as follows: $GL(E)_x:=\{\text{ordered bases of }E_x\}$ (for $x\in M$). $GL(E):=\bigcup_{x\in M} GL(...
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37 views

Making a principal bundle into a covering space

Suppose $\pi : P\rightarrow M$ is a principal $G$-bundle. I want to make this into a covering map by changing the topology of $P$. By local triviality we can find for each $x\in M$ an open $U\subset ...
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17 views

Definition of isomorphism classes of vector bundles with reduced structure group

I'm looking for a definition of the notion of isomorphism between vector bundles $E$ and $F$, over the same base $X$, whose structure groups have been reduced from $GL(n)$ to some subgroup $G\subseteq ...
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29 views

What is $P\times_G E$?

I know what is principal bundle and associated bundle according Wiki.But I am not understand what is $P\times_G E$ .Seemly it is bundle,but I am not sure what structure is it . Below picture is from ...
1
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29 views

Sectional category (Schwarz genus) of the Milnor join construction

Assume topological spaces to be normal and paracompact. Following the article: "The genus of a fiber space" by A. Schwarz, we call the sectional category (or Schwarz genus) of a locally trivial fiber ...
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80 views

Action on algebraic variety and adjoint bundles

Let $X$ be a complex algebraic variety and let $G$ be a complex algebraic group; I mean that $X$ is a reduced, separated scheme of finite type on $Spec\mathbb{C}$, and the underlying set of $G$ is a ...
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33 views

Description of complex conjugate $\operatorname{Spin}^c$ structure without cocycles

The following uses exclusively cocycle descriptions for spin and spinc structures which I would like to avoid. See for example Nicolaescu "Notes on Seiberg-Witten invariants", pages 40-41 for their ...
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32 views

When is a fibration (canonically) a principal fibration over its group of automorphisms?

The question is inspired by the following observation: Let $p: X'\to X$ be a connected covering space where both spaces are suitably nice (say they are CW complexes), then $p: X' \to X$ is a principal ...
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17 views

$G$-invariant vector field coming from a principal bundle?

Let $\pi:P\longrightarrow M$ be a $G$-principal bundle. If $(U, \phi)$ is a local trivialization of this bundle then for every $x\in M$ we have a diffeomorphism $$\phi_x:P_{x}\longrightarrow G, \phi_x:...
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35 views

Decomposition of tangent space of principal bundle

A connection on a principal bundle $\pi:P\rightarrow M$ is a choice of horizontal subspace $H_p$ at each $p\in P$, such that $T_p P = H_p + V_p$ where $V_p = \ker((\pi_*)_p)$. It is very common to ...
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42 views

Bundle map is isomorphism iff it covers a homeomorphism

Consider $P_0$ and $P_1$ principal G-bundles with projection maps $\pi_0, \pi_1$, respectively; $f:P_0 \rightarrow P_1$ a continuous G-equivariant map (i.e. a bundle map) and $g:X_0 \rightarrow X_1$ ...
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84 views

Motivation for the name “vertical subspace” in the context of fiber bundles.

Let $p:E\to B$ be a smooth fiber bundle with fiber $F$. Consider the vector spaces $V_u=\{x\in T_uE: p_*(x)=0\}$. We call $V_u$ the vertical subspace of the tangent space $T_uE$. How can we see that $...
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0answers
51 views

Certain principle bundle structure on $\mathbb{R}^{n}\setminus \{0\}$

Is there a right action of $\mathbb{H}^{2}$ on some $\mathbb{R}^{n}\setminus \{0\}$ such that this action gives us a principle fibre bundle. Here $\mathbb{H}^{2}$ is the Poincare upper plane ...
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0answers
30 views

How to obtain the metric tensor of a principal bundle total space given a connection (assuming it is metric compatible) in the total space?

The title says it all. In a principal bundle I know the connection defined in the total space. How can I calculate the metric that would be compatible with this connection.
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31 views

Trouble proving identity - Gauge theory/Maurer-Carton one-form/Adjoint representation

The Identity I am trying to prove is the one in this already asked question how to show that ${ad}_{g_{\alpha\beta}} \circ g_{\alpha\beta}^{\star}\theta=-g_{\beta\alpha}^{\star}\theta$? The author ...
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28 views

extension of a principal connection

I am trying to prove the following: Suppose that $\alpha:H\to G$ is a Lie group homomorphism and let $P\to M$ be a principal $H$-bundle and $Q\to M$ a principal $G$-bundle. Suppose further that there ...
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104 views

Relation between principal bundle automorphisms and maps in $\Lambda^0(M,Ad(P))$

Let $ G \hookrightarrow P \xrightarrow{ \pi } M $ be a principal $ G $-bundle over $M$. Denote by $\mathrm{Ad} (P) = P \times _{ \mathrm{Ad} } G $ the non-linear adjoint bundle. It seems to me that ...
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0answers
43 views

Equivalence between pullback connections of smoothly homotopic maps

Let $f,g:M\rightarrow N$ be smooth maps between smooth manifolds such that there exist a smooth homotopy $H:M\times [0,1]\rightarrow N$ between them. If we have a principal bundle $P\rightarrow N$, we ...
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0answers
84 views

morphisms of principal bundles with different structure groups

Let $f \,: X \to Y$ be a continuous map between spaces. Let $G$ and $H$ be topological groups. Consider the diagram: \begin{equation} \label{} \begin{array}{ccccccccccccccccccccccccccccccc} E_G &...
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99 views

extending maps from spaces to their whitehead towers

Let $f \,: X \to Y$ be a map between connected spaces. Let: $$ X^{(k)} \to \ldots \to X^{(0)} \approx X $$ and $$ Y^{(k)} \to \ldots \to Y^{(0)} \approx Y $$ be whitehead towers for $X$ and $Y$. What ...
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33 views

Obstruction of such gauge choice

Suppose we consider $\operatorname{ad}P_G \to T^k$ as the associated adjoint bundle (maybe this is not the correct name, but I just mean with the associated vector bundle ${\rm Lie}G$ as standard ...
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45 views

$\mathbb{RP}^3 \rightarrow \mathbb{CP}^1$ defines a principal U(1)-bundle

I have to show that the map $\pi: (x_o : x_1 : x_2 : x_3) \in \mathbb{RP}^3 \rightarrow (x_0 + i x_1) : (x_2+ ix_3) \in \mathbb{CP}^1$ defines a principal U(1)-bundle. The two standard coordinate ...
0
votes
0answers
10 views

local sections transformation formula

I am trying to prove a formula which states the relations between two local sections in a principal bundle: Let $P(M,G)$ be a principal bundle let $\{ U_\alpha \}_{\alpha \in I}$ be an open cover for ...
0
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0answers
15 views

Sign disagreement in curvature form on $U(1)$-bundle

this is rather trivial question to masters, but as I'm totally disoriented by my computation, so deciding to ask. I'm trying to solve the following exercise, 1.1, showing the curvature 2-form $F(X,Y)=...