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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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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3 votes
1 answer
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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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1 answer
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Non existence of a preferred Horizontal subspace on a bundle. Why not ? (Basics) [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 choose to put my finger on the identity element of the group over a ...
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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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4 votes
1 answer
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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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1 vote
1 answer
242 views

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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6 votes
1 answer
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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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2 votes
1 answer
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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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1 answer
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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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1 answer
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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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3 votes
0 answers
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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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3 votes
1 answer
277 views

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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4 votes
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104 views

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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3 votes
1 answer
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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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2 answers
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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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4 votes
0 answers
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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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5 votes
1 answer
305 views

$1$-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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1 vote
0 answers
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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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1 answer
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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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17 votes
1 answer
403 views

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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1 vote
1 answer
265 views

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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4 votes
0 answers
77 views

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 ...
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4 votes
0 answers
150 views

Relative Thom isomorphism

Let $\tilde{X}$ be a space with an action of the symmetric group $\mathfrak{S}_k$ and define $X:=\tilde{X}/\mathfrak{S}_k$ to be the quotient. On the other hand, $\mathfrak{S}_k$ acts on $(\mathbb{R}^...
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5 votes
1 answer
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Curvature as infinitesimal holonomy 2

This question may be seen as a follow up of this original question. I'm learning Cheeger-Simons differential characters (reading Differential Characters of Bär and Becker). If I understand correctly, ...
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3 votes
1 answer
153 views

Is the irreducible locus of the character variety a principal bundle in Zariski topology?

Let $\Sigma$ be a compact orientable surface and let $G$ be a reductive algebraic group (say, $G=\mathrm{SL}_n(\mathbb{C})$ for simplicity). The representation variety is $$ X_G(\Sigma) = \mathrm{Hom}(...
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9 votes
0 answers
286 views

Milnor's universal bundle as a colimit?

I have had Milnor's construction of the classifying space of a topological group explained to me on multiple occasions, and seen it described briefly in various places. But only now am I reading the ...
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2 votes
0 answers
93 views

Classifying space induces a equivalence of categories between $\operatorname{PBun}_G(M)$ and $\Pi(M,BG)$ for finite groups $G$

Let $G$ be a finite group, $BG$ its classifying space and M a manifold. Then it is mentioned in Schweigert and Woike - Orbifold construction for topological field theories (Remark 2.3 d) that there is ...
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3 votes
0 answers
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On the construction of principal $S^1$-bundles with prescribed characteristic form

I am trying to understand an argument made by Kobayashi (https://projecteuclid.org/download/pdf_1/euclid.tmj/1178245006, page 35) in his construction of a principal $S^1$-bundle with connection $1$-...
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1 vote
0 answers
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On prequantization bundles over integral symplectic manifolds

I am trying to clarify certain subtleties regarding prequantization bundles over symplectic manifolds, for which I haven't found any clear explanation so far. Let me fix some definitions first. ...
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3 votes
0 answers
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Evaluating the Euler class of a circle bundle on fibers

I am trying to understand what kind on information the Euler class provides about certain submanifolds of a given circle bundle. This might be completely obvious, but I don't see how to answer the ...
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1 vote
0 answers
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Definition of universal bundle in category of smooth manifolds

Let $(P, M, G)$ a smooth principal bundle over smooth manifold $M$ with Lie group $G$. We denote by $k_{G}(M)$ the set of all classes of principal G-bundle isomorphisms. If i want to define universal ...
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2 votes
0 answers
235 views

Differences between induced vector fields on a smooth manifold and on a principal bundle

In the context of the connections on fibre bundle, I have found some difficulties trying to understand the fundamental vector field (my reference is Nakahara, but I'm having some problems with the ...
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4 votes
0 answers
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Lifting a metric on a principal bundle with non-compact fibre

Let $\pi \colon P \to M$ be a $G$-principal bundle. If $G$ is compact, we may lift any metric $m$ on $M$ to a $G$-invariant metric $\bar{m}$ on $P$ such that $\pi_\ast\bar{m} = m$. Doing so accounts ...
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2 votes
0 answers
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Principal bundles over marked Riemann surface $\mathcal{M}_{g,h,r,s}$

Let $\mathcal{M}_{g,h,r,s}$ be a Riemann surface with genus $g$, $h$ boundary components, r interior marked points, and $s$ marked points on the boundary $\partial \mathcal{M} = \Sigma$. In the case ...
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  • 161
13 votes
1 answer
414 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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