In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every point x of the space X we associate (or "attach") a vector space V(x) in such a way that these vector spaces fit together to form another space of the same kind as X (e.g. a topological space, manifold, or algebraic variety), which is then called a vector bundle over X.
The simplest example is the case that the family of vector spaces is constant, i.e., there is a fixed vector space V such that V(x) = V for all x in X: in this case there is a copy of V for each x in X and these copies fit together to form the vector bundle X × V over X. Such vector bundles are said to be trivial. A more complicated (and prototypical) class of examples are the tangent bundles of smooth (or differentiable) manifolds: to every point of such a manifold we attach the tangent space to the manifold at that point. Tangent bundles are not, in general, trivial bundles. For example, the tangent bundle of the sphere is non-trivial by the hairy ball theorem. In general, a manifold is said to be parallelizable if, and only if, its tangent bundle is trivial.
Vector bundles are almost always required to be locally trivial, however, which means they are examples of fiber bundles. Also, the vector spaces are usually required to be over the real or complex numbers, in which case the vector bundle is said to be a real or complex vector bundle (respectively). Complex vector bundles can be viewed as real vector bundles with additional structure. In the following, we focus on real vector bundles in the category of topological spaces.
My apologies if this question is trivial. I have searched the forum and haven't found an existing answer to this question.
I've been working through differential geometry problem sets I found online (associated with MATH 481 at UIUC) and am struggling to show that T(MxN) is diffeomorphic to TM...
Hopefully, I am in the right forum.
I am trying to get an intuitive understanding of how fiber bundles can describe gauge theories. Gauge fields transform in the adjoint representation and can be decomposed as:
Wμ = Wμata
Gauge field = Gauge group x generators in the adjoint...
I read about connections on principal bundles. I don't have the knowledge nor the time to learn about principal bundles in the first place. Never the less this makes me wonder if such connections are the same as those talked about in the context of tangent vector spaces. Are they the same thing?
I am reading "Road to Reality" by Rogen Penrose. In chapter 15, Fibre and Gauge Connection ,while going through Clifford Bundle, he says:
.""""...Of course, this in itself does not tell us why the Clifford bundle has no continuous cross-sections. To understand this it will be helpful to look at...
Hi everyone,
Does anyone know of a good intuitive resource for learning Yang-Mills theory and Fiber Bundles? Ultimately my goal is to gain a geometric understanding of gauge theory generally. I have been studying differential forms and exterior calculus. Thanks!
Hello! I am reading about connections on principal bundles and the book I read introduces the connection one form as ##\omega \in \mathfrak{g} \otimes T^*P##, where ## \mathfrak{g}## is the Lie Algebra associated with the principle bundle P. I am a bit confused about what does this mean...
Hello,
Is the Berry connection compatible with the metric(covariant derivative of metric vanishes) in the same way that the Levi-Civita connection is compatible with the metric(as in Riemannanian Geometry and General Relativity)?
Also, does it have torsion? It must either have torsion or not be...
Hello! I am having some troubles understanding fibre bundles and I would be really grateful if someone can explain them to me in layman terms (at least how to visualize them). To begin with, I am not sure what is the fibre bundle, is it the projection function, or the total space (or something...
Let ##P## be a ##U(1)## principal bundle over base space ##M##.
In physics there are phenomenons related to a loop integration in ##M##, such as the Berry's phase
##\gamma = \oint_C A ##
where ##C(t)## is a loop in ##M##, and ##A## is the gauge potential (pull back of connection one-form of...
Hi, I am looking for textbooks in QFT. I studied QFT using Peskin And Schroeder + two year master's degree QFT programme.
I want to know about the next items:
1) Lorentz group and Lie group (precise adjectives, group representation and connection between fields and spins from the standpoint of...
I've been studying TQFT and gauge theory. Dijkgraaf-Witten theory in particular. One learns that a topological field theory applied to a manifold outputs the number of principal G bundles of a manifold.
My question is for the Physicists in the room, why do you want to know the number of...
Hello!
I am studying the mentioned topic I am confused in calculating the average value of friction factor in sub channels ... please help ??
thank you in advance
A textbook or even better some openly available pdf would be preferred.
I have no idea where to start a search and I normally prefer recommendations over just picking any book from some publisher.
Thank you.
1 Let A → N, B → N be two vector bundles over a manifold N. How to show that there is a vector bundle Hom(A, B) whose fiber above x ∈ N is Hom(A, B)x := Hom(Ax, Bx)?
2 Let A → N, B → N be two vector bundles over a manifold N. Let C∞(A, B) denote the space of maps of vector bundles from A to B...
Hi All,
Please let me set things up before the actual question:
I have a Lefschetz fibration ## f: W^4 \rightarrow S^2 ## , where ## W^4## is a closed, oriented manifold, and
## S^2## is the 2-disk. This f is a smooth map with finitely-many singularities {## x_1, x_2,..,x_n ##}, so that we have...
For a map between vector bundles (which commute with a certain Lie groups like Sl2R or GL2R), what does it mean exactly for a fiber to be multiplicity free?
Eplanations would be good, but examples would be even better. Thanks in advance, Gauss bless you!
CM
I'm trying to learn how to think about principal bundles where the fibre is a lie group with local trivialization ϕ^{-1}_i:π(U_i)→U_i×G . For example ϕ^{-1}_i:π(S^2)→S^2×U(1) (if that makes sense) . But I don't know how to think of this (and other products with lie groups like that)...
I would like a good source from which I can study fiber bundles (mainly their application in Yang-Mills gauge theories, but also in differential geometry)... I tried to study them from the advanced differential geometry (note)book of 1 of my professors but it was a mess and it confused me even...
As I understand it, Felix Klein sought to classify geometries with respect to what groups G that respected the structure of the given space X. Lately i read in an article on "the history of connections" by Freeman Kamielle that Cartan wished to generalize this notion. Is it correct to think of...
I read in a differential geometry paper that Maxwell's equations can be formulated in terms of a connection on a Hermitian line bundle on Minkowski space.
I understand the derivation of the field strength 2 form,the proof that Maxwell's equations say that its exterior derivative is zero and its...
Hi everyone!
I would like to ask you some clarifications on an explicit example of local trivializations and transition functions of fibre bundles: namely on the [-1,1]\hookrightarrow E\rightarrow S^1 bundle (which I guess is the simplest possible example).
Following Nakahara (chapter 9...
The circle is the fiber over Minkowski spacetime for electromagnetism?
I want to make connection to the classical vector potential via some " picture" involving this circle. Does the following come close?
Can I consider a 3 dimensional surface in C_1XMinkowski space that at a given slice of...
Homework Statement
If M and N are smooth manifolds, then T(MxN) is diffeomorphic to TM x TN
Homework Equations
The Attempt at a Solution
So I'm here
let ((p,q),v) \in T(MxN)
then p \in M and q \in N and v \in T(p,q)(MxN).
so T(p,q)(MxN) v = \sum_{i=1}^{m+n}...
This question asks whether every circle bundle comes from a 2 plane bundle. Paracompact space please - preferably a manifold.
By circle bundle I mean the usual thing, a fiber bundle with fiber, a circle, that is locally a product bundle. The transition functions lie in some group of...
Hi, I was working through a Twistor paper and it was explaining the link between holomorphic vector bundles and anti self dual gauges and it had an equation like this, for electro-magnetism.
\lambda^a \lambda^b(\frac{\partial A_{b\dot{b}}}{\partial x^{a\dot{a}}}-\frac{\partial...
Hey guys,
I am confused about the concept of sections of vector bundles. Mobius band is nontrivial line bundle over circle so we can not find any nonzero section from circle to mobius strip. However a plane which is twisted once is a trivial line bundle over line. That means there is a nonzero...
Ok, so I don't have much of an intuition for frame bundles, so I have some basic questions.
A frame bundle over a manifold M is a principle bundle who's fibers are the sets of ordered bases for the vector fields on M right.
1) This means that any point in the fiber (say, over a point m in M)...
Suppose (P,M,\pi,G) is a G-principal bundle. With this I mean a locally trivial fibration (G acts freely on P) over M=P/G with total space P and typical fibre G, as well as a differentiable surjective submersion \pi\colon P\to M. In this case M is nearly a manifold, but may be non-Hausdorff...
Hello all !
My question:
Does fibre bundles are necessary for describing gauge invariance in electromagnetic case? Or fibre bundles uses only for describing gauge invariance in cases of weak, electroweak and strong interactions?
Thanks
This might go into stats, I'm not sure. But I'll throw it out there. You are at the grocery store and they have two product bundles:
Four bananas and three limes for $10.
Two grapefruits and five limes for $12.
You want to come up with a way to compute the average cost of a lime, the...
Hi, All:
I am reading a paper in which , if I understood well, a spin structure in a manifold M
is equivalent to M admitting a trivialization of the tg.bundle over the 1-skeleton of M
( I guess M is assumed to be "nice-enough" so that it is a simplicial complex )
so that...
I would like to know if the following correct.
Suppose I start with a connection on a real vector bundle and extend it to the complexification of the bundle.
The curvature forms of the complexification seem to be the same as curvature 2 forms of the real bundle.
From this it seems that the...
Is there an example of a real vector bundle over a compact smooth manifold with all zero characteristic classes (Euler class,Stiefel-Whitney classes and Pontryagin classes) that is non-trivial?
I have a question regarding light bundles and the diffraction of waves. I've been trying to wrap my head around the processes that govern how diffraction works and it all seems to make sense to me regarding water waves and sound. If I just apply Huygens' principle that every point in a wave is...
http://www.kabculus.com/capacitance-and-inductance-matrices/node7.html
this short page describes conductor bundles, which are power transmission lines hung parallel to each other. i think the page is trying to explain why, when they are hung that way that they reduce impedance per meter, but i...
Hi, everyone:
I am trying to find a result for the number of bundles (up to bundle iso.) over a fixed
base and fixed fiber. For example, for B=S<sup>1</sup> , and fiber I=[0,1]
I think that there are two; the cylinder and the Mobius strip.
I think that the reason there are...
Is it true in complete generality that every fiber bundle with contractible fiber have a global section? Or do some sort of restrictions on the bundle need to be made? I ran across a mention of this fact in Guillemin and Sternberg's "Supersymmetry..." and I'm not sure how to prove it.
Hello,
suppose you start with Yang Mills theory with some gauge group G, for example SU(5). Then you turn on a gauge bundle, say a U(1) bundle, and the group breaks down. I know that from hearsay but I wonder how would you describe that explicitly in formulas?
meha
Becuase of geometric phase,I'm looking for a good book on fiber bundles, with a minimum of prerequistes and that takes a more intutive rather than formal approach.I am reading a book called modern differential geometry for physicists. It is a good book but sometimes abstract.
I know about...
Hello,
Say you have a function f on the domain R^n, and an integral transform P which integrates f over all possible straight lines in R^n. I am lead to believe that the range of this is R^(2n), or a tangent bundle, which I am having MASSIVE problems visualising!
Am I right in saying the...
An oriented surface with a Riemannian metric has a natural action of the unit circle on its tangent bundle. Rotate the tangent vector through the angle theta in the positively direction.
Is there a natural action of SO(n) on the tangent bundle of an oriented Riemannian n-manifold?
Same...
In trying to understand why not all tangent bundles are trivial, I've attempted to prove that they are all trivial and see where things go wrong. Unfortunately, I finished the proof and cannot find my mistake. Here it is:
Let M be an n-manifold with coordinate charts (U_\alpha...
This is a question from Hirsch's Differential Topology book: show that there is a bijective correspondence between
K^k(S^n) \leftrightarrow \pi_{n-1}(GL(k)) ,
where K^k(S^n) denotes the isomorphism classes of rank k vector bundles over the sphere. The basic idea is that any vector bundle...
Hi:
I am trying to understand more geometrically the relation between triviality
of bundles and existence of global sections. This is what I have for now. Please
comment/critique:
Let p:E-->B be a fiber bundle :
consider E embedded in B as the 0 section. Then...
Hi, I have a small question about exterior derivatives d on defined on principal bundles P.
We have the Ehresmann connection on a principal bundle P, represented by a Lie-algebra valued one-form omega. We can use the section sigma to pull this one-form back to our basemanifold, where the...
Hi, I'm a little stuck on Nakahara's treatment about fibre bundles. I hope someone can give me a clear answer on this; they are quite elementary questions, I guess.
We have:
* A principal bundle P(M,G)
* A fibre G_{p} at p= \pi(u)
Then the vertical subspace V_{u}P is defined as a...
Homework Statement
Mrs. K has just received a large shipment of candy at the local Blockbuster. Wanting to get out of the store as fast as possible (so that she can return to her loving husband) she uses the shrink wrap machine to bundle the candy into two blocks, one of 8 kg and the other...
As far as I know, principal bundle is a fiber bundle with a fiber beeing a principal homogeneous space (or a topological group). According this definition vector bundle is a special principal bundle, because vector space with vector addition as group operation is a topological group.
But I feel...