# I Curvature forms and Riemannian curvatures of connections

#### lavinia

Gold Member
Corrected now.
The principal bundles of the local gauge in QFT? I thought that was a well known fact about interacting QFTs. You don't think they are? It has to do with the requirement of locality, with having all interactions invariant under independent choices of local gauge at all the infinite spacetime points.
Can you in precise detail define this principal bundle?

#### RockyMarciano

Can you in precise detail define this principal bundle?
Consider the Yang-Mills field bundle, the space of sections of this bundle is infinite-dimensional because it is considered independently at each spacetime point in the manifold.

#### lavinia

Gold Member
Consider the Yang-Mills field bundle, the space of sections of this bundle is infinite-dimensional because it is considered independently at each spacetime point in the manifold.

But the space of sections of any bundle is infinite dimensional. You were saying that the principal bundle is infinite dimensional.

Differences of gauge fields may be viewed as 1 forms with values the adjoint bundle - the bundle where the representation of the structure group is the adjoint representation. This representation is finite dimensional as is the structure group. The space of 1 forms is infinite dimensional as is the space of connections. The space of connections is affine since a convex combination of two connections is another connection.

For the principal bundle to be infinite dimensional, the structure group of the bundle must be an infinite dimensional Lie group. What group did you have in mind?

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#### RockyMarciano

As a disclaimer to what I saying I'm referring below to the global, non-perturbative case where everything is mathematically a bit nebulous. In the perturbative local case, i.e. locally at each spacetime point and order by order, where the great computational precision in predictions is reached, things are mathematically different and the appropriate object is the finite dimensional jet bundle , with dimension n determined by the finite order of field derivatives of the Lagrangian density one considers, that is itself depending on the perturbative order one wishes to work with. This follows from Noether's second theorem.

But the space of sections of any bundle is infinite dimensional. You were saying that the principal bundle is infinite dimensional.
Yes, but in physics this space of sections is itself a principle bundle with physical relevance(choice of gauge, nontrivial local sections). Take a look at page 21 and following of the notes I linked. Where they talk about the Gribov ambiguity. And also the talk section in wikipedia about the Gribov ambiguity.
Differences of gauge fields may be viewed as 1 forms with values the adjoint bundle - the bundle where the representation of the structure group is the adjoint representation. This representation is finite dimensional as is the structure group. The space of 1 forms is infinite dimensional as is the space of connections. The space of connections is affine since a convex combination of two connections is another connection.
For the principal bundle to be infinite dimensional, the structure group of the bundle must be an infinite dimensional Lie group. What group did you have in mind?
Yes, you are absolutely right in the usual mathematical setting. But the thing is that in physics in this particular nonperturbative setting there is a somewhat different standard of rigor, here we should be referring to infinite dimensional Lie algebras rather than groups, due to the Gribov ambiguity mentioned above, and that is what is done in the perturbative case. But usually in the formal presentation it is assumed that no ambiguity exists and therefore a finite dimensional bundle with finite dimensional fibre is possible.

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#### lavinia

Gold Member
There is nothing in the link about the Gribov ambiguity that talks about infinite dimensional Lie groups.

I can see from the link on Noether's theorem that there may be an infinite dimensional Lie algebra in what you are talking about. But where is the principal bundle and where is the connection?

I agree with you that this thread is pointless. There is too much vagueness to have a clarification of your questions.

Also you have diverged completely from you original question which itself was vague.

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#### RockyMarciano

There is nothing in the link about the Gribov ambiguity that talks about infinite dimensional Lie groups.
Did you check the notes, where it says: "A is itself an infinite dimensional principle bundle over the space B := A/G where the group G of all gauge transformations plays the role of the structure group."
EDIT: I edited my previous post.
I can see from the link on Noether's theorem that there may be an infinite dimensional Lie algebra in what you are talking about. But where is the principal bundle and where is the connection?
The Noether theorem refers to the perturbative approach.

#### RockyMarciano

I agree with you that this thread is pointless. There is too much vagueness to have a clarification of your questions.

Also you have diverged completely from you origin question which itself was vague.
I think we can clarify some of the vagueness(either here or in a new thread), but I agree there is some confusion coming basically from not distinguishing the local form the global aspects, and these are key when talking about Yang-Mills gauge fields in terms of principal (and associated) bundles.

This confusion was patent also in the previous discussion with Ben Niehoff, where he was referring to the non-triviality of the local gauge bundle, while I was thinking about the triviality of the gobal gauge group bundle determined by the base manifold being contractible; unless one noticed and warned about this confusion the conflict was served.

It seems we are now again falling for the same mistake, except you are now thinking about the global bundle and group(the global gauge symetry that is hopefully recovered in the end), wich is of course finite dimensional, while I was now speaking instead about the local gauge group(as in the local gauge groups in the standard model of particle physics $(U(1)×SU(2)×SU(3))$) that is infinite-dimensional(due to the problem with fixing choices of gauge groups as explained in wikipedia:
"In gauge theory, especially in non-abelian gauge theories, global problems at gauge fixing are often encountered. Gauge fixing means choosing a representative from each gauge orbit, that is, choosing a section of a fiber bundle. The space of representatives is a submanifold (of the bundle as a whole) and represents the gauge fixing condition. Ideally, every gauge orbit will intersect this submanifold once and only once. Unfortunately, this is often impossible globally for non-abelian gauge theories because of topological obstructions and the best that can be done is make this condition true locally. A gauge fixing submanifold may not intersect a gauge orbit at all or it may intersect it more than once. The difficulty arises because the gauge fixing condition is usually specified as a differential equation of some sort, e.g. that a divergence vanish (as in the Landau or Lorenz gauge). The solutions to this equation may end up specifying multiple sections, or perhaps none at all. This is called a Gribov ambiguity (named after Vladimir Gribov).
Gribov ambiguities lead to a nonperturbative failure of the BRST symmetry, among other things.."

In the absence of this issue, that is if the situation was ideal and there were no obstructions of course the space of local sections being infinite dimensional as usual wouldn't determine an infinite dimensional group as you explained in #28 and as it is supposed to happen when recovering the global gauge symmetry case(only there is no mathematically rigurous formulation of it for the 4 dimensional case so far, but it is routinely assumed that it exists).

There is a recent thread where this global, local distinction came up and I think the posts #13, #15 and #17 by samalkhaiat referring to the infinite dimesnional local gauge groups are relevant . Please take a look and maybe let me know if things are a bit less vague after this.

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#### RockyMarciano

Now for the relevance to my original question, it lies also in this local-infinite dimensional/global-finite dimensional difference, both the Yang-Mills non-abelian connections(that when quantized correspond to the gauge bosons) and their physical field strengths belong in the local gauge infinite-dimensional setting, while the curvatures in the usual Riemannian manifolds are always in the usual mathematical setting where the ambiguity from physics commented above doesn't come up, and we only have the finite dimensional associated bundles and their equivalence between principal and tangent bundle always holds(as is also expected in the physical theory when the global gauge is recovered even if it hasn't been formalized completely in 4 dimensions). I wasn't really asking about this equivalence that both Lavinia and Ben Niehoff thought I was questioning, but about the distinction introduced in physics by the infinite dimensional local gauge groups.

Now, admittedly most mathematicians are not aware of it because they don't work with it and don't need it, hopefully most theoretical physicists are, I'm not sure.

#### Ben Niehoff

Gold Member
The gauge groups of the standard model require a principle bundle with a fiber of no more than 7 dimensions. Conveniently just enough to fit into 11-dimensional M theory. The space $\mathbb{CP}^2 \times S^3$ has isometry group $SU(3) \times SO(4)$; one can then break some of the symmetry of the $S^3$ to break the $SO(4)$ to $SU(2) \times U(1)$.

#### RockyMarciano

The gauge groups of the standard model require a principle bundle with a fiber of no more than 7 dimensions. Conveniently just enough to fit into 11-dimensional M theory. The space $\mathbb{CP}^2 \times S^3$ has isometry group $SU(3) \times SO(4)$; one can then break some of the symmetry of the $S^3$ to break the $SO(4)$ to $SU(2) \times U(1)$.
Where did you get the number seven? I count 12 dimensions in the SM global principal bundle's fibre, 1+3+8=12

#### Ben Niehoff

Gold Member
Where did you get the number seven? I count 12 dimensions in the SM global principal bundle's fibre, 1+3+8=12
I literally just showed you how to get 7 in the very post you are quoting!

#### RockyMarciano

I literally just showed you how to get 7 in the very post you are quoting!
No, what I am asking is where did you get the notion that only seven dimensions are required? required by whom?

#### Ben Niehoff

Gold Member
No, what I am asking is where did you get the notion that only seven dimensions are required? required by whom?
Required in order to get the Standard Model gauge group by KK reduction.

#### RockyMarciano

What on earth has the Kaluza-Klein dimensional reduction to do with anything in this thread is something I would like to know.

#### Ben Niehoff

Gold Member
It's a way to get gauge groups from bundles. Perhaps "principal bundle" is the wrong choice of words. But the point is, if the isometry group is $G$, then you have vector fields available that generate $G$. Call those vector fields $X_i$. Now you can easily define some connection one-forms

$$A_\mu \equiv A_\mu^i \, X_i$$
on the base manifold. So, a small displacement in the base results in some action on the fiber by isometries.

#### RockyMarciano

It's a way to get gauge groups from bundles.
Looks somewhat disconnected from the theme we were discussing and glossing over my posts #32 and #33, so I take you don't disagree with what's stated there.

#### Ben Niehoff

Gold Member
Most of this thread is pretty disconnected from whatever you originally seemed to be asking about. You still haven't explained clearly.

#### RockyMarciano

That is mostly true, basically because when I asked I wasn't fully aware of the possible answer(that's why I asked), and admittedly it is hard from my original question to get to the answer, so I understand to a great extent the puzzlement in this thread.

So actually my question was about a difference I perceived about Yang-Mills connections and their curvatures and connections and curvatures in the usual mathematical setting of Riemannian geometry. Now I understand that the difference I perceived came from the physics of Yang-Mills fields specified by something called Gribov ambiguities that acts locally. I know this issue is basically ignored nowadays and in general is not considered of much importance when assuming the principle of global gauge redundancy so the local/global distinction is not stressed much either, certainly not in the usual QFT textbooks.

Mathematically the idea is that Yang-Mills fields must have the same information about the bundle as any local trivialization of the bundle, but in the physical non-abelian case with local gauge fixing, due to the absence of global section(that leads to the nontrivial bundle you spoke about at the beginning of the thread) there is a topological obstruction to this that gives rise to the Gribov ambiguity, but this isn't supposed to affect the global trivial bundle gauge groups, as recovering in the end the gauge redundancy is basic for the consistence of the theory, so somehow it must be still true that the Yang-Mills fields must have the same information about the bundle as the local trivialization..

You still haven't explained clearly.
Point to anything(or the whole thing) you see as unclear or wrong in this post, that'd be helpful.

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#### Ben Niehoff

Gold Member
Point to anything(or the whole thing) you see as unclear or wrong in this post, that'd be helpful.
Your entire post is unclear, basically. Pretty much all of them. Your writing is full of run-on sentences which seem grammatically incomplete and can't seem to make a definite point. I agree with the suggestion that you should provide a concrete example of what you think the problem is.

#### RockyMarciano

Your entire post is unclear, basically. Pretty much all of them. Your writing is full of run-on sentences which seem grammatically incomplete and can't seem to make a definite point. I agree with the suggestion that you should provide a concrete example of what you think the problem is.
Thanks for the feed-back. So I'll try simple questions: Do you know what a Gribov ambiguity is? Do you agree with samalkhaiat that local gauge groups are infinite-dimensional? Do you agree that the global gauge bundle is trivial?

#### Ben Niehoff

Gold Member
Thanks for the feed-back. So I'll try simple questions: Do you know what a Gribov ambiguity is? Do you agree with samalkhaiat that local gauge groups are infinite-dimensional? Do you agree that the global gauge bundle is trivial?
I have some idea what a Gribov ambiguity is. Samalkhaiat has not participated in this thread. I'm not sure "local" and "global" mean what you think they mean. Also, you've posted this thread under "Differential Geometry", so I am not sure why you are surprised to get answers in terms of differential geometry rather than quantum field theory.

#### RockyMarciano

I have some idea what a Gribov ambiguity is.
Great, then you know the answer to my original question.

Samalkhaiat has not participated in this thread.
I gave a link in #32, surely you can read it, referencing information from outside a thread is routinely done.
I'm not sure "local" and "global" mean what you think they mean.
I give them the meaning of local as used in the usual "local gauge" concept, and global as in the usual global symmetry.

Also, you've posted this thread under "Differential Geometry", so I am not sure why you are surprised to get answers in terms of differential geometry rather than quantum field theory.
I'm not surprised by that, I'm a bit surprised by your passive-agresive attitude in this thread,but just because I don't know why any sane person would do that in a science forum, I'm that naive. Why would you come back to the thread out of the blue with some totally unrelated stringy comment that according to your strict view is not differential geometry proper? Usually people give answers in these foums to offer help(like Lavinia for example), otherwise they don't participate.

#### Ben Niehoff

Gold Member
I'm not surprised by that, I'm a bit surprised by your passive-agresive attitude in this thread,but just because I don't know why any sane person would do that in a science forum, I'm that naive. Why would you come back to the thread out of the blue with some totally unrelated stringy comment that according to your strict view is not differential geometry proper? Usually people give answers in these foums to offer help(like Lavinia for example), otherwise they don't participate.
You have been quite rude to those offering you help in this thread, both by your attitude and by your refusal to explain what you are trying to ask about.

#### RockyMarciano

Certainly you haven't been offering help in the last part of the thread. I have not been rude at all with the other two people that did offer help in this thread, I value highly their help.

#### RockyMarciano

I've explained it several times, the last one your answer was that you didn't understand my english, curiously in my last post you didn't have that complain about my grammar and the point I was making.

"Curvature forms and Riemannian curvatures of connections"

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