Curvature forms and Riemannian curvatures of connections

[RockyMarciano]

I'm trying to think of the curvature form of a connection on a tangent frame pricipal bundle as an alternative description of the Riemannian curvature of the connection(see i.e. https://en.wikipedia.org/wiki/Curvature_form)

One thing I want to confirm is does a non-vanishing curvature-form imply loss of the information about the vector direction when parallel transporting it like in the Riemannian curvature case?

 

[Ben Niehoff]

One thing I want to confirm is does a non-vanishing curvature-form imply loss of the information about the vector direction when parallel transporting it like in the Riemannian curvature case?

The curvature 2-form contains exactly the same information as the Riemann tensor; it's just a change of basis:

$$\Omega^a{}_b = \frac12 e_\mu{}^a e^\nu{}_b \, R^\mu{}_{\nu\rho\sigma} \, dx^\rho \wedge dx^\sigma$$
I'm not sure what you mean by "loss of information about vector direction when parallel transporting", though.

 

[RockyMarciano]

The curvature 2-form contains exactly the same information as the Riemann tensor; it's just a change of basis:

$$\Omega^a{}_b = \frac12 e_\mu{}^a e^\nu{}_b \, R^\mu{}_{\nu\rho\sigma} \, dx^\rho \wedge dx^\sigma$$
I'm not sure what you mean by "loss of information about vector direction when parallel transporting", though.

I mean the usual path dependence of tangent vectors in the presence of curvature.

 

[Ben Niehoff]

I mean the usual path dependence of tangent vectors in the presence of curvature.

I wouldn't call that "loss of information". And yes, of course it's still there.

 

[RockyMarciano]

I wouldn't call that "loss of information". And yes, of course it's still there.

Ok, thanks.
To give you more context about what I have in mind. I'm thinking about curvature forms of connections in principal G-bundles. In order to keep this correspondence with Riemann curvature in the general case, if we are modelling a theory with gauge invariance and we need path independence so that we can define a gauge covariant derivative, we must impose a trivial principal bundle and a base manifold with topology compatible with this triviality, right?

And in this special context of a trivial bundle the curvature form and path independence is just a special case of the Riemannian path dependent more general case in the same way that a (pseudo)Riemannian manifold with flat curvature preserves path independence trivially as a special case of general Riemannian curvature.

So for example in the specific physical case of Yang-Mills fields we have a connection with curvature due to the structure group being non-abelian, and also path independence as it is a gauge theory, the pricipal bundle is trivial and the base manifold is Minkowskian with contractible topology and all works fine.
But in a more general mathematical case where we didn't demand bundle triviality(therefore no gauge invariance) and a specific base manifold compatible with it in general we wouldnt' have both path independence and non-vanishing curvature (just from the information of the formula in #2), correct?

 

[Ben Niehoff]

Ok, thanks.
To give you more context about what I have in mind. I'm thinking about curvature forms of connections in principal G-bundles. In order to keep this correspondence with Riemann curvature in the general case, if we are modelling a theory with gauge invariance and we need path independence so that we can define a gauge covariant derivative, we must impose a trivial principal bundle and a base manifold with topology compatible with this triviality, right?

But gauge connections are path-dependent! Unless your gauge group is Abelian.

There is no need for your gauge bundle to be trivial; in fact the instanton number (i.e. second Chern class) computes its non-triviality.

And in this special context of a trivial bundle the curvature form and path independence is just a special case of the Riemannian path dependent more general case in the same way that a (pseudo)Riemannian manifold with flat curvature preserves path independence trivially as a special case of general Riemannian curvature.

Except that they're really the same thing. The Riemann tensor is the field strength of an $SO(p,q)$-connection on the tangent bundle.

So for example in the specific physical case of Yang-Mills fields we have a connection with curvature due to the structure group being non-abelian, and also path independence as it is a gauge theory, the pricipal bundle is trivial and the base manifold is Minkowskian with contractible topology and all works fine.
But in a more general mathematical case where we didn't demand bundle triviality(therefore no gauge invariance) and a specific base manifold compatible with it in general we wouldnt' have both path independence and non-vanishing curvature (just from the information of the formula in #2), correct?

Again, I'm not sure why you think gauge invariance has to do with path-dependence.

I should point out that the "charge" carried by a particle charged under a non-Abelian gauge group is in fact not gauge-invariant, but merely gauge-covariant.

 

[RockyMarciano]

I think we are basically on the same pagen as far as I can see from your replies. It's just terminology.

But gauge connections are path-dependent! Unless your gauge group is Abelian.

True, the connection in Yang-Mills is non-trivial in this sense because the structure group G is not-commutative, I'm using the path independent notion in the sense that you refer to covariance of the gauge above. What I was trying to convey by path independence is the covariance of the gauge covariant derivative, that is able to parallel transport not only information about the change in the field due to translation of the vector from one point to another, but is also able to keep track of the rotation of the local coordinate axes on each point of the base manifold(oriented differently on each point in a non-commutative way), and it can do these because the actual vector is pointing actually at the same point if we abstract it of the particular local coordinates at each point, due to the actual flatness of Minkowski space, and the triviality of the principal bundle that has it as base manifold. See below for more on what I meant by path independence/dependence in relation with gauge covariance.

There is no need for your gauge bundle to be trivial; in fact the instanton number (i.e. second Chern class) computes its non-triviality.

Well, yes with qualifiers that go beyond the level I'm keeping this so far. This requires Euclidean spacetime, the non-perturbative regime, restriction to a point in spacetime... I know they use this in QCD, but this discussion is more in the perturbative side(although I'd be interested in maybe starting another thread to talk about this if I have the chance)

Except that they're really the same thing. The Riemann tensor is the field strength of an $SO(p,q)$-connection on the tangent bundle.

Yes, I know what you mean but let me explain what I meant: This "field strength" measures the deviation from flatness in the orientation of the vector being transported(that would have kept pointing in the same direction if the manifold had been flat).

In the Yang-Mills case we are keeping track not only of this but of the change in the field as it is trasported from point to point. And we have the same number of degrees of freedom since the gauge covariance is about redundance of degrees of freedom, so we achieve it by having a flat manifold and a trivial bundle.

We would achieve the equivalent thing in the simple Riemannian manifold setting by choosing a flat manifold that allows as to preserve both lengths and angles(path independence), not just lengths or just angles like in the general case in the presence of curvature.

It is just that the more flexible structure of the principal bundle let's us switch between connections and gauge groups G, like abelian U(1) that has vanishing curvature, or SU(n) that is non-abelian and therefore has non-vanishing curvature form and keep gauge covariance thanks to the flatness of the base manifold and the triviality of the bundle.

 

[RockyMarciano]

For reference on the distinction between the usual triviality imposed by the Minkowski space' homotopy, as discussed for instance in Nakahara, and the more subtle notion of nontriviality and instantons see the answers here: http://physics.stackexchange.com/qu...potential-and-principal-bundles/291105#291105

 

[Ben Niehoff]

Sorry, I still have no idea what you're trying to get at with your use of "path dependence". I would recommend you stick to standard terminology.

Gauge connections parallel-propagate vectors in an "internal" vector space in exactly the same way that the Levi-Civita connection parallel-propagates vectors in the tangent space. There is no fundamental difference between the two concepts.

I don't believe we are on the same page at all, because you're continuing to say things that are obviously false:

True, the connection in Yang-Mills is non-trivial in this sense because the structure group G is not-commutative, I'm using the path independent notion in the sense that you refer to covariance of the gauge above. What I was trying to convey by path independence is the covariance of the gauge covariant derivative, that is able to parallel transport not only information about the change in the field due to translation of the vector from one point to another, but is also able to keep track of the rotation of the local coordinate axes on each point of the base manifold(oriented differently on each point in a non-commutative way), and it can do these because the actual vector is pointing actually at the same point if we abstract it of the particular local coordinates at each point, due to the actual flatness of Minkowski space, and the triviality of the principal bundle that has it as base manifold. See below for more on what I meant by path independence/dependence in relation with gauge covariance.

Gauge bundles do not have to be trivial, and gauge invariance does not require that the base manifold be flat Minkowski!

In the Yang-Mills case we are keeping track not only of this but of the change in the field as it is trasported from point to point. And we have the same number of degrees of freedom since the gauge covariance is about redundance of degrees of freedom, so we achieve it by having a flat manifold and a trivial bundle.

Achieve what? You're being very unclear here.

We would achieve the equivalent thing in the simple Riemannian manifold setting by choosing a flat manifold that allows as to preserve both lengths and angles(path independence), not just lengths or just angles like in the general case in the presence of curvature.

OK, you need to drop this terminology "path independence" because you're using it in a totally nonsensical way.

Also, ALL metric-compatible connections (of which the Levi-Civita connection is one) preserve both lengths AND angles. That's what metric-compatible means.

It is just that the more flexible structure of the principal bundle let's us switch between connections and gauge groups G, like abelian U(1) that has vanishing curvature, or SU(n) that is non-abelian and therefore has non-vanishing curvature form and keep gauge covariance thanks to the flatness of the base manifold and the triviality of the bundle.

Of course a U(1) gauge bundle can have curvature! That's what electromagnetism is!

Also gauge covariance has absolutely nothing to do with flatness of the base or triviality of the gauge bundle.

 

[RockyMarciano]

Sorry, I still have no idea what you're trying to get at with your use of "path dependence". I would recommend you stick to standard terminology.

Gauge connections parallel-propagate vectors in an "internal" vector space in exactly the same way that the Levi-Civita connection parallel-propagates vectors in the tangent space. There is no fundamental difference between the two concepts.

I don't believe we are on the same page at all, because you're continuing to say things that are obviously false:

Fair enough, I actually agree with what you explain here so I'm obviously doing a bad job trying to get across my point. Let me try and clarify once more. Also please be a bit indulgent with me here:I have received certain physics instruction long ago but I'm basically trying to express complex things from a layman point of view, and it is not easy at all. I'm here to correct my misunderstandings relying on people that is professionally dedicated to this like may be your case so the last thing I'd want is to appear as argumentative.

Gauge bundles do not have to be trivial, and gauge invariance does not require that the base manifold be flat Minkowski!

You are of course right, actually, for instance, global(spacetime) anomalies in QFT are examples of nontrivializations. But let me give you the context in which I refer to trivial bundles in physics(wich I also tried earlier with the reference to the stack.exchange page) and please let me know if I'm misunderstanding it. I'm referring only to QFT, and the standard model of particle physics, and I understand this gauge QFT does use flat minkowski space as base manifold. If this were so(please correct it if this is not so), there is a theorem that says: "Given a fibre bundle with base space X and structure group G, if either X or G is contractible then the bundle is trivial." Since Minkowski space is contractible I understand that the bundle is trivial, but I'm open to be corrected if I'm wrong following this reasoning.

Also, ALL metric-compatible connections (of which the Levi-Civita connection is one) preserve both lengths AND angles. That's what metric-compatible means.

EDITED: I originally wrote something confusing and that deviates from the main discussion.
Yes, the inner product is preserved by the connectio along the specific curve, and depending on the specific path. But only line lengths are preserved independent of curve, position or direction(path).
The distinction I was doing was because in flat space the inner product is preserved independently of path.I was referring to what can be preserved independent of path.
I think an important distinction here between the Riemannian covariant derivative and the gauge covariant derivative is that the connection in the first case is not a tensorial object on the manifold, the Christoffel symbols are not tensors, the curvature is tensorial; while the pricipal connection used in gauge theory is an Ehresmann connection and is constructed as a tensorial object in the bundle(in the total space instead of in the manifold), and in this sense it is frame independent already from the parallel transport moment, this was the difference I was trying to stress when referring to path dependence/independence, is this any clearer?

.

Of course a U(1) gauge bundle can have curvature! That's what electromagnetism is!

Yes, what I meant was that in the U(1) case being abelian there is no contribution to the curvature from non-commutativity, as it happens for Yang-Mills fileds. So the curvature is computed only with the term corresponding to the exterior derivative of the connection $dA$.

Also gauge covariance has absolutely nothing to do with flatness of the base or triviality of the gauge bundle.

Right. I was again referring to the special case that doesn't enter into nontrivialities of the bundle.

But I'd rather wait till I get some feed-back before I attempt to explain again the link I see and how it relates to path dependence/independence in order to avoid saying wrong things if it turns out I'm actually misunderstanding the basics.

Thanks for the assistance.

 

[RockyMarciano]

Gauge bundles do not have to be trivial,

Ok, I think I see what you mean now. Gauge principal bundles are by definition local(local gauge), and they can of course be nontrivial like for instantons.
When I was thinking of trivial bundle I was going unconsciously to the global case, whis is the global symmetry of spacetime rather than a trivial bundle.

and gauge invariance does not require that the base manifold be flat Minkowski!

Ok, it's just that I switched from the local gauge to the global gauge(global symmetries) unconsciously, of course "global nontrivializations"(the ones I was mistakenly thinking of when speaking about nontrivial bundles) are not allowed in physics as long as we are only considering Minkowski spacetime global symetry.

Do you agree with this?

 

[RockyMarciano]

Gauge connections parallel-propagate vectors in an "internal" vector space in exactly the same way that the Levi-Civita connection parallel-propagates vectors in the tangent space. There is no fundamental difference between the two concepts.

And yet, let's consider the EM gauge connection U(1), unlike the Yang-Mills case, local changes in phase (infinitesimal rotations in the "internal" space) won't affect the "curvature" you referred to above. Compare with the LC connection in the tangent space of a Riemannian manifold, an infinitesimal loop of the vector will not leave it pointing in the original direction in general and this will be measured as curvature. So i would say the quoted claim is not exact in general for any gauge connection.

 

[Ben Niehoff]

And yet, let's consider the EM gauge connection U(1), unlike the Yang-Mills case, local changes in phase (infinitesimal rotations in the "internal" space) won't affect the "curvature" you referred to above. Compare with the LC connection in the tangent space of a Riemannian manifold, an infinitesimal loop of the vector will not leave it pointing in the original direction in general and this will be measured as curvature. So i would say the quoted claim is not exact in general for any gauge connection.

The Aharonov-Bohm effect is precisely a loop around which a U(1) phase fails to end up where it started.

 

[RockyMarciano]

The Aharonov-Bohm effect is precisely a loop around which a U(1) phase fails to end up where it started.

I think we have different concepts about what "exactly the same way" and "no fundamental difference" mean.
I was counting that particular type of holonomy as a further difference, whereas the connections in internal space can make compatible curvatures (as field strengths that you referred to earlier) of the connection and holonomy as a type of monodromy(A-B effect, looping around singularities) because all those curvatures are in the context of the flat connection in flat Minkowski base space, the LC connection on tangent spaces can't: a curved connection's holonomy in a curved manifold is never a monodromy as being a curved connection is not compatible with being a flat connection in the tangent space for a given metric, and only the latter have associated monodromy as their holonomy.

 

[Ben Niehoff]

I think we have different concepts about what "exactly the same way" and "no fundamental difference" mean.
I was counting that particular type of holonomy as a further difference, whereas the connections in internal space can make compatible curvatures (as field strengths that you referred to earlier) of the connection and holonomy as a type of monodromy(A-B effect, looping around singularities) because all those curvatures are in the context of the flat connection in flat Minkowski base space, the LC connection on tangent spaces can't: a curved connection's holonomy in a curved manifold is never a monodromy as being a curved connection is not compatible with being a flat connection in the tangent space for a given metric, and only the latter have associated monodromy as their holonomy.

No, this is still not a distinction. If I parallel-transport a vector around the tip of a cone, even though my connection is locally flat, I still get a holonomy.

You don't even need the cone to be singular. Just cut off the tip and attach a spherical cap, and everything is smooth.

Furthermore, the Aharonov-Bohm effect is just a special case. The Maxwell field strength F is telling you about infinitesimal phase differences obtained by going around infinitesimal loops, just as the curvature 2-form is telling you about tangent-space vectors.

 

[RockyMarciano]

No, this is still not a distinction. If I parallel-transport a vector around the tip of a cone, even though my connection is locally flat, I still get a holonomy.

You are missing the distinction I was drawing. This holonomy is a global holonomy, a monodromy, that is the holonomy associated to flat connections. My point was that a Riemann manifold with a curved metric(so the flat cone example is irrelevant here) has a curved LC connection, and doesn't have any notion of monodromy on its tangent space, while the structure of a principal bundle admits all these different concepts of holonomy even if its base manifold is the same flat Minkowski space.

Furthermore, the Aharonov-Bohm effect is just a special case. The Maxwell field strength F is telling you about infinitesimal phase differences obtained by going around infinitesimal loops, just as the curvature 2-form is telling you about tangent-space vectors.

The Maxwell field strength, or curvature of the U(1) principal connection is telling you about the horizontal subspace, how the fiber U(1) changes around loops, not how the phase at a particular point(vertically) changes, phase changes at a point are gauge transformation that leave the physics unaltered.
For the manifold at the tangent space the curvature of the LC connection does tell you at that particular point and vertical subspace how the vector fails to recover its original orientation and this brings it closer to the Yang-Mills field curvature, that has an additional term coming from the fiber's noncommutativity already as a nontrivial property.

 

[Ben Niehoff]

You are missing the distinction I was drawing. This holonomy is a global holonomy, a monodromy, that is the holonomy associated to flat connections. My point was that a Riemann manifold with a curved metric(so the flat cone example is irrelevant here) has a curved LC connection, and doesn't have any notion of monodromy on its tangent space, while the structure of a principal bundle admits all these different concepts of holonomy even if its base manifold is the same flat Minkowski space.

I am telling you that both LC connections and gauge connections can have both types of features! I even gave examples.

The Maxwell field strength, or curvature of the U(1) principal connection is telling you about the horizontal subspace, how the fiber U(1) changes around loops, not how the phase at a particular point(vertically) changes, phase changes at a point are gauge transformation that leave the physics unaltered.

Now you're deliberately missing my point. The Maxwell field strength measures the holonomy of infinitesimal Wilson loops. These are gauge-invariant.

For the manifold at the tangent space the curvature of the LC connection does tell you at that particular point and vertical subspace how the vector fails to recover its original orientation and this brings it closer to the Yang-Mills field curvature, that has an additional term coming from the fiber's noncommutativity already as a nontrivial property.

The Riemann curvature is given by

$$R = d \omega + \omega \wedge \omega$$
The gauge curvature is given by

$$F = dA + A \wedge A$$
They are the same, just change the letters! Literally the only difference is that $F$ lives in a bundle other than the tangent bundle. ALL of the other types of properties you've mentioned (i.e., whether they can have holonomies, monodromies, etc.) are shared by both concepts.

And if you are going to reply to me again with further hangups about U(1) connections being Abelian, then look at LC connections on 2-dimensional manifolds!

Please take some time to work with these things on paper a bit.

 

[RockyMarciano]

At this point it just gets absurd to go on, I don't even know anymore what the discussion is about and it is far from the original question.

 

[lavinia]

Maybe this will help with your original question about the relation of Riemannian curvature to connections on principal bundles.

Via a local section $s$ of the principal bundle (a global section may not exist) over a domain $U$ over which the bundle is trivial, one can pull back the connection 1 form $ω$ to $U$ to get a Lie algebra valued 1 form $s^{*}ω$ on the tangent bundle. In case the principal bundle is the bundle of orthonormal frames (a principal $O(n)$ bundle) $s^{*}ω$ is a skew symmetric matrix of ordinary 1 forms defined on tangent vectors. If $s_{i}$ are the vector fields in this local frame then one can define a connection on the tangent bundle by $∇s_{i} = Σ_{j}\hat{ω_{ij}}⊗s_{j}$. This is the usual formula for a connection on a vector bundle with a Riemannian metric. The covariant derivative with respect to a tangent vector $X$ is then $∇_{X}s_{i} = Σ_{j}\hat{ω_{ij}}(X)s_{j}$.

The pull back of the Lie algebra valued curvature 2 form is the curvature of the connection on $U$. This is a skew symmetric matrix of 2 forms defined on the tangent space to $U$. In terms of the Riemann curvature tensor, this is the matrix of the linear map $R(x,y)$ and the usual skew symmetry $R(x,y)= - R(y,x)$ is just the skew symmetry of the curvature 2 form.

More generally if the principal bundle is a bundle of frames of a vector bundle (possibly a complex vector bundle and not necessarily with a metric) then for any local frame $s_{i}$ the pull back of the connection 1 form gives a connection by the same rule $∇s_{i} = Σ_{j}\hat{ω_{ij}}⊗s_{j}$ where now the $s_{i}$ are local sections of an arbitrary smooth vector bundle. In this general case the matrix $\hat{ω_{ij}}$ may not be skew symmetric since it does not necessarily take values in the Lie algebra of the orthogonal group.

As Ben suggested a good example to work through is a connection on a oriented two dimensional surface with a Riemannian metric. In this case the unit tangent circle bundle is a principal $SO(2)$ bundle. The action of $SO(2)$ on the unit circle bundle is by rotation of fiber circles. Since there is a Riemannian metric the pull back of the curvature 2 form can be written as $Ω= -K$vol where $K$ is the usual Gauss curvature and vol is the volume element of the surface.

 

[RockyMarciano]

Maybe this will help with your original question about the relation of Riemannian curvature to connections on principal bundles.

Thanks for chipping in, Lavinia! Your comments are helpful.

 

[RockyMarciano]

I think the difference I wanted to highlight is that for a finite dimensional Riemannian manifold with a given metric the curvature of its LC connection on the manifold is either constant or variable whlie when we use the principal bundle structure with a total space bundle of infinite dimensions one can have variable or constant curvature for the LC connection on the bundle and the base manifold has a constant curvature flat connection i.e. in the gauge bundle of the standard model QFTs.
The distinction is between the connection defined in the base manifold or in the bundle.
Hope this is ok.

 

[lavinia]

I think the difference I wanted to highlight is that for a finite dimensional Riemannian manifold with a given metric the curvature of its LC connection on the manifold is either constant or variable whlie when we use the principal bundle structure with a total space bundle of infinite dimensions one can have variable or constant curvature for the LC connection on the bundle and the base manifold has a constant curvature flat connection i.e. in the gauge bundle of the standard model QFTs.

I don't know what you mean by this. Can you write out an example in detail? That would be helpful.

- A principal bundle is not infinite dimensional in general. For instance, the bundle of frames is finite dimensional

The distinction is between the connection defined in the base manifold or in the bundle.

It seems that you missed the point of my previous post and also Ben's posts. A connection on the principal bundle of frames determines a connection on the tangent bundle of the manifold and visa vera. They are equivalent. More generally, a connection on an arbitrary principal bundle determines a gauge field on the manifold and visa vera. The field strength of the gauge field is just the pull back of the curvature form of the connection on the principal bundle. The two formulations are equivalent.

- It is true that one can have a bundle that is not the bundle of frames of the tangent bundle of a Riemannian manifold and this bundle may not even be compatible with a metric. In such a case, the bundle can have non-zero curvature while the underlying manifold's tangent bundle is flat. But then you have two different connections.

- I don't know what you mean by "the LC connection on the bundle" . Which bundle? An LC connection is a special connection on a vector bundle with a metric - Riemannian of semi-definite. On the principal bundle it is a connection on the bundle of orthonormal frames of the vector bundle. Many other connections on the bundle of orthonormal frames are possible. The fundamental theorem is that the LC connection is the unique connection that is torsion free.

 

[RockyMarciano]

I don't know what you mean by this. Can you write out an example in detail? That would be helpful.

- A principal bundle is not infinite dimensional in general. For instance, the bundle of frames is finite dimensional

I know, but I'm referring to the special case relevant to the renormalized gauge QFTs of the standard model(and also to the gauge formulation of GR), the principal frame bundles in these theories are infinite dimensional. So what I mean basically is to make a distinction between the finite dimensional case and the more convoluted infinite dimensional one with respect to connections.
One example, the Schur's theorem of Riemannian geometry says that for Riemannian manifolds of finite dimension n >2 the sectional curvature being constant pointwise in the manifold determines that the manifold is of constant curvature. This doesn't seem to carry over to the curvatures in gauge theory, where the base manifold can have constant curvature regardless of the different strength fields(curvature pullbacks).

It seems that you missed the point of my previous post and also Ben's posts. A connection on the principal bundle of frames determines a connection on the tangent bundle of the manifold and visa vera. They are equivalent. More generally, a connection on an arbitrary principal bundle determines a gauge field on the manifold and visa vera. The field strength of the gauge field is just the pull back of the curvature form of the connection on the principal bundle. The two formulations are equivalent.

I'm aware of that equivalence, I'm just saying that in the infinite dimensional case of the physical gauge theories with indefinite metrics things are more complicated. The group reduction to the orthonormal frame of indefinite metric requires infinite dimensional representations. I wouldn't expect a pure mathematician to be acquainted with this more esoteric aspects, I would maybe from someone working on theoretical physics, but it is not always the case it seems judging from the rest of this thread..

- I don't know what you mean by "the LC connection on the bundle" . Which bundle? An LC connection is a special connection on a vector bundle with a metric - Riemannian of semi-definite. On the principal bundle it is a connection on the bundle of orthonormal frames of the vector bundle. Many other connections on the bundle of orthonormal frames are possible. The fundamental theorem is that the LC connection is the unique connection that is torsion free.

I mean on the principal bundle in QFT gauge theory.

 

[martinbn]

That is not what Schur's theorem states. You need to read the statement more carefully.

Can you explain more why do you think those are infinite dimensional?

 

[RockyMarciano]

That is not what Schur's theorem states. You need to read the statement more carefully

Corrected now.

Can you explain more why do you think those are infinite dimensional?

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.

 

[lavinia]

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.

I think a good place to get some details about this is here.

 

[lavinia]

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.

I think a good place to get some details about this is here.

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?

 

[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.

 

[lavinia]

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.

 

[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), which 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.

 

[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]

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