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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeSep 15th 2013
• (edited Sep 15th 2013)

Started an entry in “category:motivation” on fiber bundles in physics.

(prompted by this Physics.SE question)

• CommentRowNumber2.
• CommentAuthorjim_stasheff
• CommentTimeSep 16th 2013
Imho, I think it is a disservice to say a bundle IS a field.
Bundles are the global structure in whihc physical fields live.

Yang-Mills hence E&M? Dirac's monopole as one of the earliest examples existing in a bundle, i.e. the Hopf fibration, seems to get short shrift.

Anomalies from non-trivializability might well have a reference to the Gribov ambiguity.
• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeSep 16th 2013
• (edited Sep 16th 2013)

Imho, I think it is a disservice to say a bundle IS a field.

It is a basic fact that a gauge field is a bundle equipped with a connection. For discrete gauge theory the connection disappears and then a gauge field is precisely a bundle. This fact was very much amplified back then by Freed-Quinn 93 (if it takes appeal to third party auhority) Since it is a true fact, I can’t see why it would be a disservice in a scientific context to state it.

Bundles are the global structure in whihc physical fields live.

Not gauge fields. Gauge fields are sections of gerbes. I know that this is not widely advertized fact (due to the focus on perturbation theory), but it is easily seen to be true.

• CommentRowNumber4.
• CommentAuthorjim_stasheff
• CommentTimeSep 16th 2013
It is a basic fact that a gauge field is a bundle equipped with a connection.

THEN no distinction between different sections of the same bundle with connection?
that is, they correspond to just one gauge field?
• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeSep 16th 2013
• (edited Sep 16th 2013)

Hey Jim,

let’s get on the same page here, there are two levels of bundles here:

1. a $G$-gauge field on some space $X$ is itself a $G$-principal bundle over $X$ equipped with $G$-principal connection;

A gauge transformation is an isomorphism of such $G$-principal bundles with $G$-principal connections over $X$.

2. Now what is it that a gauge field over $X$ is itself a section of? This must be some “second order” bundle over $X$, namely something such that a section of this second order bundle is itself an ordinary bundle (with connection).

Such “second order bundles” which are such that its sections are themselves ordinary $G$-principal bundles are called $G$-gerbes . Or $\mathbf{B}G$-fiber 2-bundles, to have a more conceptual term.

So if we forget the connections for a moment then:

1. the “field 2-bundle” for $G$-gauge fields on $X$ is the 2-bundle $(\mathbf{B}G) \times X \longrightarrow X$, hence the trivial $G$-gerbe on $X$;

2. a section of the “field 2-bundle” is equivalently a map $X \longrightarrow \mathbf{B}G$ and that is equivalently an ordinary $G$-principal bundle on $X$.

• CommentRowNumber6.
• CommentAuthorjim_stasheff
• CommentTimeSep 16th 2013
a G-gauge field on some space X is itself a G-principal bundle over X equipped with G-principal connection

I'm lost already - granted that, I would go on without protest

how do you write a gauge field in an action?
the simple example of the gauge field' for the Dirac monopole might could help
• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeSep 16th 2013

a G-gauge field on some space X is itself a G-principal bundle over X equipped with G-principal connection

I’m lost already - granted that, I would go on without protest

But yes, that’s what it is. What physicists call $G$-gauge fields are mathematically $G$-principal connections. What they call the “instanton sector” of the gauge field is mathematically the class of the underlying $G$-principal bundle.

how do you write a gauge field in an action?

For $\nabla$ a $G$-gauge field for $G$ a semisimple Lie group with Killing form $\langle -,-\rangle$, the canonical action functional on it (the “Yang-Mills action functional”) is the integral over Riemannian base space of the curvature 2-form wedged with its Hodge dual

$\nabla \mapsto \exp\left( \frac{i}{\hbar} \int_X \left\langle F_\nabla \wedge \ast F_\nabla\right\rangle \right) \,.$

Moreover, for $\gamma \colon \S^1 \longrightarrow X$ a path in $X$, thought of as the trajectory of a charged particle, then the interaction functional is the holonomy of the connection along this path

$(\gamma, \nabla) \mapsto hol_\nabla(\gamma) \,.$
• CommentRowNumber8.
• CommentAuthorjim_stasheff
• CommentTimeSep 17th 2013
OH `What physicists call G-gauge fields are mathematically G-principal connections. What they call the “instanton sector” of the gauge field is mathematically the class of the underlying G-principal bundle.''

Perhaps my problem is linguistic - a connection is a field - fine
but a connection IS a bundle with a connection??

even the “Yang-Mills action functional” is defined on the space of connections
• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeSep 17th 2013
• (edited Sep 17th 2013)

Not sure, Jim, where we are miscommunicating. A $G$-gauge field is a $G$-principal connection and every $G$-principal connection has an underlying $G$-principal bundle. That’s all. It’s a standard fact. (You have co-authored various articles involving this fact, if I may say so.)

• CommentRowNumber10.
• CommentAuthorjim_stasheff
• CommentTimeSep 17th 2013
As a theologian once said, the problem is with the word IS.

I agree:
A G-gauge field is a G-principal connection and every G-principal connection has an underlying G-principal bundle.

so I'm happy with
A G-gauge field is a G-principal connection ON a G-principal bundle.
but to say

A G-gauge field is G-principal bundle with a G-principal connection.

distorts my perspective.

Anyone else see my point?
• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeSep 17th 2013

Jim, if you allow then I’ll drop out of this discussion. If you go back through our exchange I think it is hard (or else unfair) to argue that I stated anything which is not standard.

• CommentRowNumber12.
• CommentAuthorjim_stasheff
• CommentTimeSep 18th 2013
Yes,let us drop the discussion since what is standard in one culture may not be in another.
• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeMay 12th 2014
• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeMar 3rd 2019

• Adam Marsh, Gauge Theories and Fiber Bundles: Definitions, Pictures, and Results (arXiv:1607.03089)

(updated on the arXiv this week) here and elsewhere

• CommentRowNumber15.
• CommentAuthorUrs
• CommentTimeOct 17th 2020

added some doi-s to the references and brought them into chronological order

• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeOct 19th 2020

Will give this book a page of its own…

• CommentRowNumber17.
• CommentAuthorUrs
• CommentTimeOct 19th 2020