added pointer to:

- Kirill Krasnov, §1.12 in:
*Formulations of General Relativity*, Cambridge Monographs on Mathematical Physics, Cambridge University Press (2020) [doi:10.1017/9781108674652, taster:pdf]

added pointer to the early discussion of

- Tai Tsun Wu, Chen Ning Yang,
*Concept of nonintegrable phase factors and global formulation of gauge fields*, Phys. Rev. D**12**(1975) 3845 [doi:10.1103/PhysRevD.12.3845]

and a scan of their Table I (here)

]]>added publication data to:

- Adam Marsh,
*Gauge Theories and Fiber Bundles: Definitions, Pictures, and Results*(arXiv:1607.03089), chapter 10 in: Adam Marsh,*Mathematics for Physics: An Illustrated Handbook*, World Scientific 2018 (doi:10.1142/10816, book webpage)

added pointer to

- Tohru Eguchi, Peter Gilkey, Andrew Hanson,
*Gravitation, gauge theories and differential geometry*, Physics Reports Volume 66, Issue 6, December 1980, Pages 213-393 (doi:10.1016/0370-1573(80)90130-1)

added pointer to

Will give this book a page of its own…

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

]]>added pointer to the exposition

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

(updated on the arXiv this week) here and elsewhere

]]>I have added a section *Basic Idea of the Definition of Fiber Bundles* prompted by this PO question

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.

]]>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? ]]>

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.)

]]>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 ]]>

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

$(\gamma, \nabla) \mapsto hol_\nabla(\gamma) \,.$ ]]>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 ]]>

Hey Jim,

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

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 $X$.

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:

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 $X$;

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 $X$.

THEN no distinction between different sections of the same bundle with connection?

that is, they correspond to just one gauge field? ]]>

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.

]]>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. ]]>

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

(prompted by this Physics.SE question)

]]>