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Started an entry in “category:motivation” on fiber bundles in physics.
(prompted by this Physics.SE question)
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.
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 $X$.
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
$(\gamma, \nabla) \mapsto hol_\nabla(\gamma) \,.$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.)
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 have added a section Basic Idea of the Definition of Fiber Bundles prompted by this PO question
added pointer to the exposition
(updated on the arXiv this week) here and elsewhere
added pointer to
Will give this book a page of its own…
added pointer to
added publication data to:
added pointer to the early discussion of
and a scan of their Table I (here)
added pointer to:
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