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From Wikipedia:

Part of the table in https://en.wikipedia.org/wiki/Gauge_theory_(mathematics)

In here, $G$-principal bundle $P$ on spacetime $\Sigma$ is given. Gauge transformation $\phi$ can be written in three ways, and it seems to use the third way. More specifically, \begin{align} \phi : \Sigma \to \mathrm{Ad}(P) \end{align} (Why is it the third? I refer this : Global vs. local gauge group in mathematical sense - physics examples?) where $\mathrm{Ad}(P)$ is defined as \begin{align} \mathrm{Ad}(P) = P \times^\mathrm{Ad}_G G = P \times G/\sim\\ (p, g) \sim (ph, h^{-1}gh) \end{align}

On the table, it said "global gauge symmetry" is covariantly constant with respect to the connection. My question is what connection is considered.

I have difficulty in connecting two concepts. The simplest $U(1)$-gauge theory E.M. on $\mathbb{R}^4$ case is too simple to understand how the connection is interwining in here.

Could you help me to understand this with some examples?

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2 Answers 2

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This is not quite right.

Given any fibre bundle $E$, the gauge group is $Aut E$, the group of of invertible maps $E \rightarrow E$ that preserve the base. When $E$ is a trivial bundle and so say a product $UF \rightarrow U$, then this group is $U \rightarrow Aut F$. Since locally all fibre bundles are trivial, this tells you what the gauge group is locally.

Now, when we specialise to a principal bundle, then the bundle comes along with an action of a group $G$ that acts fibre transitively and freely. This means that every fibre of the bundle is isomorphic to this group. Now, this group is called by mathematicians the structure group but the gauge group by physicists. It is different from the gauge group above. I'll call it the structural gauge group.

The standard model is given by a principal bundle with structural gauge group $U(1)_Y \times SU(2)_L \times SU(3)_c$. Here the subscripts stand for hyperforce (Y), isospin (L) and colour (c).

Now, it's not enough to have principal bundles around, we also require a vector bundle. This is given by a represention of the structural gauge group on some vector space, $\alpha: G \rightarrow Aut V$ and we denote this vector bundle as $E[V]$. We always have the adjoint representation of the structural group which represents the structural gauge group on its Lie algebra $G'$. This is usually written as $Ad: G \rightarrow Aut G'$. And we denote $E[G']$ by $Ad E$. This is called the adjoint bundle.

Now, connections on the principal bundle are 1-forms valued in the adjoint bundle. Physically, these are the gauge fields given by the photon, the W & Z bosons as well as the gluon fields.

So a choice of connection is a choice of gauge field. This is what you have denoted by $\Sigma \rightarrow Ad P$. It is not a gauge transformation. A gauge transformation comes from transforming the connection between two different charts.

As this is a connection we have parallel transport (covariantly constant transport) by this connection and this is what you have described as 'global gauge symmetry'. It's akin to translational symmetry on Euclidean space, except here global parallel transport is on a curved manifold, or more precisely, a curved bundle.

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    $\begingroup$ Oh, in fact, my bundle $\mathrm{Ad}(P)$ is $P \times^{\mathrm{Ad}}_G G$. I use $\mathrm{Ad}$ for the conjugate action on group $G$ and $\mathrm{ad}$ for the conjugate action on the Lie algebra $G'$. I guess there is some relation because connection, and transformation both are under such conjugate action (despite they are in different type of fiber) $\endgroup$
    – ChoMedit
    Apr 18, 2021 at 8:33
  • $\begingroup$ @ChoMedit: I've used that notation as well gor the adjoint bundle. The Wikipedia article on the adjoint action helps clarify how the various actions fit together: $ad$ is the derivative of $Ad$. I use the notation $G'$ for Lie algebras as they are the derivative of Lie groups - and besides, I don't know the latex for fraktur fonts! $\endgroup$ Apr 18, 2021 at 9:00
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I guess I finally got it.

Suppose $G$ be a structure group, and $P$ be a $G$-principal bundle on $\Sigma$. If $\phi : P \to P$ be a gauge transformation, then there exists smooth map $u : P \to G$ defined by $\phi(p) = p^\circ u(p)$.

On the other hand, the connection, or gauge field is $A \in \Omega_1(P; \mathfrak{g})$.

If we consider $\mathrm{d} u$ the derivative of $u$, it is one of the connection on $P$. Hence, we can define that $u$ is covariantly constant by following. \begin{align} \mathrm{d} u + A = 0 \end{align}

Of course, the gauge symmetry itself, is not only for the principal bundle, but also for any kind of associated bundle because physics also considered every matter field interacting with the gauge fields. However, the dictionary seems to describe whole phenomena from describing it just in a starting point(gauge field).

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