Global vs. local gauge group in mathematical sense - physics examples?

Question

Upon reading about the principal bundle picture of (quantum) field theory I encountered two different definitions of the gauge group:

• Local gauge group $G$. Corresponds to the fibers of the $G$-bundle. Local gauge transformations correspond to the change of coordinates in which fields and form are written in trivializations. In other words, as far as I understand, local gauge transformations = transition functions for trivializations of the principal bundle and the associated bundles.
• Global gauge group $\mathfrak G=Aut(P)$. Is the group of principal bundle diffeomorphisms, and is claimed to be much bigger then $G$.

I would like to understand which notions in physics correspond to these two, apparently very distinct, notions.

First, in physics, there is the notion of global symmetry, as far as I understand this has nothing to do with gauge/bundles. But on the other hand, the limit of a gauge symmetry where the parameter becomes constant is a global symmetry. Typical $U(1)$ example: $\exp(i\alpha(x))$ = $U(1)$ gauge transformation, but if $\alpha(x)=\alpha=const.$, then it is a global symmetry. Is there any deeper connection between global symmetry and gauge "symmetry"/bundles?

Second, I would appreciate some examples which clarify the distinction between local and global gauge groups mentioned above. If, for example we take a prototype $SU(2)$ gauge theory of the form $$ \mathcal L = -\frac{1}{4}(F_{\mu\nu}^a)^2+|(D_\mu\phi)^a|^2-V(\phi) $$ then we can put it in the bundle picture as a $SU(2)$ principal bundle, the fields $A_\mu^a$ living in the associated bundle $$ (P\times\mathfrak su(2))/\{(p,A)\sim(pg,Ad_{g^{-1}}A)\} $$ and the doublet field $\phi$ in fundamental representation of $SU(2)$ living in the associated bundle $$ (P\times\mathbb C^2)/\{(p,\phi)\sim(pg,\rho(g^{-1})\phi)\}. $$ So, the $SU(2)$ group which we are referring to here is what was the local gauge group $G$ above, is this correct? The gauge transformations apply to coordinate versions of the fields in the trivializations of the associated bundles to change between different coordinates.

Now if, this is the case, what is the global gauge group $\mathfrak G$ then? In particular I am seeking the answer for the following questions:

• Do global gauge transformations apply to the model $SU(2)$ theory above?
• In what sense is the global gauge group much bigger then the local gauge group?
• Is there a model example where one can clearly write down the local gauge group and the global gauge group?
• Are there cases where the two groups are the same?
• How does this discussion relate to quantization of classical fields (if it does at all)?

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Edit

I'll post here answers to various questions in the comments/answers.

About the definitions I used in my text. I took them from my lecture notes from a course on mathematical gauge theories taught by a mathematician. Although there is no on-line version of this course, I can provide more detailed definitions if needed.

Specifically, concerning the distinction global/local gauge, I have found this piece of information on nLab, which seems to be agreement with my definition: http://ncatlab.org/nlab/show/gauge+group. Although this page provides some explanations, it uses some mathematical terminology which I am not entirely familiar with. The problem both with the nLab page and with the lecture I attended is to make the connection to physics (I am a physicist).

Ok, I'll just write down the definitions from my lecture notes.

Local gauge

Let $P\to M$ a principal $G$-bundle, $\omega$ a connection 1-form on $P$. $U_i, U_j\subset M$ such that $U_i\cap U_j\neq\emptyset$ over which P is trivial: $$ \psi_i:\pi^{-1}(U_i)\to U_i\times G\\ \psi_j:\pi^{-1}(U_j)\to U_j\times G $$ with transition functions $$ \psi_i\circ\psi_j^{-1}:(U_i\cap U_j)\times G\to (U_i\cap U_j)\times G\\ \phantom{\psi_i\circ\psi_j^{-1}:XXX}(m,g)\mapsto (m,\psi_{ij}(m)g) $$ where $\psi_{ij}:U_i\cap U_j\to G$.

Using these trivializations we can construct preferred sections as follows: $$ \sigma_i:U_i\to\pi^{-1}(U_i)\\ \phantom{\sigma_i:x}m\mapsto \psi_i^{-1}(m,e)\,. $$ Define $$ \omega_i:=\sigma_i^*\omega\,, $$ which is a $\mathfrak g$-valued 1-form on $U_i$. Changing between coordinates on $U_i$ and $U_j$ is what is called choice of local gauge. We would like to write down a transformation prescription for $\omega_{i/j}$.

On G we have the canonical 1-form $\theta$ with values in $\mathfrak g$ defined as follows: $$ \theta_g(X_g)=A\in\mathfrak g\quad\text{if }(A^*)_g=X_g $$ where by $A^*$ we mean the fundamental vector field on $G$ corresponding to $A$. One can show that the 1-forms transform in the following way: $$ \omega_j=Ad_{\psi_{ij}}^{-1}\omega_i+\psi_{ij}^*\theta\,. $$ This looks to me like a gauge transformation of guage fields $A=A_\mu dx^\mu$ in physics where we write $$ A'=gAg^{-1}+gdg^{-1} $$ and I even believe that the above transformation for $\omega_j$ can be rewritten as something like this $$ \omega_j=Ad_{\psi_{ij}}^{-1}\omega_i+\psi_{ij}^{-1}d\psi_{ij}\,, $$ Then it looks like the transformation for $A$ with $g=\psi_{ij}^{-1}$.

So much about the local gauge transformation.

Global gauge group

The group of global gauge transformation $\mathfrak G$ is the set of automorphisms of the principal bundle P: $$ \mathfrak G = Aut(P)\,. $$ This must not be confused with the structure group $G$, which is sometimes called gauge group in leterature, but is much smaller.

The (global) gauge group can be described in three ways:

1. $\mathfrak G=\{\phi:P\to P|\phi\text{ is a diffeo., }\pi\circ\phi=\pi, \phi(pg)=\phi(p)g\,\forall g\in G\}$
2. $\mathfrak G=\{u:P\to G|u\text{ smooth s.t. }u(pg)=g^{-1}u(p)g\,\forall g\in G\}$. Note that $\phi(p)=p\cdot u(p)$.
3. $\mathfrak G=\{\text{sections }s:M\to F\}$, where $F=(P\times G)/\sim$ with $(p,h)\sim(pg,g^{-1}pg)\forall g\in G$.

If $\omega$ is a connection 1-form which corresponds to a choice to the horizontal tangent space $H$ on $P$, then $\phi^*\omega$ is also a connection 1-form, defining the pull-back connection $$ (\phi^*H)_p:=(D_p\phi)^{-1}H_{\phi(p)}\quad\Leftrightarrow\quad D_p\phi((\phi^*H)_p):=H_{\phi(p)} $$ We conclude that $\mathfrak G$ acts on connections, but the explicit form of this action if fairly complicated. In contrast, the action of $\mathfrak G$ on the curvature is easy to understand.

From the last equation above we conclude that $\phi$ maps $\phi^*H$ to $H$, and so it maps $\tilde\Omega$, the curvature of $\phi^*H$, to $\Omega$, the curvature of $H$: $$ \phi(\tilde\Omega(X,Y))=\Omega(X,Y)\quad\forall X,Y\in T_mM\,. $$ One can show that the following transformation relation holds: $$ \tilde\Omega=Ad_{u^{-1}}\Omega\,. $$ where $u:P\to G$ is the map from the second definition of $\mathfrak G$ above corresponding to $\phi$.

Two connections $H_1$ and $H_2$ on P are called gauge equivalent if there is a $\phi\in\mathfrak G$ with $\phi^*H_2=H_1$.

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One of the things which I am curious about is that from the local gauge transformations the transformation law for the 1-form $\omega$ seems to be what we call the gauge transformation of gauge fields $A$ (see above), but from the global gauge group transformation, the transformation of the curvature is what we see in physics, namely if $F$ is the field strength corresponding to $A$, and $g$ an element of the gauge group then $$ F\to F'=gFg^{-1}\,. $$ Also, as I said at the very beginning, I would like to pin down what these mathematical definitions correspond to in what we learn in physics. In physics, when we discuss a certain gauge theory, there is THE gauge group, such as $U(1)$, $SU(2)$, etc. under which fields transform in certain representations, as well as the gauge fields themselves, which transform in the adjoint representation. Now in mathematics I see the distinction local/global gauge group. The essence of my question is do understand this distinction and to relate it to phyics.

Answer 1

I admit I am a bit confused by your terminology, but here is how I learned it: Let $P$ be a $G$-principal bundle and $\Sigma$ a spacetime.

• gauge group: The fibers of the $G$-principal bundle over the spacetime, i.e. the group $G$.
• (Local) group of gauge transformations: The group of diffeomorphisms $t : P \rightarrow P$, which are fiber-preserving and $G$-equivariant, i.e. if $\pi : P \rightarrow \Sigma$ is the projection then $\pi \circ t = \pi$, and $t$ commute with the group action on $P$.

One can now, by transitivity of the group action on the fibers, define a function $g_t: P \rightarrow G$ by $t(p) = pg_t(p) \forall p \in P$, and such functions $g : P \rightarrow G$ conversely define a gauge transformation by $t_g(p) = pg(p)$ as long as they fulfill $g_t(ph) = h^{-1}g_t(p)h \forall h \in G$, so we have two alternative characterizations of local gauge transformations:

$\mathcal{G} = \{t |t \in \mathrm{Diff}(P) \wedge \pi \circ t = t \wedge t(ph) = t(p)h \forall h \in G\} = \{ g| g \in \mathrm{Maps}(P,G) \wedge g(ph) = h^{-1}g(p)h \forall h \in G \}$

The equivariant diffeomorphisms of $P$ are called local, since they apply a different group element to every spacetime point.

Now, the associated bundles are affected as follows: Let $\phi : \Sigma \rightarrow P \times_G V$ be a section of the associated bundle, i.e. a field. By a similar argument to the above, these are in bijection to $G$-equviariant functions $f_\phi : P \rightarrow V$ satisfying $f_\phi(pg) = \rho(g^{-1})f_\phi(p)$. This is esentially the reason why, in $\mathrm{U}(1)$ symmetry, a gauge transformation $\mathrm{e}^{\mathrm{i}\alpha(x)}$ acts on fields as $\phi(x) \mapsto \mathrm{e}^{-\mathrm{i}\alpha(x)} \phi(x)$.

So, you see, the local group of gauge transformations is much bigger that the global gauge group since it allows far more functions than just the constant ones. You can always clearly write down the global gauge group (it defines your theory!), but writing down the local one more explicit than I did above is hard. For $\mathrm{U}(1)$, however, it is just $\{x \mapsto \mathrm{e}^{\mathrm{i}\alpha(x)} x | \alpha : P \rightarrow \mathrm{U}(1) \text{is smooth (enough)}\}$, I think. Cases where the two groups coincide demand a spacetime that is a point, I would guess, but I am not wholly confident in that.

Also, all of this can be done classically, nothing about gauge theories is inherently quantum.

EDIT:

Alright, your edit was very helpful in discerning what is actually going on here.

Your global gauge group is what physicists call the group of gauge transformations. The gauge group of a a gauge theory is what you call a local gauge group (and what the nLab also calls the local gauge group). When physicists say the gauge group $\mathrm{SU}(N)$, they mean it is what you call the local gauge group.

The global gauge group of the nLab is just the group of transformations (not necessarily gauge transformations, terminology is terrible here, I know) that leaves all observables invariant, i.e. it is the group of symmetries of the theory (not the group of symmetries of the Lagrangian), the group of gauge transformations is naturally a subgroup of this. The difference is that this global gauge group can contain transformations that have not really something to do with the structure of the local gauge group, and can contain things which are not gauge transformations. This global gauge group can even exist if you have no explicit gauge theory, and is inherently a QFT concept.

In other news, you are right, your connection form $\omega$ is the gauge field $A$ of a physical gauge theory, and it transforms exactly like you wrote. Now, the problem with the gauge field is exactly that ugly transformation, so we construct the curvature transforming in the adjoint rep and call it the field strength $F$. The action of a pure (Yang-Mills) gauge theory is then (up to prefactors) given by

$$ \int_\Sigma \mathrm{Tr}_{ad}(F \wedge \star F)$$

since the action must be invariant under gauge transformations and the $\mathrm{Tr}_{ad}(F \wedge \star F)$ is pretty much the only object we can construct out of the gauge fields that is invariant and can be integrated over the spacetime.

Answer 2

The names of these creatures are a true mess and there are mainly two independent notation schemes: the mathematical and the physical one.

Let $P \to M$ be a $G$-principal bundle. Then

• $G$ is called the structure group by mathematicians and the gauge group by physicians
• The (infinite-dimensional) group of automorphism of $P$, or equivalently the group of sections of $Ad(P)$, is called the gauge group (maths) or the group of gauge transformation (physics)
• If you have a trivial bundle $P = M \times G$, the automorphism group can be identified with maps from $M$ to $G$ (since $Ad(P) = M \times G$ in this case). Hence it make sense to speak of constant gauge transformations and these are often called global gauge transformation.

Note that there is no difference between what you call local and global gauge transformations. They are the same thing just looked at from different viewpoints. A gauge transformation is by definition an automorphism of your bundle $P$. If you look at this transformation in a local trivialization $(U, \tau)$ then you see that a gauge transformation exactly corresponds to a function $U \to G$ which act on the trivialization (and on the other creatures living on $U$ like connections, curvature forms and local sections). Hence a gauge transformation can be interpreted as a change of trivialization (with the same open covering) or in physics lingo as a coordinate change. Conversely a family of maps $U_i \to G$ on trivializing sets $U_i \subseteq M$ satisfying certain compatibility relations on the intersections give rise to a global section of $Ad(P)$, that is a gauge transformation. So your "local gauge transformations" are just "global transformations" in coordinate patches.

Answer 3

This is not quite a complete answer, but more of an overly large comment on terminology.

The definitions from nLab don't agree with Giovanni Giachetta, Luigi Mangiarotti, Gennadi Sardanashvily: Advanced Classical Field Theory, which I'll summarize briefly:

The authors call the group $G$ of a (principal) $G$-bundle the structure group. This is standard terminology that can also be found in Kobayashi, Nomizu or other textbooks on differential geometry. Physics literature may call this group the 'gauge group'.

The group of gauge transformations is the group of equivariant bundle automorphism and the gauge group the group of vertical equivariant bundle automorphisms (ie covering the identity on the base space). The latter is what you called 'global gauge group'. Making this distinction may be non-standard - I believe most authors require gauge transformations to be vertical. As the gauge group yields the gauge symmetries of the corresponding Lagrangian field theory, this seems like a reasonable definition.

As you correctly stated, the gauge group is isomorphic to the group of global sections of the $G$-bundle associated to the principal bundle via conjugation. Now, transition maps between local trivializations are also given by sections of the same bundle, but local ones. This could motivate the term 'local gauge group', which would be distinct from the (normally finite-dimensional) structure group.

Locally, a 'global gauge transformation' of course needs to be expressed as a family of 'local gauge transformations'. If the physical theory is gauge-invariant, it doesn't really matter if you're dealing with a 'local' or a 'global' gauge transformation. For the same reason, general covariance of general relativity means nothing bad will happen if we conflate coodinate transformations and diffeomorphisms.