Gauge transformations on frame bundles

Recall that a gauge transformation on a vector bundle \({E}\) is an active transformation of the bases underlying the components defining a local trivialization, which is equivalent to a new set of local trivializations and transition functions (and is not a transformation on the space \({E}\) itself). On the frame bundle \({F(E)}\), we perform the same basis change for the fixed frames associated with each trivializing neighborhood

\(\displaystyle e_{i}^{\prime}=e_{i}\gamma_{i}^{-1}, \)

which also defines the new identity sections, and is equivalent to new local trivializations where

\(\displaystyle f_{i}^{\prime}(p)=\gamma_{i}f_{i}(p), \)

giving us new transition functions

\(\displaystyle g_{ij}^{\prime}=\gamma_{i}g_{ij}\gamma_{j}^{-1}, \)

which are the same as those in the associated vector bundle \({E}\). We will call this transformation a neighborhood-wise gauge transformation.

An alternative (and more common) way to view gauge transformations on \({F(E)}\) is to transform the actual bases in \({\pi^{-1}(x)}\) via a bundle automorphism

\(\displaystyle p^{\prime}\equiv\gamma^{-1}(p), \)

and then change the fixed bases in each trivializing neighborhood to

\(\displaystyle \begin{aligned}e_{i}^{\prime} & =\gamma^{-1}(e_{i})\\ & \equiv e_{i}\gamma_{i}^{-1} \end{aligned} \)

in order to leave the maps \({f_{i}(p)}\) the same (which also leaves the identity sections and transition functions the same). This immediately implies a constraint on the basis changes in \({U_{i}\cap U_{j}}\): since \({g_{ij}^{\prime}=\gamma_{i}g_{ij}\gamma_{j}^{-1}}\), requiring constant \({g_{ij}}\) means we must have

\(\displaystyle \gamma_{i}^{-1}=g_{ij}\gamma_{j}^{-1}g_{ij}^{-1}. \)

We will call this transformation an automorphism gauge transformation.

Δ Note that this constraint means that automorphism gauge transformations are a subset of neighborhood-wise gauge transformations, which allow arbitrary changes of frame in every trivializing neighborhood. Also note that for automorphism gauge transformations, the matrices \({\gamma_{i}^{-1}}\) (and therefore the new identity section elements \({e_{i}^{\prime}}\)) are determined by the automorphism \({\gamma^{-1}}\), while neighborhood-wise gauge transformations are defined by arbitrary matrices \({\gamma_{i}^{-1}}\) in each neighborhood.
As with the associated vector bundle, for either type of gauge transformation the gauge group is the same as the structure group, and a gauge transformation \({\gamma_{i}^{-1}}\) is equivalent to the transition function \({g_{i^{\prime}i}}\) from \({U_{i}}\) to \({U_{i}^{\prime}}\), the same neighborhood with a different local trivialization.

We now define the matrices \({\gamma_{p}^{-1}}\) to be those which result from the transformation \({\gamma^{-1}(p)}\) on the rest of \({\pi^{-1}(x)}\), i.e.

\(\displaystyle e_{p}^{\prime}\equiv e_{p}\gamma_{p}^{-1}. \)

Note that \({\gamma_{p}^{-1}}\) is determined by \({\gamma_{i}^{-1}}\): since we require that \({f_{i}^{\prime}=f_{i}}\), we have

\(\displaystyle \begin{aligned}e_{i}^{\prime}f_{i}(p) & =e_{p}^{\prime}\\ \Rightarrow e_{i}\gamma_{i}^{-1}f_{i}(p) & =e_{p}\gamma_{p}^{-1}\\ & =e_{i}f_{i}(p)\gamma_{p}^{-1}\\ \Rightarrow\gamma_{p}^{-1} & =f_{i}(p)^{-1}\gamma_{i}^{-1}f_{i}(p), \end{aligned} \)

or more generally, using the definition of a right action \({f_{i}(g(p))=f_{i}(p)g}\) we get

\(\displaystyle \gamma_{g(p)}^{-1}=g^{-1}\gamma_{p}^{-1}g. \)

Δ It is important to remember that the matrices \({\gamma_{i}^{-1}}\) are dependent upon the local trivialization (since they are defined as the matrix acting on the element \({e_{i}\in\pi^{-1}(x)}\) for \({x\in U_{i}}\)), but the matrices \({\gamma_{p}^{-1}}\) are independent of the local trivialization, and are the action of the automorphism \({\gamma^{-1}}\) on the basis \({e_{p}}\).

The above depicts how an automorphism gauge transformation on \({F(E)}\) transforms the actual elements of the fiber over \({x}\), including the identity section elements corresponding to the fixed bases in each local trivialization, thus leaving the local trivializations unchanged.

This result can be understood as \({\gamma^{-1}}\) being a transformation on the internal space \({V_{x}}\) itself, applied to all the elements of \({\pi^{-1}(x)}\), each of which is a basis of \({V_{x}}\). For example, in the figure above, \({\gamma^{-1}}\) rotates all bases clockwise by \({\pi/2}\). To see why this is so, note that the matrix in the transformation \({v_{i}^{\prime\mu}=(\gamma_{i})^{\mu}{}_{\lambda}v_{i}^{\lambda}}\) has components which are those of \({\gamma_{i}\in GL(V_{x})}\) in the basis \({e_{i\mu}}\). Therefore in a different basis \({e_{p\mu}\in\pi^{-1}(x)}\) we must apply a different matrix \({v_{p}^{\prime\mu}=(\gamma_{p})^{\mu}{}_{\lambda}v_{p}^{\lambda}}\) which reflects the change of basis \({e_{p\mu}=f_{i}(p)^{\lambda}{}_{\mu}e_{i\lambda}}\) via a similarity transformation

\(\displaystyle \begin{aligned}\gamma_{p} & =f_{i}(p)^{-1}\gamma_{i}f_{i}(p)\\ \Rightarrow\gamma_{p}^{-1} & =f_{i}(p)^{-1}\gamma_{i}^{-1}f_{i}(p). \end{aligned} \)

Viewed as a transformation on \({V_{x}}\), \({\gamma^{-1}}\) will then commute with any fixed matrix applied to the bases, which as we saw is the right action; as we see next, this corresponds to the equivariance of \({\gamma^{-1}}\) required by it being a bundle automorphism.

We now check that \({\gamma^{-1}}\) is a bundle automorphism with respect to the right action of \({G}\), i.e. that \({\gamma^{-1}\left(g(p)\right)=g\left(\gamma^{-1}(p)\right)}\):

\(\displaystyle \begin{aligned}\gamma^{-1}\left(g(p)\right) & =e_{g(p)}\gamma_{g(p)}^{-1}\\ & =e_{g(p)}g^{-1}\gamma_{p}^{-1}g\\ & =e_{p}\gamma_{p}^{-1}g\\ & =e_{\gamma^{-1}(p)}g\\ & =g\left(\gamma^{-1}(p)\right) \end{aligned} \)

Δ A possible source of confusion is that a local gauge transformation (different at different points) can be defined globally on \({F(E)}\); meanwhile, a global gauge transformation (the same matrix \({\gamma_{i}^{-1}}\) at every point) can only be defined locally (unless \({F(E)}\) is trivial).

Consider the associated bundle to \({F(E)}\) with fiber \({GL(\mathbb{K}^{n})}\), where the local trivialization of the fiber over \({x}\) is defined to be the possible automorphism gauge transformations \({\gamma_{i}^{-1}}\) on the identity section element over \({x}\) in the trivializing neighborhood \({U_{i}}\). Then recalling that \({\gamma_{i}^{-1}=g_{ij}\gamma_{j}^{-1}g_{ij}^{-1}}\), we see that the action of the structure group on the fiber is by inner automorphism. Since the values of \({\gamma^{-1}}\) on \({F(E)}\) are determined by those in the identity section, we can thus view automorphism gauge transformations as sections of the associated bundle \({(\mathrm{Inn}F(E),M,GL(\mathbb{K}^{n}))}\).

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