# 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 which are not necessarily consistent in $${U_{i}\cap U_{j}}$$.
 ◊ 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 $${\left(v_{i}^{\mu}\right)^{\prime}=(\gamma_{i})^{\mu}{}_{\lambda}v_{i}^{\lambda}}$$ 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 $${\left(v_{p}^{\mu}\right)^{\prime}=(\gamma_{p})^{\mu}{}_{\lambda}v_{p}^{\lambda}}$$ 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 potential 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 $${(\mathrm{Inn}F(E),M,GL(\mathbb{K}^{n}))}$$.