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 ${(\mathrm{Inn}F(E),M,GL(\mathbb{K}^{n}))}$ .

Mathematics for Physics | An Illustrated Handbook

An Illustrated Handbook