# Vertical tangents and horizontal equivariant forms

A smooth bundle $${(E,M,\pi)}$$ is a manifold itself, and thus has tangent vectors. A tangent vector $${v}$$ at $${p\in E}$$ is called a vertical tangent if $${\mathrm{d}\pi(v)=0}$$, i.e. if it is tangent to the fiber over $${x}$$ where $${\pi(p)=x}$$, so the projection down to the base space vanishes. The vertical tangent space $${V_{p}}$$ is then the subspace of the tangent space $${T_{p}}$$ at $${p}$$ consisting of vertical tangents, and viewing the vertical tangent spaces as fibers over $${E}$$ we can form the vertical bundle $${(VE,E,\pi_{V})}$$, which is a subbundle of $${TE}$$. We can also consider differential forms on a smooth bundle, which take arguments that are tangent vectors on $${E}$$. A form is called a horizontal form if it vanishes whenever any of its arguments are vertical.

On a smooth principal bundle $${(P,M,G)}$$, we have a consistent right action $${\rho\colon G\rightarrow\mathrm{Diff}(P)}$$, and the corresponding Lie algebra action $${\mathrm{d}\rho\colon\mathfrak{g}\rightarrow\mathrm{vect}(P)}$$ is then a Lie algebra homomorphism. The fundamental vector fields corresponding to elements of $${\mathfrak{g}}$$ are vertical tangent fields; in fact, at a point $${p}$$, $${\mathrm{d}\rho\left|_{p}\right.}$$ is a vector space isomorphism from $${\mathfrak{g}}$$ to $${V_{p}}$$:

$$\displaystyle \mathrm{d}\rho\left|_{p}\right.\colon\mathfrak{g}\overset{\cong}{\rightarrow}V_{p}$$

In addition, the right action $${g\colon P\rightarrow P}$$ of a given element $${g}$$ corresponds to a right action $${\mathrm{d}g\colon TP\rightarrow TP}$$, which maps tangent vectors on $${P}$$ via

$$\displaystyle \mathrm{d}g(v)\colon T_{p}P\rightarrow T_{g(p)}P.$$

This map is an automorphism of $${TP}$$ restricted to $${\pi_{P}^{-1}(x)}$$, which we denote $${T_{\pi^{-1}(x)}P}$$, and it preserves vertical tangent vectors. We can then consider the pullback $${g^{*}\varphi(v_{1},\ldots,v_{k})=\varphi(\mathrm{d}g(v_{1}),\ldots,\mathrm{d}g(v_{k}))}$$ as a right action on the space $${\Lambda^{k}P}$$ of $${k}$$-forms on $${P}$$.

If we have a bundle $${(E,M,\pi_{E},F)}$$ associated to $${(P,M,\pi_{P},G)}$$, we can define an $${F}$$-valued form $${\varphi_{P}}$$, which can be viewed on each $${\pi_{P}^{-1}(x)}$$ as a mapping

$$\displaystyle \varphi_{P}\colon T_{\pi^{-1}(x)}P\otimes\cdots\otimes T_{\pi^{-1}(x)}P\rightarrow F\times\pi_{P}^{-1}(x),$$

where $${g\in G}$$ has a right action $${\mathrm{d}g}$$ on $${T_{\pi^{-1}(x)}P}$$ and a left action $${g}$$ on the abstract fiber $${F}$$ of $${E}$$. The form $${\varphi_{P}}$$ is called an equivariant form if this mapping is equivariant with respect to these actions, i.e. if

$$\displaystyle g^{*}\varphi_{P}=g^{-1}\left(\varphi_{P}\right).$$

If $${\varphi_{P}}$$ is also horizontal, then it is called a horizontal equivariant form (AKA basic form, tensorial form). If we pull back a horizontal equivariant form to the base space $${M}$$ using the identity sections, we get forms

$$\displaystyle \varphi_{i}\equiv\sigma_{i}^{*}\varphi_{P}$$

on each $${U_{i}\subset M}$$. Using the identity section relation $${\sigma_{i}=g_{ij}^{-1}(\sigma_{j})}$$ and the pullback composition property $${\left(g(h)\right)^{*}\varphi=h^{*}\left(g^{*}\varphi\right)}$$, we see that the values of these forms satisfy

\displaystyle \begin{aligned}\varphi_{i} & =\left(g_{ij}^{-1}(\sigma_{j})\right)^{*}\varphi_{P}\\ & =\sigma_{j}^{*}\left(\left(g_{ij}^{-1}\right)^{*}\varphi_{P}\right)\\ & =\sigma_{j}^{*}\left(g_{ij}\left(\varphi_{P}\right)\right)\\ & =g_{ij}\left(\varphi_{j}\right), \end{aligned}

where in the third line $${g_{ij}}$$ is acting on the value of $${\varphi_{P}}$$. This means that at a point $${x}$$ in $${U_{i}\cap U_{j}}$$, the values of $${\varphi_{i}}$$ and $${\varphi_{j}}$$ in the abstract fiber $${F}$$ correspond to a single point in $${\pi_{E}^{-1}(x)\in E}$$, so that the union $${\bigcup\varphi_{i}}$$ can be viewed as comprising a single $${E}$$-valued form $${\varphi}$$ on $${M}$$. Such a form is sometimes called a section-valued form, since for fixed argument vector fields its value on $${M}$$ is a section of $${E}$$. It can be shown that the correspondence between the $${E}$$-valued forms $${\varphi}$$ on $${M}$$ and the horizontal equivariant $${F}$$-valued forms on $${P}$$ is one-to-one. Equivariant $${F}$$-valued 0-forms on $${P}$$ are automatically horizontal (since one cannot pass in a vertical argument), and are thus one-to-one with sections on $${E}$$.

The above depicts how the differential of the right action of $${G}$$ on $${\pi_{P}^{-1}(x)\in P}$$ creates an isomorphism to the vertical tangent space $${\mathfrak{g}\cong V_{p}}$$. A horizontal equivariant form $${\varphi_{P}}$$ on $${P}$$ maps non-vertical vectors to the abstract fiber $${F}$$ of an associated bundle, and pulling back by the identity sections yields an $${E}$$-valued form $${\varphi}$$ on $${M}$$. Although denoted identically, the $${f_{i}}$$ are those corresponding to each bundle.

On the frame bundle $${(P,M,\pi_{P},GL(n,\mathbb{K}))}$$ associated with a vector bundle $${(E,M,\pi_{E},\mathbb{K}^{n})}$$, a $${\mathbb{K}^{n}}$$-valued form $${\vec{\varphi}_{P}}$$ is then equivariant if

$$\displaystyle g^{*}\vec{\varphi}_{P}=\check{g}^{-1}\vec{\varphi}_{P},$$

where $${\check{g}^{-1}}$$ is a matrix-valued 0-form on $${P}$$ operating on the $${\mathbb{K}^{n}}$$-valued form $${\vec{\varphi}_{P}}$$. The pullback of a horizontal equivariant form on $${P}$$ to the base space $${M}$$ using the identity sections satisfies

$$\displaystyle \vec{\varphi}_{i}=\check{g}_{ij}\vec{\varphi}_{j},$$

where $${\check{g}_{ij}}$$ is now a matrix-valued 0-form on $${M}$$. At a point $${x}$$ in $${U_{i}\cap U_{j}}$$, the values of $${\vec{\varphi}_{i}}$$ and $${\vec{\varphi}_{j}}$$ in the abstract fiber $${\mathbb{K}^{n}}$$ correspond to a single abstract vector in $${V_{x}=\pi_{E}^{-1}(x)\in E}$$, so that the union $${\bigcup\vec{\varphi}_{i}}$$ can be viewed as comprising a single $${V}$$-valued form $${\vec{\varphi}}$$ on $${M}$$. Thus an equivariant $${\mathbb{K}^{n}}$$-valued 0-form on $${P}$$ is a matter field on $${M}$$.

 ◊ This correspondence can be viewed as follows. The right action of $${g}$$ on $${P}$$ is a transformation on bases, so that the equivalent transformation of vector components is $${g^{-1}}$$. The left action of $${g^{-1}}$$ on the abstract fiber of $${E}$$ is also a transformation of vector components. Thus the equivariant property can be viewed as “keeping the same value when changing basis on both bundles,” so that the values of $${\vec{\varphi}_{P}}$$ on $${\pi_{P}^{-1}(x)\in P}$$ correspond to a single point in $${\pi_{E}^{-1}(x)\in E}$$, i.e a single abstract vector over $${M}$$. In other words, $${\vec{\varphi}\in T_{x}M}$$ is determined by the value of $${\vec{\varphi}_{P}}$$ at a single point in $${\pi_{P}^{-1}(x)\in P}$$. The horizontal requirement means we do not consider forms which take non-zero values given argument vectors which project down to a zero vector on $${M}$$.

Under an automorphism gauge transformation, the transformation of a horizontal equivariant form on the frame bundle $${P}$$ is defined by the pullback of the automorphism

$$\displaystyle \vec{\varphi}_{P}^{\prime}\equiv\left(\gamma^{-1}\right)^{*}\vec{\varphi}_{P}.$$

The automorphism does not give us a right action on $${T_{\pi^{-1}(x)}P}$$ by a fixed element, but it does give a right action when acting on the element in the identity section, so since the identity sections remain constant we have

\begin{aligned}\vec{\varphi}_{i}^{\prime} & =\sigma_{i}^{*}\left(\gamma^{-1}\right)^{*}\vec{\varphi}_{P}\\
& =\left(\gamma^{-1}\sigma_{i}\right)^{*}\vec{\varphi}_{P}\\
& =\left(\gamma_{i}^{-1}\sigma_{i}\right)^{*}\vec{\varphi}_{P}\\
& =\sigma_{i}^{*}\left(\gamma_{i}^{-1}\right)^{*}\vec{\varphi}_{P}\\
& =\sigma_{i}^{*}\check{\gamma}_{i}\vec{\varphi}_{P}\\
& =\check{\gamma}_{i}\vec{\varphi}_{i},
\end{aligned}

where we have used $${\left(g(h)\right)^{*}\varphi=h^{*}\left(g^{*}\varphi\right)}$$ twice, and in the penultimate line we used the equivariance of $${\vec{\varphi}_{P}}$$. Under neighborhood-wise gauge transformations, there is no change in $${\vec{\varphi}_{P}}$$ but we have new identity sections $${\sigma_{i}^{\prime}(x)=\gamma_{i}^{-1}(\sigma_{i}(x))}$$, so that we get

\displaystyle \begin{aligned}\vec{\varphi}_{i}^{\prime} & =\sigma_{i}^{\prime*}\vec{\varphi}_{P}\\ & =\left(\gamma_{i}^{-1}\left(\sigma_{i}\right)\right)^{*}\vec{\varphi}_{P}\\ & =\sigma_{i}^{*}\left(\gamma_{i}^{-1}\right)^{*}\vec{\varphi}_{P}\\ & =\check{\gamma}_{i}\vec{\varphi}_{i}, \end{aligned}

matching the behavior for both automorphism gauge transformations and for gauge transformations as previously defined directly on $${M}$$ in the section on matter fields.

Note that if a horizontal equivariant form takes values in the abstract fiber $${F}$$ of another bundle associated to the frame bundle, the same reasoning applies, but with $${\check{\gamma}_{i}}$$ applied using the left action of $${G}$$ on $${F}$$. In particular, recalling that the adjoint rep $${\rho=\mathrm{Ad}}$$ of $${G}$$ on $${\mathfrak{g}}$$ defines an associated bundle $${(\mathrm{Ad}P,M,\mathfrak{g})}$$ to $${P}$$, we can consider a $${\mathfrak{g}}$$-valued horizontal equivariant form $${\check{\Theta}_{P}}$$ on $${P}$$, whose pullback by the identity section under a gauge transformation satisfies

$$\displaystyle \check{\Theta}_{i}^{\prime}=\check{\gamma}_{i}\check{\Theta}_{i}\check{\gamma}_{i}^{-1},$$

and which similarly across trivializing neighborhoods also undergoes a gauge transformation

$$\displaystyle \check{\Theta}_{i}=\check{g}_{ij}\check{\Theta}_{j}\check{g}_{ij}^{-1}.$$