Gauge Groups and Representation-Theoretic Interfaces

Let $G$ be a finite-dimensional real Lie group, $M$ a smooth manifold, and $P$ a smooth manifold. A smooth right principal $G$-bundle is a right principal $G$-bundle $\pi :P\to M$ in the topological sense of Principal g bundle and associated fiber bundle, with the following additional smooth structure: $\pi$ and the right $G$-action are smooth, and the principal-bundle charts can be chosen as equivariant diffeomorphisms ${\pi }^{-1}(U)\cong U\times G$. Thus each chart respects the projection to $U$ and the right action on the $G$ factor. The smoothness convention is that of Smooth maps between manifolds with boundary; boundaryless manifolds are the default in this item.

The topological principal-bundle clause retains its free, transitive right action on every fibre and its local triviality. Smoothness adds compatible differentiable structure; it does not select a connection or assert that a global trivialization exists. The empty-base case is allowed and then $P$ is empty by the cited topological definition.

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ as recorded by The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), and let $\pi :P\to M$ be a smooth right principal $G$-bundle in the sense of Smooth principal $G$-bundle. The tangent bundle $TP$ has the canonical smooth structure supplied by Assuming countable choice, the tangent bundle has a canonical smooth 2n-manifold structure. Its vertical subbundle is $VP:=\ker (d\pi )\subseteq TP\mathrm{.}$ In each principal chart $U\times G$, this is the tangent bundle in the $G$ factor, so it is a smooth vector subbundle of $TP$ in the sense of Smooth vector bundles, rank, fibres, and trivial bundles.

A principal connection on $P$ is a smooth vector subbundle $H\subseteq TP$ such that $T_{p}P=V_{p}P\oplus H_p$ for every $p\in P$ and $d(R_g{)}_p(H_p)=H_{pg}(p\in P,g\in G),$ where $R_g(p)=pg$ is the principal right action. Thus $H$ is a smoothly varying horizontal complement to the vertical tangent directions, invariant under the structure-group action. This definition is conditional on a specified $H$; it is not an existence theorem for connections on every bundle.

Let $h:TP\to H$ and $v:TP\to VP$ be the projections given by this direct-sum decomposition. For smooth vector fields $X,Y$ on $P$, define the vertical-valued curvature by $K_H(X,Y):=-v([hX,hY]),$ where the vector-field bracket is the commutator The Lie bracket of smooth vector fields. The bracket definition gives the commutator on functions; Smooth vector fields form a Lie algebra under the Lie bracket proves that it is again a smooth vector field. In a local frame adapted to the smooth splitting $TP=VP\oplus H$, the projections $h$ and $v$ are smooth. Therefore $v([hX,hY])$ is a smooth vertical vector field whenever $X,Y$ are smooth. This formula is tensorial in $X,Y$: for a smooth function $f$, $v([fhX,hY])=fv([hX,hY])-hY(f)v(hX)=fv([hX,hY]),$ because $v(hX)=0$, and the same calculation in the second input gives $v([hX,fhY])=fv([hX,hY])$, since $v(hY)=0$. It is alternating because the bracket is alternating. To see the induced bundle map is smooth, choose local smooth frames $E_1,\dots ,E_n$ of $TP$ and $F_1,\dots ,F_r$ of $VP$. Tensoriality gives, for $X={\sum }_i{f}_i{E}_i$ and $Y={\sum }_j{g}_j{E}_j$, $K_H(X,Y)={\sum }_{i,j}{f}_i{g}_j{K}_H(E_i,E_j)\mathrm{.}$ Each $K_H(E_i,E_j)$ is a smooth section of $VP$, so its components in the $F_a$ frame are smooth; the displayed formula therefore has smooth local coefficient functions. Thus the formula defines a smooth alternating bundle map $TP{\times }_{P}TP\to VP$, independent of the local vector-field extensions representing the tangent vectors.

This item uses the vertical-valued convention. It does not identify $VP$ with a Lie-algebra-valued form or assert a Maurer–Cartan formula. Its axiom base contains ${\mathrm{A}\mathrm{C}}_{\omega }$ exactly through the canonical tangent-bundle and vector-field suppliers above; the displayed splitting and curvature formula themselves select no objects from a family.

Let $\pi :P\to M$ be a smooth right principal $G$-bundle as in Smooth principal $G$-bundle. A smooth gauge transformation of $P$ is a smooth diffeomorphism $\mathrm{\Phi }:P\to P$ satisfying $\pi \circ \mathrm{\Phi }=\pi ,\mathrm{\Phi }(pg)=\mathrm{\Phi }(p)g(p\in P,g\in G)\mathrm{.}$ Here diffeomorphism has the meaning of Diffeomorphisms and local diffeomorphisms of manifolds. The gauge group $\mathcal{G}(P)$ is the set of these transformations under composition. The identity is a smooth diffeomorphism and composites of smooth maps are smooth by Identity maps and composites of smooth maps are smooth; composites of diffeomorphisms are diffeomorphisms because their inverses are composites of the smooth inverses. The identity covers $M$ and commutes with the right action. If $\mathrm{\Phi },\mathrm{\Psi }\in \mathcal{G}(P)$, then $\mathrm{\Phi }\circ \mathrm{\Psi }$ covers $M$ and $(\mathrm{\Phi }\circ \mathrm{\Psi })(pg)=\mathrm{\Phi }(\mathrm{\Psi }(p)g)=\mathrm{\Phi }(\mathrm{\Psi }(p))g$, so the conditions are closed under composition. The inverse of such a $\mathrm{\Phi }$ also covers $M$: if $q=\mathrm{\Phi }(p)$ then $\pi ({\mathrm{\Phi }}^{-1}(q))=\pi (p)=\pi (q)$. Equivariance gives ${\mathrm{\Phi }}^{-1}(qg)={\mathrm{\Phi }}^{-1}(\mathrm{\Phi }(p)g)={\mathrm{\Phi }}^{-1}(\mathrm{\Phi }(pg))=pg\mathrm{.}$ Its smoothness is part of the diffeomorphism condition. Hence these maps form a group under the composition law of Group and abelian group.

The structure group $G$ acts on each fibre of $P$, while $\mathcal{G}(P)$ acts on the total bundle by automorphisms over the identity map of $M$. They are different groups: a gauge transformation may vary from fibre to fibre, and this definition assigns no topology or finite-dimensional Lie-group structure to $\mathcal{G}(P)$.

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $P\to M$ be a smooth right principal $G$-bundle with a principal connection $H$, and let $\mathrm{\Phi }\in \mathcal{G}(P)$. Then ${\mathrm{\Phi }}_{\ast }H:=d\mathrm{\Phi }(H)$ is a principal connection. If $K_H$ and $K_{{\mathrm{\Phi }}_{\ast }H}$ are the vertical-valued curvatures from Principal connections and vertical curvature, then for $X,Y\in T_{p}P$, $K_{{\mathrm{\Phi }}_{\ast }H}(d{\mathrm{\Phi }}_{p}X,d{\mathrm{\Phi }}_{p}Y)=d{\mathrm{\Phi }}_p(K_H(X,Y))\mathrm{.}$ Composition makes this a left action of $\mathcal{G}(P)$ on the set of principal connections.

Let $G$ be a finite-dimensional real Lie group, let $P\to M$ be a smooth right principal $G$-bundle, let $V$ be a finite-dimensional real vector space, and let $\rho :G\to \mathrm{GL}(V)$ be a smooth representation, where $\mathrm{GL}(V)$ is the group of invertible linear maps $V\to V$. Write $\mathcal{G}(P)$ for the gauge group of Gauge group of a smooth principal bundle. Put $E_{\rho }:=P{\times }_{\rho }V$ for the associated smooth vector bundle of Associated bundles and Associated vector bundles are well-defined. For every $\mathrm{\Phi }\in \mathcal{G}(P)$ define $\bar{\mathrm{\Phi }}:E_{\rho }\to E_{\rho },[p,v]⟼[\mathrm{\Phi }(p),v]\mathrm{.}$ Here a smooth vector-bundle automorphism means a smooth fibrewise-linear diffeomorphism covering ${\mathrm{id}}_M$. The map $\bar{\mathrm{\Phi }}$ is such an automorphism, and $\overline{\mathrm{\Phi }\circ \mathrm{\Psi }}=\bar{\mathrm{\Phi }}\circ \bar{\mathrm{\Psi }},\overline{{\mathrm{id}}_P}={\mathrm{id}}_{E_{\rho }}\mathrm{.}$ Thus $\mathrm{\Phi }\mapsto \bar{\mathrm{\Phi }}$ is a group homomorphism to the group of smooth vector-bundle automorphisms of $E_{\rho }$. If ${\mathrm{\Gamma }}^{\mathrm{\infty }}(E_{\rho })$ denotes the smooth sections as in Smooth sections, local sections, and support, then $\mathrm{\Phi }\cdot s:=\bar{\mathrm{\Phi }}\circ s$ is a linear left action of $\mathcal{G}(P)$ on ${\mathrm{\Gamma }}^{\mathrm{\infty }}(E_{\rho })$. No injectivity or surjectivity of either induced action is asserted.

The square below records the quotient descent, where $q:P\times V\to E_{\rho }$ is the orbit projection. It commutes because $\mathrm{\Phi }$ is $G$-equivariant; the pointwise verification appears in step 1.1.