Non-Abelian gauge data and bundle sectors

Assume the axiom of countable choice ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $M$ be a finite-dimensional, Hausdorff, second-countable smooth manifold without boundary. A non-Abelian gauge datum on $M$ consists of the following specified objects.

First choose a finite-dimensional real Lie group $G$ that is compact in its manifold topology (every open cover has a finite subcover) and noncommutative (there are $g,h\in G$ with $gh\mathrm{\ne }hg$). The actual group $G$, including its global form, is part of the datum. Its Lie algebra is $\mathfrak{g}=\mathrm{Lie}(G)=T_{e}G$ with the bracket transported from left-invariant smooth vector fields.

Choose a smooth right principal $G$-bundle $\pi :P\to M$: the projection and right action are smooth, and the action is free and transitive on each fibre. There is an open cover $\{U_{\alpha }{\}}_{\alpha \in A}$ of $M$ with a family of equivariant diffeomorphisms ${\varphi }_{\alpha }:{\pi }^{-1}(U_{\alpha })\to U_{\alpha }\times G$ for $\alpha \in A$, each over $U_{\alpha }$. Explicitly, ${\mathrm{pr}}_1\circ {\varphi }_{\alpha }=\pi$, and if ${\varphi }_{\alpha }(p)=(x,g)$ then ${\varphi }_{\alpha }(ph)=(x,gh)$ for every $h\in G$.

Choose a specified set $I$ of representation labels and, for each $a\in I$, a finite-dimensional complex vector space $V_a$, a positive-definite Hermitian form $h_a$, and a smooth group homomorphism ${\rho }_a:G\to U(V_a,h_a)$. Here $U(V_a,h_a)$ is the group of complex-linear isometries of $h_a$, and a representation is the corresponding finite-dimensional group action. Its associated vector bundle is the quotient $E_a=P{\times }_{{\rho }_a}{V}_a$ with relation $[pg,v]=[p,{\rho }_a(g)v]$.

Two smooth right principal $G$-bundles over $M$ are isomorphic over $M$ when a smooth diffeomorphism between their total spaces covers ${\mathrm{id}}_M$ and commutes with the right $G$-actions. To represent their isomorphism types by sets, encode a bundle by the ordered tuple of its total-space set, topology, maximal smooth atlas, projection, and right action; $M$ and $G$ are fixed parameters. Write $q{\cong }_{M}P$ when $q$ is such a bundle code isomorphic to $P$. In the cumulative hierarchy $V_0=\mathrm{\varnothing }$, $V_{\alpha +1}=\mathcal{P}(V_{\alpha })$, and $V_{\lambda }={\bigcup }_{\alpha <\lambda }{V}_{\alpha }$ at limit ordinals; $\mathrm{rank}(q)$ is the least ordinal $\alpha$ with $q\subseteq V_{\alpha }$. Let $r(P)$ be the least rank of a code $q{\cong }_{M}P$, and define the bundle sector of $P$ to be its set code $[P]:=\{q\in V_{r(P)+1}:q{\cong }_{M}P\text{ and }\mathrm{rank}(q)=r(P)\}\mathrm{.}$ The given bundle supplies a code, so minimizing the attained ranks at most its rank defines $r(P)$. Minimum-rank selection and Separation in $V_{r(P)+1}$ make $[P]$ a nonempty set. Isomorphic bundles have the same code, and equality of codes gives a common representative and hence a bundle isomorphism. This construction requires no representative choice and applies to every bundle allowed above. The datum specifies a set $\mathrm{\Sigma }$ of admitted bundle-sector codes and requires $[P]\in \mathrm{\Sigma }$; a model may take $\mathrm{\Sigma }=\{[P]\}$. References to the isomorphism class as a sector use this set code. The Lie algebra alone does not replace the chosen group or sector. This definition does not choose a connection, an action functional, or a quantum theory.

Minimum-rank set coding in ZF

Full mathematical treatment of right principal and associated bundles

Full lecture notes on principal and associated bundles

Countable-choice assumption used to form the Lie algebra

Finite-dimensional real Lie group

Lie algebra of a Lie group

Topological right principal bundle and quotient convention

Smooth right principal bundle

Finite-dimensional group representation

Associated representation bundle

Smooth vector-bundle structure on the associated quotient