Finite-dimensional gauge slices and Faddeev–Popov operators

Assume the axiom of countable choice ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $M$ be a finite-dimensional, Hausdorff, second-countable smooth manifold without boundary. Fix a compact noncommutative finite-dimensional real Lie group $G$, whose Lie algebra is $\mathfrak{g}=T_{e}G$ with bracket transported from left-invariant vector fields. Let $\pi :P\to M$ be a smooth right principal bundle: the right action is smooth, free and transitive on each fibre, and each point has a local equivariant principal chart. Let $H\subset TP$ be a supplied smooth right-invariant horizontal complement to $VP=\ker (d\pi )$, and write $v$ for projection onto $VP$ along $H$. For $Z\in \mathfrak{g}$, the fundamental vertical vector is $Z_P(p)={\frac{d}{dt}\mid }_{t=0}p{\exp }_G(tZ)$. The principal connection form $\omega$ is a smooth $\mathfrak{g}$-valued one-form satisfying $\omega (Z_P)=Z$, $\ker \omega =H$, and $R_g^{\ast }\omega ={\mathrm{Ad}}_{g^{-1}}\omega$, where ${\mathrm{Ad}}_g=d(C_g{)}_e$ and $C_g(h)=gh{g}^{-1}$. In a local section $s:U\to P$ (a smooth map with $\pi \circ s={\mathrm{id}}_U$), the potential $A=s^{\ast }\omega$ changes under a section change $s^{\mathrm{\prime }}=su$, for smooth $u:U\to G$, by $A^{\mathrm{\prime }}={\mathrm{Ad}}_{u^{-1}}A+{\theta }_L(u)$, where ${\theta }_L(u)=u^{\ast }{\theta }_L$, $({\theta }_L{)}_g=d(L_{g^{-1}}{)}_g$, and $L_g(h)=gh$. The full gauge group consists of smooth diffeomorphisms $\mathrm{\Phi }:P\to P$ satisfying $\pi \circ \mathrm{\Phi }=\pi$ and $\mathrm{\Phi }(pg)=\mathrm{\Phi }(p)g$; this definition gives it no finite-dimensional Lie-group structure.

A finite-dimensional regulated gauge-slice datum consists of integers $1\le k\le n$; a finite-dimensional Lie group $K$ of dimension $k$, with Lie algebra $\mathfrak{k}:=T_{e}K$ and bracket transported from left-invariant vector fields; a chosen smooth positive left-invariant density ${\mu }_K$, meaning $(L_h^K{)}^{\ast }{\mu }_K={\mu }_K$ for $L_h^K(g)=hg$; a finite-dimensional smooth manifold $X$ of dimension $n$ with a chosen smooth positive density ${\mu }_X$; and a smooth right action $\alpha :X\times K\to X$, written $x\cdot g$, with $x\cdot e=x$ and $(x\cdot g)\cdot h=x\cdot (gh)$. A smooth positive density on an $m$-manifold has coordinate form $\rho (x)\mathrm{\mid }d{x}^1\cdots d{x}^m\mathrm{\mid }$ with smooth $\rho >0$, and transforms by the absolute coordinate Jacobian. The group $K$ and its action on $X$ are separate regulator data. They are not identified with the full gauge group or the full space of connections unless a particular finite model explicitly constructs such an action and verifies that it preserves $X$.

Choose an embedded smooth submanifold $S\subset X$ of codimension $k$, a smooth positive density ${\mu }_S$, and an open neighborhood $U\subset X$ of a point $s\in S$. Embedded means that near each point of $S$, a smooth chart on $X$ identifies $S$ with ${\mathbb{R}}^{n-k}\times \{0\}$. A gauge condition on $U$ is a smooth map $\chi :U\to {\mathbb{R}}^k$ such that $U\cap S={\chi }^{-1}(0)$ and, for every $s^{\mathrm{\prime }}\in U\cap S$, $d{\chi }_{s^{\mathrm{\prime }}}$ is surjective with kernel $T_{s^{\mathrm{\prime }}}S$. For every $x\in U$, the infinitesimal orbit map is $R_x:=d({\alpha }_x{)}_e:\mathfrak{k}\to T_{x}X$, where ${\alpha }_x(g)=x\cdot g$; call its kernel the infinitesimal stabilizer at $x$. The Faddeev–Popov operator at $x$ is the ordinary linear map $M_x:=d{\chi }_x\circ R_x:\mathfrak{k}\to {\mathbb{R}}^k$. At a slice point $s^{\mathrm{\prime }}\in U\cap S$, call the condition transverse when $M_{s^{\mathrm{\prime }}}$ is an isomorphism. Choose a positive density $\nu$ on $\mathfrak{k}$ equal to $({\mu }_K{)}_e$, and the standard positive density $\lambda$ on ${\mathbb{R}}^k$. For every $x\in U$, the finite-dimensional Faddeev–Popov Jacobian ${\mathrm{\Delta }}_{\mathrm{F}\mathrm{P}}(x)$ is the nonnegative density ratio defined by $M_x^{\ast }\lambda ={\mathrm{\Delta }}_{\mathrm{F}\mathrm{P}}(x)\nu$; in chosen bases it is $\mathrm{\mid }\det M_x\mathrm{\mid }$, and it is zero when $M_x$ is singular.

This datum defines only finite-dimensional Jacobians and local coordinate slices. It neither defines a determinant of an infinite-dimensional operator nor constructs a functional measure or a continuum path integral. A finite-mode or lattice application must supply the regulator manifold $X$, the finite-dimensional group $K$, their smooth action, and the domains and densities above.

Skinner, finite-dimensional Faddeev–Popov model and its limitations

Mitter and Viallet, Sobolev gauge slices and local orbit-space geometry

Countable-choice assumption inherited by the gauge data

Compact group, right principal bundle, and bundle sector

Right-action principal connection form and local potential

Local section-change convention for the potential

Full gauge group as base-fixing equivariant diffeomorphisms

Finite-dimensional smooth Lie group

Left Maurer–Cartan form