The geometric source domain

Definition 2.2 (Global geometric configuration). A global geometric configuration is a tuple $$\Psi=\bigl(M,\;g,\;P,\;A,\;\Phi,\;\mathcal{B}\bigr),\tag{2}$$ in which

1. $g$ is a Lorentzian metric on $M$;

2. $\pi\colon P\to M$ is a principal $G$-bundle;

3. $A\in\Omega^1(P,\mathfrak{g})$ is a principal connection on $P$ [6, 3];

4. $\Phi$ is a section of the associated bundle $P\times_\rho W\to M$;

5. $\mathcal{B}$ denotes the boundary, asymptotic, causal, and global data required to make the model well posed. Among these data we always include a choice of time orientation $\mathfrak{t}$ for $g$; the remaining components of $\mathcal{B}$ are model-specific and may comprise, for instance, fixed induced data on $\partial M$, a decay class at infinity, a fixed conformal boundary structure, a global hyperbolicity requirement, a spin structure, or a fixed topological sector.

Definition 2.4 (Admissibility class). An admissibility class $\mathcal{C}_{\mathrm{geo}}$ is a specified class of tuples (2) satisfying declared conditions on regularity, signature, causal structure, boundary and asymptotic behaviour, and topological sector, and in which every $g$ is time-orientable with a distinguished time orientation $\mathfrak{t}\in\mathcal{B}$. The class $\mathcal{C}_{\mathrm{geo}}$ is part of the data of the model. All subsequent statements are relative to a fixed $\mathcal{C}_{\mathrm{geo}}$.

Definition 2.5 (Ambient allowed automorphisms). Fix a principal $G$-bundle $P\to M$. Let $\operatorname{Diff}_{\mathcal{B}}(M)$ denote the group of diffeomorphisms of $M$ preserving the orientation of $M$ and the non-metric structural data in $\mathcal{B}$ (boundary, asymptotic regime, topological sector), and let $$\operatorname{Aut}_{\mathcal{B}}(P) :=\bigl\{\hat F\in\operatorname{Aut}(P)\;:\;\hat F \text{ covers some } f\in\operatorname{Diff}_{\mathcal{B}}(M)\bigr\}\tag{3}$$ be the group of $G$-equivariant diffeomorphisms of $P$ covering such diffeomorphisms. Define $$\operatorname{Diff}_{P,\mathcal{B}}(M)\;:=\;\text{image of }\ \operatorname{Aut}_{\mathcal{B}}(P)\ \text{ in }\ \operatorname{Diff}_{\mathcal{B}}(M),\tag{4}$$ that is, the subgroup of allowed base diffeomorphisms that admit an equivariant lift to $P$.

Definition 2.8 (Redundancy groupoid).

Let $\mathfrak{G}_{\mathrm{geo}}$ be the groupoid whose objects are the elements of $\mathcal{C}_{\mathrm{geo}}$. Given two objects $$\Psi=(M,g,P,A,\Phi,\mathcal{B}), \qquad \Psi'=(M,g',P',A',\Phi',\mathcal{B}'),$$ the arrows $F\colon\Psi\to\Psi'$ are the pairs $F=(\hat F,f)$ consisting of

• an allowed diffeomorphism $f\in\operatorname{Diff}_{\mathcal{B}}(M)$, and

• a $G$-equivariant bundle isomorphism $\hat F\colon P\to P'$ covering $f$,

subject to $$f_*g=g', \qquad f_*\mathfrak{t}=\mathfrak{t}', \qquad (\hat F^{-1})^{*}A = A', \qquad \hat F_{\rho}\circ\Phi = \Phi'\circ f, \qquad f_*\mathcal{B}=\mathcal{B}',\tag{7}$$ where $\hat F_\rho$ is the induced isomorphism $P\times_\rho W\to P'\times_\rho W$. The second condition is the requirement that arrows preserve the chosen time orientation; it is not implied by the first, since $f_*g=g'$ alone permits $f$ to exchange the two components of the timelike cone bundle. Composition is composition of bundle maps; identities are $(\mathrm{id}_P,\mathrm{id}_M)$. The isotropy group of $\mathfrak{G}_{\mathrm{geo}}$ at $\Psi$ is $\operatorname{Aut}(\Psi)$ of (6).

Definition 2.9 (Physical equivalence and the reduced source).

Write $$\Psi\simeq_{\mathrm{geo}}\Psi' \quad\Longleftrightarrow\quad \exists\,F\colon\Psi\to\Psi' \text{ in } \mathfrak{G}_{\mathrm{geo}}.\tag{8}$$ This is an equivalence relation on $\mathcal{C}_{\mathrm{geo}}$ because $\mathfrak{G}_{\mathrm{geo}}$ is a groupoid. The reduced geometric source is the orbit space $$S_{\mathrm{geo}}\;:=\; \mathcal{C}_{\mathrm{geo}}/\!\simeq_{\mathrm{geo}} ,\tag{9}$$ with quotient map $q\colon\mathcal{C}_{\mathrm{geo}}\to S_{\mathrm{geo}}$, $q(\Psi)=[\Psi]$.