• Definitions / Notation used
• $\mathcal{A}$ is affine modeled on $N$, hence translations are forced
• Assumptions vs Consequences
  • Definitional (geometry you don’t get to negotiate)
  • Ansatz (choice of this instantiation)
  • Consequences
• Why this matters
• Key takeaway
• Technical takeaway

A connection is not a tensor, so it isn’t a linear object you can add and subtract at will. And yet, in actual calculations, we constantly take differences of gauge potentials, linearize around a background, and treat variations like honest 1-forms. That’s not hypocrisy — it’s geometry: the space of connections is an affine space, and its model vector space is precisely the space of adjoint-valued 1-forms. It follows that “translations” by adjoint 1-forms are not an optional convenience; they are the native linear structure of connection space. Any symmetry principle that pretends otherwise will eventually get tangled when you try to make torsion-like objects transform covariantly.

§Definitions / Notation used

• $X=X^4,$ $Y=Y^{14},$ immersion $\iota : X \hookrightarrow Y,$ pullback $\iota^*.$
• Along $\iota(X): TY|_X\simeq TX\oplus N_{\iota},$ indices $\mu,\nu$ on $TX,$ $a,b$ on $N_{\iota},$ $M,N$ on $Y.$
• Metric bookkeeping: $g_X := \iota^* g_Y,$ and near $\iota(X), g_Y \simeq g_X \oplus \sigma(x)^2\delta_{ab},$ and $n^a \otimes n^b.$
• Hodge stars $*_{X}$ and $*_{Y}$ are distinct.
• Ambient structure: $Y$ carries a Spin$(7,7)$ structure.
• Gauge bundle: principal $P_H \to Y$ with gauge group $H$ (as in the anchor), and adjoint bundle $\mathrm{ad}(P_H)$.
• Translation module (fixed): $N:=\Omega^1(Y,\mathrm{ad}(P_H)).$
• Shorthand (local): write $\mathcal{A}:=\mathrm{Conn}(P_H)$ for the set of all connections on $P_H.$
• Rule: never treat a connection as a tensor; tensorial objects are differences like $\delta A:=A-A_0$ once a background $A_0$ is chosen (background itself is “coming next,” not developed here).

§$\mathcal{A}$ is affine modeled on $N$, hence translations are forced

Let’s start with the one statement that keeps everything honest:

Lemma. The space of connections $\mathcal{A}=\mathrm{Conn}(P_H)$ is an affine space modeled on the vector space

$$ N=\Omega^1(Y,\mathrm{ad}(P_H)). $$

What this means operationally is simple: for any two connections $A_1, A_2\in\mathcal{A},$ their difference is a well-defined adjoint-valued 1-form on $Y,$

$$ A_1 - A_2 \in \Omega^1(Y,\mathrm{ad}(P_H))=N, $$

and conversely, for any $A \in \mathcal{A}$ and any $\phi \in N,$ there is a uniquely defined “shifted” connection $A + \phi \in \mathcal{A}.$ You do not get a canonical origin in $\mathcal{A}$ (no preferred “zero connection”), but you do get canonical displacements between connections.

That is the affine geometry: points are connections; vectors are adjoint 1-forms. It’s the same relationship as “events vs displacement vectors” in an affine spacetime. The crucial distinction is that “adding two points” is meaningless, but “subtracting two points” is meaningful.

Now the forcing move: if $N$ is the model vector space of $\mathcal{A},$ then $N$ comes with a canonical translation action on $\mathcal{A}:$

$$ \text{(translation)} \tau_{\phi}:\mathcal{A}\to\mathcal{A}, \tau_{\phi}(A)=A+\phi, \phi\in N. $$

This isn’t a new physical postulate; it’s the intrinsic linear structure of the configuration space you already committed to by working with connections. You can refuse to call these translations “symmetries,” but you cannot refuse that they exist and that all linearized and perturbative statements are statements in $N$.

So why bring in an inhomogeneous group at all? Because the usual gauge group $H$ (vertical automorphisms of $P_H$) acts on $\mathcal{A}$ by the standard formula

$$ A \mapsto A^\epsilon := \epsilon^{-1}A\epsilon + \epsilon^{-1}d\epsilon, \epsilon \in H, $$

and this action is not linear on $A$ (the $\epsilon^{-1}d\epsilon$ term is inhomogeneous). That inhomogeneity is fine as long as you only ever build observables out of curvature. But the torsion-first program in this instantiation insists on using a tensorial 1-form built from connection data (later: an augmented torsion coordinate), and to do that, you must work in the affine language: pick an origin $A_0$ to talk about $\delta A:=A-A_0 \in N,$ and then control how such $N$-valued objects transform. The clean way to make the affine structure equivariant is to enlarge the symmetry group so that it contains both the “rotations” ($H$) and the “translations” ($N$) that are already built into $\mathcal{A}$. That enlargement is precisely the inhomogeneous group $G=H \ltimes N$ introduced next (and it is explicitly developed in the GU draft’s discussion of the inhomogeneous gauge group).

§Assumptions vs Consequences

§Definitional (geometry you don’t get to negotiate)

• $\mathcal{A}=\mathrm{Conn}(P_H)$ is an affine space; for $A_1, A_2 \in \mathcal{A}$, $A_1 - A_2 \in N=\Omega^1(Y,\mathrm{ad}(P_H))$.
• $H$ acts on connections by $A\mapsto A^\epsilon=\epsilon^{-1}A\epsilon+\epsilon^{-1}d\epsilon$ (standard gauge theory).
• Connections are not tensors; only differences like $\delta A$ are tensorial.

§Ansatz (choice of this instantiation)

• We will treat the affine translations by $N$ as part of the fundamental gauge/transport symmetry, i.e., we will work with an inhomogeneous symmetry group rather than only $H$.

§Consequences

• The “missing” linear degrees of freedom live in $N$ and must be tracked explicitly; they are not reducible to ordinary $H$-gauge rotations.
• To write covariant, torsion-first variables, you will inevitably introduce a background $A_0$ as an affine origin (in future parts) and package transformations as a combined rotation + translation.

§Why this matters

Because torsion-first dynamics wants a bona fide $N$-valued field (a tensorial 1-form built from connection data) while ordinary $H$-gauge transformations act inhomogeneously on connections, the only way to keep covariance without cheating is to respect the affine geometry of $\mathcal{A}$: you introduce an affine origin $A_0$ (next), rewrite all “connection fields” as displacements $\delta A\in N$, and enlarge the symmetry so that both rotations and translations are tracked together; that enlargement is exactly what makes the later augmented torsion transform covariantly and what allows the Shiab/Lagrangian sector downstream to pair curvature and a torsion-like 1-form without ever pretending the connection itself is a tensor.

§Key takeaway

Connections live in an affine space, so the “linear directions” of gauge theory live in $N=\Omega^1(Y,\mathrm{ad})$. If you want a torsion-first, covariant formulation, you have to treat those directions as first-class, not as informal perturbations.

§Technical takeaway

$\mathrm{Conn}(P_H)$ is an affine space modeled on $N=\Omega^1(Y,\mathrm{ad}(P_H))\ \Rightarrow\ A_1-A_2 \in N.$