• Definitions / Notation used
• Main technical argument
  • Argument sketch
• Concrete example: what becomes well-defined only after fixing \(A_0\)
• Assumptions vs Consequences
  • Assumptions
  • Consequences
• Why this matters
• Key takeaway
• Technical takeaway

The space of gauge connections is not linear — it’s affine. If we want to talk about differences of connections, we must choose an origin. In this torsion-first, transport-based instantiation, that origin is the distinguished background connection \(A_0\). Choosing \(A_0\) is not new physics; it is coordinate choice in field space. It’s what makes “transport displacement” and “augmented torsion” into legitimate, pullback-safe variables.

§Definitions / Notation used

• \(X = X^4\), \(Y = Y^{14}\), immersion \(\iota: X \to Y\), pullback \(\iota^\*\).
• Along \(\iota(X): TY|\_X \simeq TX \oplus N\_{\iota},\) with indices \(\mu,\nu\) on \(TX\); \(a,b\) on \(N\_{\iota}\); \(M,N\) on \(Y\).
• \(Y\) has split signature \((7,7)\); Spin\((7,7)\) is structural.
• Principal \(H\)-bundle \(P_H \to Y\); \(\mathrm{ad}(P_H)\) its adjoint bundle.
• \(N := \Omega^1(Y, \mathrm{ad}(P_H))\) (translation space).
• \(G := H \ltimes N\), with \(\omega = (\varepsilon, \\))$, \(\varepsilon \in H\), \(\\) \in N$.
• Distinguished background connection \(A\_0 \in \mathrm{Conn}(P\_H)\); covariant exterior derivative \(d\_{A\_0}\).
• Rotated connection \(B\_{\omega} := A\_0 \cdot \varepsilon\); curvature \(F\_B := dB\_{\omega} + B\_{\omega} \wedge B\_{\omega}\).
• “Connections aren’t tensors”: pull back \(\delta A\) or \(T\), not \(A\) itself.

Native vs invasive reminder: Fields are native to \(Y\); \(X\) only receives invasive data via pullback \(\iota^\*\) of covariant/tensorial objects.

§Main technical argument

Lemma ( \(A\_0\) as affine origin makes a covariant displacement coordinate). Fix \(A\_0 \in \mathrm{Conn}(P\_H)\). Then every connection \(A\) is uniquely expressible as

$$ A = A\_0 + \delta A, \delta A \in \Omega^1(Y, \mathrm{ad}(P\_H)) = N. $$

Moreover, under the \(A\_0\)-preserving (tilted) gauge/transport bookkeeping, \(\delta A\) transforms homogeneously (adjointly), hence \(\iota^\*(\delta A)\) is well-defined on \(X\).

§Argument sketch

\(\mathrm{Conn}(P_H)\) is an affine space modeled on \(N\). Without an origin, “the connection field” is not a vector variable, and any attempt to treat \(A\) as a tensor causes inhomogeneous transformation terms to show up.

Once \(A_0\) is fixed, \(\delta A := A - A_0\) is a bona fide \(\mathrm{ad}(P_H)\)-valued 1-form. The point of the transport group \(G = H \ltimes N\) is that it packages the inhomogeneous “connection disease term” into a compensator. Restricting to the tilted embedding of \(H\) inside \(G\) (the \(A_0\)-preserving subgroup), the inhomogeneous pieces cancel, leaving \(\delta A\) transforming as

$$ \delta A \mapsto h^{-1} (\delta A) h. $$

That makes \(\delta A\) pullback-safe:

$$ \iota^\*(\delta A) \in \Omega^1(X, \mathrm{ad}(\iota^\* P\_H)). $$

§Concrete example: what becomes well-defined only after fixing \(A_0\)

• Illegal (not covariant): \(\iota^\*(A)\).
• Correct: \(\iota^\*(A - A\_0) = \iota^\*(\delta A)\).

The cancellation is exactly “subtract the same disease term twice.”

§Assumptions vs Consequences

§Assumptions

• Split signature on \(Y\) is fixed (Spin\((7,7)\) context).
• \(A_0\) is chosen once and held fixed.
• Use \(G = H \ltimes N\) so translation and rotation bookkeeping is explicit.

§Consequences

• Displacements \(\delta A = A - A_0\) are legitimate fields in \(N\).
• Pullback to \(X\) is legal for \(\delta A\) (and later \(T\)), not for raw \(A\).
• The rotated connection \(B\_{\omega} = A\_0 \cdot \varepsilon\) becomes operational in a way that is consistent with the transport bookkeeping.

§Why this matters

A torsion-first instantiation lives or dies on whether you can write down variables that are both gauge-covariant and pullback-safe. Fixing \(A_0\) is the minimal move that makes “difference-from-background” a meaningful field. It’s not adding structure to \(Y\); it’s selecting an affine origin so transport variables exist. Without \(A_0\), the torsion coordinate you actually want cannot even be stated cleanly.

§Key takeaway

Choosing \(A_0\) is choosing an affine origin, not adding physics.

§Technical takeaway

The only pullback-safe “connection-like” data are differences from \(A_0\) (\(\delta A\)) or compensated combinations (\(T\)). Raw \(A\) is not tensorial.