• Definitions / Notation used
• Main technical argument
  • Explicit transformation line showing cancellation
• Geometric meaning
• Assumptions vs Consequences
  • Assumptions
  • Consequences
• Why this matters
• Key takeaway
• Technical takeaway

Naive torsion is usually where gauge covariance goes to die. GU’s transport move is to stop trying to “fix torsion” after the fact and instead define torsion as a compensated difference from the outset. The augmented torsion

$$ T := \eta - \varepsilon^{-1} d_{A_0} \varepsilon $$

is engineered so the bad (inhomogeneous) terms cancel. This gives a torsion variable that transforms adjointly and can be pulled back to $X$ without illegal moves.

§Definitions / Notation used

• $\omega = (\varepsilon, \eta) \in G = H \ltimes N$, with $\eta \in \Omega^1(Y, \mathrm{ad}(P_H))$.
• $A_0$ fixed; $d_{A_0}$ is the covariant exterior derivative.
• $B_{\omega} := A_0 \cdot \varepsilon$, curvature $F_B := dB_{\omega} + B_{\omega} \wedge B_{\omega}$.
• Augmented torsion:

$$ T(\omega) := \eta - \varepsilon^{-1} d_{A_0} \varepsilon \in \Omega^1\big(Y, \mathrm{ad}(P_H)\big). $$

§Main technical argument

Lemma (covariance of augmented torsion).

Under the tilted right action by $h \in H$, augmented torsion transforms adjointly:

$$ T(\omega \cdot h) = h^{-1} T(\omega) h. $$

§Explicit transformation line showing cancellation

Use the tilted ($A_0$-aware) bookkeeping where the components transform as

$$ \varepsilon^\prime = \varepsilon h, \eta^\prime = h^{-1} \eta h + h^{-1} d_{A_0} h. $$

Compute:

$$ T^\prime = \eta^\prime - (\varepsilon^\prime)^{-1} d_{A_0} (\varepsilon^\prime) $$

$$ = \big(h^{-1} \eta h + h^{-1} d_{A_0} h\big) - (h^{-1} \varepsilon^{-1}) d_{A_0}(\varepsilon h) $$

$$ = h^{-1} \eta h + h^{-1} d_{A_0} h - h^{-1} \varepsilon^{-1} (d_{A_0} \varepsilon) h - h^{-1} d_{A_0} h $$

$$ = h^{-1} \big( \eta - \varepsilon^{-1} d_{A_0} \varepsilon \big) h $$

$$ = h^{-1} T h. $$

The inhomogeneous term $h^{-1} d_{A_0} h$ cancels exactly.

§Geometric meaning

$T$ is the affine difference between two connection-building routes from the same $\omega$:

• translate: $A_0 + \eta$
• rotate: $B_{\omega} = A_0 \cdot \varepsilon$

Then $T$ is the “difference” in the model space $N$:

$$ T = (A_0 + \eta) - (A_0 \cdot \varepsilon), $$

which is precisely $\eta - \varepsilon^{-1} d_{A_0} \varepsilon$.

§Assumptions vs Consequences

§Assumptions

• $A_0$ fixed, $d_{A_0}$ used in compensators.
• Tilted $H$-action includes the inhomogeneous $h^{-1} d_{A_0} h$ term on $\eta $.

§Consequences

• $T$ transforms covariantly: $T \mapsto h^{-1} T h$.
• Therefore $\iota^*(T)$ is well-defined on $X$.
• $B_{\omega}$ and $F_B$ are operational objects compatible with transport bookkeeping.

§Why this matters

$T$ is the first transport-native torsion variable that survives gauge covariance. It is linear (lives in $N$), pullback-safe (can be observed on $X$ via $\iota^*$), and built to avoid “connections-aren’t-tensors” pitfalls. Later, when we build Einstein-like curvature contractions, we will not take naive Ricci traces on $\mathrm{ad}$-valued curvature; those contractions are replaced by Shiab $\bullet_{\varepsilon}$. $T$ is designed to be compatible with that gauge-aware contraction discipline.

§Key takeaway

Augmented torsion is the right torsion variable because it is covariant by construction.

§Technical takeaway

$T := \eta - \varepsilon^{-1} d_{A_0} \varepsilon$ transforms as $T \mapsto h^{-1} T h$ because the inhomogeneous terms cancel line-by-line.