• 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.