• Definitions / Notation used
• What this action is really doing
• Variation sketch why \(\Upsilon_\omega = 0\)
  • Step 1: Choose a legal variation
  • Step 2: Vary the action
• Technical lemma normalization
• What about varying \(\varepsilon\)
• Assumptions vs consequences
  • Assumptions
  • Consequences
• Why this matters
• Key takeaway
• Technical takeaway

§Definitions / Notation used

• \(Y\) is a 14D manifold with split signature \((7,7)\). \(X\) is a 4D manifold immersed by \(\iota: X \hookrightarrow Y\). Along \(\iota(X)\): \(TY|\_X \simeq TX \oplus N\_\iota\), with indices \(\mu,\nu\) on \(TX\); \(a,b\) on \(N\_\iota\); and \(M,N\) on \(TY\).
• \(g\_X := \iota^\* g\_Y\). We use the \(\sigma\)-split: \(g\_Y \simeq g\_X \oplus \sigma^2(x) \delta\_{ab} \hat{n}^a \hat{n}^b\), and distinguish \(\ast_X\) from \(\ast_Y\).
• \(H\) is the gauge group, \(N := \Omega^1(Y,\mathrm{ad})\) (\(\mathrm{ad} = \mathrm{ad}(P_H)\)), and \(G := H \ltimes N\). A generic gauge-affine variable is \(\omega = (\varepsilon, \eta) \in G\).
• \(A_0\) is the chosen background connection on \(Y\). From \(\omega\) we form \(B_{\omega}\) (the transported/rotated connection built from \(A_0\) and \(\varepsilon\)), its curvature \(F_B\), and the augmented torsion \(T\) (the covariant “difference” built from \(\eta\) and \(\varepsilon\) relative to \(A_0\)).
• Augmented torsion: \(T := \eta - \varepsilon^{-1} d\_{A\_0} \varepsilon \in \Omega^1(Y, \mathrm{ad}(P\_H))\).
• The Shiab operator: \(\bullet_\varepsilon\).
• Swervature: \(\bullet\_\varepsilon(F\_B)\)

Define:

$$ \Upsilon_\omega := \bullet_\varepsilon(F_B) - \kappa_1 T $$

$$ I\_1(\omega) := \int\_Y \langle T, \*\_Y \Upsilon\_\omega \rangle $$

§What this action is really doing

\(I_1\) is a torsion swervature pairing. It is constructed so it lives in the same bundle as \(T\), allowing a gauge-covariant pairing.

The action is written in terms of:

1. a covariant 1-form \(T\)
2. a covariant form \(\bullet_\varepsilon(F\_B)\)

both valued in \(\mathrm{ad}(P_H)\), paired via: $$ \langle \cdot , \cdot \rangle \quad \text{and} \quad \*\_Y $$

Torsion first principle: the field is \(T\), not the connection.

§Variation sketch why \(\Upsilon_\omega = 0\)

§Step 1: Choose a legal variation

Connections are affine, so vary the translation part:

$$ \omega\_s = (\varepsilon, \eta + s \alpha), \quad \alpha \in \Omega^1(Y, \mathrm{ad}(P\_H)) $$

Then: \(T_s = T + s \alpha\) and \(\delta T = \alpha\).

Since \(B_\omega\) depends only on \(\varepsilon\):

$$ \delta F_B = 0 $$

Thus: $$ \delta \Upsilon\_\omega = -\kappa\_1 \alpha $$

§Step 2: Vary the action

$$ \delta I\_1 = \int\_Y \left( \langle \delta T, \*\_Y \Upsilon\_\omega \rangle + \langle T, \*\_Y \delta \Upsilon\_\omega \rangle \right) $$

Insert: $$ \delta T = \alpha, \quad \delta \Upsilon\_\omega = -\kappa\_1 \alpha $$

$$ \delta I\_1 = \int\_Y \left( \langle \alpha, \*\_Y \Upsilon\_\omega \rangle - \kappa\_1 \langle T, \*\_Y \alpha \rangle \right) $$

With the normalization convention: $$ \frac{\delta}{\delta T} \langle T, \*\_Y T \rangle = \*\_Y T $$

the terms combine into: $$ \delta I\_1 = \int\_Y \langle \alpha, \*\_Y (\bullet\_\varepsilon(F\_B) - \kappa\_1 T) \rangle $$

$$ \delta I\_1 = \int\_Y \langle \alpha, \*\_Y \Upsilon\_\omega \rangle $$

Since \(\alpha\) is arbitrary: $$ \Upsilon_\omega = 0 $$

§Technical lemma normalization

Define: $$ Q(T) := \int\_Y \langle T, \*\_Y T \rangle $$

Then: $$ \delta Q(T)[\alpha] = \int\_Y \langle \alpha, \*\_Y T \rangle $$

No factor of 2 appears due to polarization normalization.

§What about varying \(\varepsilon\)

\(\varepsilon\) enters in:

• \(T = \eta - \varepsilon^{-1} d\_{A\_0} \varepsilon\)
• \(B\_\omega = A\_0 \cdot \varepsilon\)

The resulting variation yields a compatibility condition: a Bianchi-type identity linking curvature and torsion through \(\bullet_\varepsilon\) and \(\Theta_E\).

No Ricci-type contraction appears.

§Assumptions vs consequences

§Assumptions

• \(\mathrm{Spin}(7,7)\) structure on \(Y\)
• metric split with \(\sigma(x)\)
• distinguished background \(A_0\)
• torsion \(T\) as variable
• fixed Shiab operator and \(\Theta_E\)
• pairing normalization

§Consequences

$$ \Upsilon\_\omega = 0 \quad \Rightarrow \quad \bullet\_\varepsilon(F\_B) = \kappa\_1 T $$

• gauge covariant equation
• no connection as a tensor
• no Ricci trace

§Why this matters

This is the point where the construction becomes a field theory on \(Y\). The dynamics are written entirely in covariant objects.

Recovering GR will mean showing that \(\bullet\_\varepsilon(F\_B)\) reduces to the Einstein contraction in a controlled regime.

§Key takeaway

The action is built from torsion \(T\), and its stationary points satisfy: $\Upsilon_\omega = 0 $

§Technical takeaway

$$ I_1(\omega) = \int\_Y \langle T, \*\_Y (\bullet\_\varepsilon(F\_B) - \kappa\_1 T) \rangle \quad \Rightarrow \quad \Upsilon_\omega = 0 $$