• 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

$$ I_1(\omega) = \int_Y \langle T, *_Y (\bullet_\varepsilon(F_B) - \kappa_1 T) \rangle \quad \Rightarrow \quad \Upsilon_\omega = 0 $$