• 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

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