• Definitions / Notation used
• The Technical Heart: Where Torsion Lives
• One Diagram in Words
• Assumptions vs Consequences
• Why This Matters
• Key Takeaway
• Technical Takeaway

What happens when geometry isn’t just curved, but twisted? In conventional setups, torsion is either zero or perturbative—something you switch on when convenient. In GU, torsion is built-in. It’s not a feature; it’s the default.

Why axial? Because the totally antisymmetric part of the torsion tensor—the part you can wedge into a 3-form—is the only one that survives all the project’s constraints: gauge covariance, pullback consistency, and compatibility with chiral fermions.

More specifically, the augmented torsion variable

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

lives in the translation module $N = \Omega^1(Y, \mathfrak{ad})$ and enters directly into the dynamics via the action:

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

In this post, we spell out where this torsion lives in the decomposition $TY|_X \simeq TX \oplus N_\iota$, how it acts, and why axial torsion is the only one that does the job without cheating.

§Definitions / Notation used

• $\omega = (\varepsilon, \xi)$: transport variable (gauge + translation)
• $A_0$: fixed background connection on $Y$
• $T := \xi - \varepsilon^{-1} d_{A_0} \varepsilon$: augmented torsion
• $F_B$: curvature of $B_\omega := A_0 \cdot \varepsilon$
• $\Upsilon_\omega := \bullet_\varepsilon(F_B) - \kappa_1 T$: dynamic variable for curvature-minus-torsion
• $I_1(\omega)$: action $\int_Y \langle T, *_Y \Upsilon_\omega\rangle$
• $TY|_X \simeq TX \oplus N_\iota$: tangent/normal decomposition
• $TX$: tangential bundle (directions along $X$)
• $N_\iota$: normal bundle (directions orthogonal to $X$)
• Axial torsion: totally antisymmetric part, expressible as a 3-form

§The Technical Heart: Where Torsion Lives

Let’s decompose torsion into its components under the $TX \oplus N_\iota$ split. A 1-form $T$ with values in $\mathfrak{ad}(P_H)$ can have three kinds of components:

1. Tangential: $T_\mu dx^\mu$ — observable directions
2. Mixed: $T_\mu^a dx^\mu \otimes n^a$ — hybrid
3. Normal: $T^a n^a$ — internal-only

But torsion enters dynamics not linearly but quadratically, via wedge products and $*_Y$ duals. That means only combinations that result in gauge-invariant 3-forms matter. The only such 3-form that is:

• gauge-covariant
• nontrivial under pullback
• consistent with spinor coupling

is the totally antisymmetric axial torsion:

$$ T = T_{abc} n^a \wedge n^b \wedge n^c $$

This is a 3-form fully in the normal bundle—nothing tangential survives. It is “purely internal” but has real effects: through its coupling to curvature (via the Shiab operator $\bullet_\varepsilon$) and through fermion overlap, it governs both chirality selection and mass-like effects.

Why axial? Because it’s the only torsion component that:

• is manifestly antisymmetric
• wedges into $\Theta_E$ consistently
• survives under $\mathrm{Spin}(7,7)$ projection
• doesn’t violate pullback rules

§One Diagram in Words

Imagine the 10D normal space $N_\iota$ at each point of $X$ as a flexible internal frame. Axial torsion is a twisting of this frame that wraps three normal directions at once. It doesn’t affect $X$ directly—but it skews how fields pull back, how chiral modes survive, and how overlaps induce mass scales.

§Assumptions vs Consequences

Assumptions (Ansatz):

• Torsion is present by default in the connection.
• Only the totally antisymmetric (axial) component is turned on.
• Torsion is valued in $\mathfrak{ad}(P_H)$, and transforms covariantly.
• Dynamics are written in terms of augmented torsion $T = \xi - \varepsilon^{-1} d_{A_0} \varepsilon$.

Consequences (Operational):

• Only internal (normal-normal-normal) components of torsion survive in the action.
• Tangential or mixed torsion components vanish or violate the pullback structure.
• The action couples $T$ to curvature via Shiab: no naive Ricci-like traces.
• $T$ acts as the geometric selector for chiral fermion modes on $X$.

§Why This Matters

• Fermion chirality is not inserted—it is selected by the geometry of axial torsion.
• Mass scales from overlap depend on torsion-localized Hermite modes.
• Gauge invariance of the action $I_1(\omega)$ hinges on using $T$, not raw connection differences.
• Cosmological terms inherit $\sigma$-weighted suppression from the wedge structure of $T$.

§Key Takeaway

In this program, axial torsion isn’t a tweak—it’s the backbone of chirality, mass generation, and internal geometry. It lives entirely in the normal bundle and defines the gravitational interaction structure from the inside out.

§Technical Takeaway

• $T = \xi - \varepsilon^{-1} d_{A_0} \varepsilon$
• Axial torsion: $T_{abc} n^a \wedge n^b \wedge n^c \in \Omega^3(Y, \mathfrak{ad})$
• Only totally antisymmetric components enter the action $I_1(\omega)$