Spinors and Immersion Geometry

Let $Y$ be the total (ambient) space and $X$ the observed $4$-manifold. An immersion $\iota: X \hookrightarrow Y$ induces a splitting along $\iota(X)$: $$ \left.TY\right|_{\iota(X)} \simeq TX \oplus N_{\iota} $$ so the spin bundle on $Y$ decomposes as $S_{Y} \simeq S_{X} \otimes S_{N_{\iota}}$. Spinor fields native to $Y$ are non-chiral and appear on $X$ only through pullback: $$\Psi_{X}(x)=(\iota^{\ast} \Psi_{Y})(x).$$ At this stage, there is no intrinsic handedness; the product structure merely organizes external and internal spinor degrees of freedom.

Dirac Spinors with Torsion

A general spin connection on $Y$ in GU incorporates gauge-equivariant torsion: $$ \nabla_{\mu} = \partial_{\mu} + \frac{1}{4}\omega_{\mu}^{ab}\gamma_{a}\gamma_{b}, \qquad \omega_{\mu}^{ab}=\omega_{\mu}^{ab(\mathrm{LC})}+K_{\mu}^{ab}. $$ Here $\omega^{(\mathrm{LC})}$ is the Levi–Civita connection, and $K_{\mu}^{ab}$ is the contorsion derived from the torsion tensor $$ T^{a}{}_{\mu\nu}=\Xi^{a}{}_{\mu\nu} - \big(\varepsilon^{-1} d_{A_{0}}\varepsilon\big)^{a}{}_{\mu\nu}, $$ with $\Xi$ an affine shift, $\varepsilon\in H$ a gauge transformation, and $A_{0}$ a reference connection. The contorsion is defined as $$ K^{ab}{}_{\mu}=\tfrac{1}{2}\Big(T^{ab}{}_{\mu}-T^{ba}{}_{\mu}-T_{\mu}{}^{ab}\Big). $$ The Dirac Lagrangian for a spinor $\psi$ on $Y$ is $$ \mathcal{L}=\bar\psi i\gamma^{\mu} \nabla_{\mu} \psi = \bar\psi i\gamma^{\mu}\left(\partial_{\mu}+\tfrac{1}{4}\omega_{\mu}^{ab}\gamma_{a}\gamma_{b}\right) \psi. $$ Substituting $\omega=\omega^{(\mathrm{LC})}+K$ yields a torsion-free term plus an equivariant interaction: $$ \mathcal{L}_{\text{int}} =\frac{i}{8} \bar\psi \gamma^{\mu} K_{\mu ab}[\gamma^{a} \gamma^{b}] \psi. $$ This ensures that the full connection remains gauge-covariant under the inhomogeneous group $G=H\ltimes N$, providing a unified geometric description of gravitational and gauge structure.

Torsion and the $\gamma_{5}$ Coupling

The torsion tensor naturally isolates an axial component responsible for chirality, without assuming total antisymmetry. We can decompose $$ T_{\lambda\mu\nu}=T_{[\lambda\mu\nu]}+\Phi(B)_{\lambda\mu\nu}, $$ where $\Phi(B)$ is an equivariant compensator ensuring covariance under $G$. The Clifford contraction of $K_{\mu\nu\rho}$ selects the axial vector part. Using the identity (up to conventions) $$ \gamma^{ab}\gamma_{\mu} \sim \epsilon^{ab}{}_{\mu\nu} \gamma^{\nu}\gamma_{5}, $$ the torsion contribution reduces to the axial interaction $$ \mathcal{L}_{\text{int}} =\bar\psi \gamma^{\mu}\gamma_{5} T^{(A)}{\mu} \psi \qquad T^{(A)}_{\mu}=\tfrac{1}{6} \epsilon_{\mu\nu\rho\sigma} T^{\nu\rho\sigma}. $$ Thus, the appearance of $\gamma_{5}$ is not imposed but emerges naturally from the gauge-equivariant torsion of GU. Its Clifford image acts on spinors as an axial vector field, uniting the chirality mechanism with gauge covariance.

Orientation and Handedness Selection

The equation above shows that axial torsion couples oppositely to left- and right-handed components: $$ \bar\psi_{L} \gamma^{\mu}\psi_{L} T^{(A)}_{\mu} = -\bar\psi_{R} \gamma^{\mu}\psi_{R} T^{(A)}_{\mu}. $$ Reversing signature or immersion orientation flips $\epsilon_{\mu\nu\rho\sigma}$ and hence $T^{(A)}_{\mu}\to -T^{(A)}_{\mu}$, interchanging the favored handedness. The immersion thus acts as a geometric parity selector: orientation and signature fix which chirality is energetically realized.

Pullback to $X$: Emergent Chirality

Observation corresponds to pullback, $\Psi_{X}=\iota^{\ast} \Psi_{Y}$. The handedness bias encoded in torsion is inherited. The chiral asymmetry of $X$ arises from torsion on $Y$, inherited equivariantly via pullback. Although $S_{Y}$ is non-chiral, the decomposition $$ S_{Y} \simeq S_{X} \otimes S_{N_{\iota}} $$ together with torsion and orientation makes the $S_{X}$ factor effectively chiral.

Conclusion

GU derives chirality not by imposing a Weyl projection or breaking an internal symmetry, but geometrically, through torsion: $$ \text{Chirality}=\text{Immersion}+\text{Torsion}+\text{Signature/Orientation}. $$ A fundamentally non-chiral spinor theory on $Y$ becomes chiral on $X$ because observation pulls back a torsion-induced $\gamma_{5}$ bias, whose sign is fixed by the immersion’s orientation and inherited equivariantly under the inhomogeneous gauge symmetry.