nusa
Nusa
nusa@gitcitadel.com,nusa
Jun 20, 2026

Spinors and fermions

How spinors on $Y$ become fermions on $X$. The observer never gets 14D fermions on $X$; they get a pulled-back slice of a 14D spinor, decomposed by the tangent/normal split.

If you let spinors live on $Y$, then “fermions on spacetime” becomes a derived notion, not a starting axiom. The immersion $\iota : X \hookrightarrow Y$ does two jobs at once: it induces the observed metric $g_X = \iota^\ast g_Y$ , and it restricts every ambient field to what the observer can actually sample. For spinors, that restriction translates into a spectral filter, because the normal directions $N_\iota$ come with their own mode structure. In this instantiation, we use the Hermite basis to turn “internal degrees of freedom” into normal-bundle oscillator modes, and it tells us what can survive on $X$.

Definitions / Notation used

  • $X = X^4$, $Y = Y^{14}$, $\iota : X \hookrightarrow Y$, and pullback $\iota^\ast$.
  • Along $\iota(X)$: $TY|X \simeq TX \oplus N\iota,$ with indices $\mu,\nu$ on $TX$, indices $a,b$ on $N_\iota$, and indices $M,N$ on $Y$.
  • The metric split is $g_X := \iota^\ast g_Y, g_Y \simeq g_X \oplus \sigma(x)^2 \delta_{ab} n^a \otimes n^b.$
  • Spinors are native to $Y$: $\Psi \in \Gamma(Y,S_Y).$ The observed restriction is $\Psi_\iota := \Psi \circ \iota.$
  • Local symbol: $n=(n^a)$ denotes normal coordinates, or a local normal section, along $N_\iota$ near $\iota(X)$.

Main technical argument: factorization is the natural basis adapted to $TY|X \simeq TX \oplus N\iota$

Along $\iota(X)$, the observer distinguishes tangent directions from normal directions. The tangent directions become spacetime directions on $X$. The normal directions are transverse to the observer’s world-volume and are not identified with $TX$.

In a local tubular neighborhood of $\iota(X)$ inside $Y$, points can be written schematically as $(x,n)$, with $x \in X$ and $n$ in the normal fiber $N_\iota|_x$.

Apply this to spinors. A spinor on $Y$ is native to $Y$, but near $\iota(X)$ the tangent/normal split lets us organize the ambient Clifford action into tangent and normal pieces. Globally, the spinor bundle can be twisted, so this should not be read as a global product decomposition without qualification. But locally, and spectrally, the split acts as an organizing principle.

The normal dependence is then expanded in a Hermite-Gaussian basis. These are harmonic-oscillator-like modes in the normal directions. The basic factorization is

$$ \Psi(x,n) = \phi_k(n) \otimes \psi_k(x). $$

Here:

  • $\phi_k(n)$ is a Hermite-Gaussian mode on the normal bundle sector $N_\iota$.
  • $k$ is a multi-index labeling normal oscillator excitations.
  • $\psi_k(x)$ is the corresponding spinor factor on $X$ after pullback/restriction.

An ambient spinor admits a normal-mode expansion near the immersed submanifold, and the coefficients of that expansion are spinorial fields along $X$.

Operationally, the observer samples

$$ \Psi_\iota(x) = \Psi(\iota(x)). $$

In local normal coordinates this is the restriction of $\Psi(x,n)$ to the normal profile selected by the immersion. That is how $\iota$ determines which normal profile is sampled.

The metric ansatz explains why the Hermite basis is natural. The normal part of the metric is

$$ \sigma(x)^2 \delta_{ab} n^a \otimes n^b. $$

So the normal sector is locally Euclidean, scaled by $\sigma(x)$. In the simplest spectral approximation, the normal operator behaves like a harmonic oscillator in these transverse directions. The Hermite-Gaussian modes are therefore the natural basis for transverse localization.

This also gives a precise interpretation of “internal structure.” In this construction, internal labels are not assumed to be Standard Model group representations. They arise as labels in the normal-mode sector, combined with the Clifford and transport structure native to $Y$.

The observer’s fermion content on $X$ is not the full ambient space $\Gamma(Y,S_Y)$. It is the pullback-stable, symmetry-respecting part of the ambient spinor spectrum. In low-energy language, it is dominated by the lowest normal modes.

Assumptions vs Consequences

Definitional

Spinors are native to $Y$:

$$ \Psi \in \Gamma(Y,S_Y). $$

The observer sees the restricted field

$$ \Psi_\iota = \Psi \circ \iota. $$

The tangent/normal split along the immersion is

$$ TY|X \simeq TX \oplus N\iota. $$

Ansatz

The normal bundle sector admits a harmonic-oscillator-like spectral organization.

The normal basis is taken to be Hermite--Gaussian:

$$ {\phi_k(n)}. $$

The low-energy observed sector is dominated by low-$k$ modes.

Consequence

The natural basis for observed fermions is

$$ \Psi(x,n) = \phi_k(n) \otimes \psi_k(x). $$

Each normal mode $\phi_k$ carries a spinor coefficient $\psi_k(x)$ on $X$.

The internal labels of the observed fermion sector are not imposed as Standard Model representations. They arise from normal-mode labels, Clifford orientation, and the transport geometry.

Why this matters

  • What “survives on $X$” is not arbitrary 14D spinor data, but a filtered low-$k$ spectrum.
  • It makes chirality selection a spectral question rather than a kinematic projection.
  • It lets torsion act differently across normal-mode sectors, which is the opening needed for mass hierarchy and selection rules.
  • It sets the stage for BRST, and anomaly questions must be asked about the surviving pulled-back spectrum, not about unconstrained ambient spinors.

Key takeaway

Spinors live on $Y$.

The observer on $X$ sees a restricted, mode-filtered spinor sector.

In the Hermite basis, that sector is organized as normal modes times a 4D spinor.

Technical takeaway

$$ TY|X \simeq TX \oplus N\iota, \qquad g_Y \simeq g_X \oplus \sigma(x)^2 \delta_{ab} n^a \otimes n^b. $$

$$ \Psi \in \Gamma(Y,S_Y), \qquad \Psi_\iota = \Psi \circ \iota. $$

$$ \Psi(x,n)=\phi_k(n)\otimes\psi_k(x). $$

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