Once bosons live on $Y$, the next practical question is simple: why do observers on $X$ experience anything like gauge bosons on spacetime? The answer is not that we add gauge fields to $X$. The answer is that an adjoint-valued 1-form on $Y$, when restricted to the immersed submanifold and pulled back along $\iota$, has tangential components that become genuine 1-forms on $X$. The normal components remain geometrically present, but they do not pull back as spacetime vector fields.

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 $TY$.
• $g_X := \iota^\ast g_Y$. 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 $A_0$).
• The Shiab operator: $\bullet_\varepsilon$.
• Local notation used only here: $P_\parallel$ and $P_\perp$ denote the projections of $TY|{\iota(X)}$ onto $TX$ and $N\iota$.

Main technical argument: restriction, split, pullback

Take an adjoint-valued 1-form on $Y$:

$$ \alpha\in\Omega^1(Y,\mathrm{ad}(P_H)). $$

Locally, this could stand for $\vartheta$, for a tensorial displacement such as $T$, or for a variation of connection data. We use $\alpha$ only as a local placeholder.

Along $\iota(X)$, the tangent bundle of $Y$ splits as $TY|{\iota(X)}\simeq TX\oplus N\iota.$

So every tangent direction along the immersed copy of $X$ decomposes into a tangential part and a normal part. Therefore the restricted 1-form decomposes as

$$ \alpha|{\iota(X)} = \alpha\parallel + \alpha_\perp, $$

where

$$ \alpha_\parallel(\cdot):=\alpha(P_\parallel\cdot), \qquad \alpha_\perp(\cdot):=\alpha(P_\perp\cdot). $$

This is still a statement along $\iota(X)\subset Y$. It is not yet a field on $X$.

Now pull back.

By definition,

$$ (\iota^*\alpha)x(u) = \alpha{\iota(x)}(d\iota_x(u)), \qquad u\in T_xX. $$

But

$$ d\iota_x(u)\in T_{\iota(x)}\iota(X)\simeq T_xX. $$

It has no normal component. Therefore the normal part of $\alpha$ is never evaluated by the pullback. Hence

$$ \boxed{\iota^\alpha=\iota^\alpha_\parallel,\qquad \iota^*\alpha_\perp=0.} $$

This is the basic selection rule.

Tangential components of adjoint-valued 1-forms on $Y$ become adjoint-valued 1-forms on $X$. Those are exactly the kinematic type of gauge-boson-like fields on spacetime.

Normal components do not become spacetime 1-forms. They are not necessarily irrelevant. They can affect the $X$-physics through curvature, torsion, localization, overlap integrals, and the normal geometry controlled by $\sigma(x)$. But they do not appear as ordinary gauge vector fields on $X$.

Gauge-boson-like fields on $X$

The clean tensorial objects to pull back are not raw connections treated as tensors. They are

$$ T\in\Omega^1(Y,\mathrm{ad}(P_H)), \qquad F_B\in\Omega^2(Y,\mathrm{ad}(P_H)). $$

Therefore $X$ receives

$$ \iota^*T\in\Omega^1(X,\mathrm{ad}(P_H)), $$

and

$$ \iota^*F_B\in\Omega^2(X,\mathrm{ad}(P_H)). $$

The first has the type of a spacetime adjoint-valued 1-form. The second has the type of a spacetime adjoint-valued field strength.

Thus the gauge-boson-like sector on $X$ is not introduced independently. It is the tangential pullback of the ambient adjoint geometry.

Schematically:

$$ \text{ambient adjoint 1-form on }Y \quad\longrightarrow\quad \text{tangential component along }X \quad\longrightarrow\quad \text{adjoint-valued 1-form on }X. $$

That is what makes it look like a gauge boson.

Spin-connection-like pieces

The spin-connection-like sector is subtler.

A spin connection on $X$ is not simply “some component of $B_\omega$” unless we specify which adjoint directions act gravitationally on the induced tangent and spinor structures. In this instantiation, that role is handled by the fixed gravitational selector $E$ and the fixed form $\Theta_E$.

The $E$-selected sector identifies the gravitational block inside the adjoint geometry. The Shiab operator then determines how curvature in that block contributes to the Einstein-like equation.

So the distinction is gauge-boson-like pieces are the non-gravitational adjoint-valued tangential 1-form components seen on $X$, while spin-connection-like pieces are the $E$-selected gravitational components of the ambient adjoint geometry, interpreted through the induced geometry of the immersion and the Shiab/$\Theta_E$ contraction.

The important constraint is that we do not take a naive Ricci trace of $F_B$. The Einstein-like contraction is only defined through $\bullet_\varepsilon(F_B),$ with $E$ and $\Theta_E$ already fixed.

One diagram in words

Start with the ambient transport variable

$$ \omega=(\varepsilon,\vartheta) $$

on $Y$. From it form the tensorial objects

$$ T=\vartheta-\varepsilon^{-1}d_{A_0}\varepsilon, \qquad F_B. $$

Restrict them to $\iota(X)$. Along $\iota(X)$, split directions using

$$ TY|{\iota(X)}\simeq TX\oplus N\iota. $$

The tangential pieces pull back to honest forms on $X$. These are the gauge-boson-like fields and field strengths seen by spacetime observers. The normal pieces do not pull back as spacetime 1-forms, but they remain part of the ambient geometry and can influence $X$-physics through torsion, curvature, localization, overlap, and the $\sigma(x)$-weighted normal geometry.

Assumptions vs Consequences

Definitional

Fields are native to $Y$. $X$ receives fields by restriction and pullback along $\iota:X\hookrightarrow Y.$

The ambient structure on $Y$ is $\mathrm{Spin}(7,7)$ with split signature $(7,7)$.

Connections are not tensors. Tensorial statements on $X$ should use objects such as $T, F_B, \delta T, \delta F_B,$ or properly induced connection structures.

Ansatz

Along the immersed spacetime,

$$ TY|{\iota(X)}\simeq TX\oplus N\iota. $$

The metric takes the split form

$$ g_Y\simeq g_X\oplus \sigma(x)^2\delta_{ab}\hat n^a\otimes \hat n^b. $$

The selectors $E$ and $\Theta_E$ fix the gravitational block and the Shiab contraction.

Axial torsion is default and non-perturbative.

Consequence

For any

$$ \alpha\in\Omega^1(Y,\mathrm{ad}(P_H)), $$

one has

$$ \iota^\alpha=\iota^\alpha_\parallel, \qquad \iota^*\alpha_\perp=0. $$

Thus tangential adjoint 1-form components become gauge-boson-like fields on $X$.

The $E$-selected adjoint sector supplies spin-connection-like gravitational data, but only through the induced geometry and the Shiab/$\Theta_E$ contraction.

Why this matters

• After the tangential/normal split, we still need to decompose the adjoint directions themselves.
• Boson masses require a precise statement of which $X$-visible fields are being massed. The pullback selection rule identifies those fields before overlap and torsion effects are introduced.
• Fermions on $X$ couple to the induced and pulled-back bosonic geometry. Chirality selection from axial torsion depends on which ambient components survive the pullback.

Key takeaway

Restrict to $\iota(X)$, split by $TX\oplus N_\iota$, then pull back.

Tangential adjoint 1-form components look like gauge bosons on $X$.

The gravitational sector is the $E$-selected adjoint geometry interpreted through Shiab/$\Theta_E$, not through a naive Ricci trace.

Technical takeaway

$$ \alpha|{\iota(X)}=\alpha\parallel+\alpha_\perp. $$

$$ \iota^\ast\alpha=\iota^\ast\alpha_\parallel, \qquad \iota^\ast\alpha_\perp=0. $$

$X$ -visible tensorial bosonic data: $\iota^\ast T,\qquad \iota^\ast F_B.$

Einstein-like contraction: $\bullet_\varepsilon(F_B)$ with fixed $E,\Theta_E.$