• Definitions / Notation used
• Main technical argument: restriction, split, pullback
  • Gauge-boson-like fields on \(X\)
  • Spin-connection-like pieces
  • One diagram in words
• Assumptions vs Consequences
  • Definitional
  • Ansatz
  • Consequence
• Why this matters
• Key takeaway
• Technical takeaway

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 \(E,\Theta_E.\)