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Appendix B – Spectral Normalization of the Electromagnetic Sector in SPU

B.1 Purpose of this appendix

This appendix clarifies the spectral and geometric origin of numerical factors appearing in the normalization of the electromagnetic sector of SPU theory, by rigorously separating:

  • what is rigorously derived from spectral geometry
  • what is standard in the literature (spectral action, heat kernel)
  • what is dynamical and external to geometry (in particular, the parameter $\delta$)

The goal is to show how spectral structure fixes the form and natural scale of the gauge term, leaving to RG dynamics the task of determining the final physical value of the coupling constant.

B.2 Spectral action and heat kernel expansion

SPU employs the spectral action in standard form:

$$S_{\text{spec}} = \mathrm{Tr}, f\left(\frac{D^2}{\Lambda^2}\right)$$

where:

  • $D$ is the generalized Dirac operator on the internal space
  • $f$ is a smooth cutoff function
  • $\Lambda$ is the spectral scale

For $\Lambda \to \infty$, the action admits an asymptotic expansion via the heat kernel:

$$\mathrm{Tr}, f\left(\frac{D^2}{\Lambda^2}\right) = \sum_{n \geq 0} f_{4-n}, \Lambda^{4-n}, a_n(D^2)$$

where the coefficients $a_n$ are local geometric invariants.

B.3 Gauge kinetic term from coefficient a₄

The kinetic term of gauge fields emerges uniquely from the coefficient $a_4$:

$$a_4(D^2) \supset \frac{1}{24\pi^2} \int d^4x, \mathrm{Tr}(F_{\mu\nu}F^{\mu\nu})$$

This result is universal and independent of the details of the function $f$.

The physical normalization of the gauge term is thus fixed, at the spectral level, by the combination:

$$S_{\text{gauge}} = \frac{1}{4 g_0^2}\int d^4x, F_{\mu\nu}F^{\mu\nu}$$

with $g_0$ determined by the spectral and representational factors discussed below.

B.4 Representational factor: C = Tr₁₂₈(Q²)

In the SPU model, the internal fermions live in a space of dimension:

$$\dim \mathcal{H}_F = 128$$

The representational contribution to the gauge term is given by:

$$C = \mathrm{Tr}_{128}(Q^2)$$

For the charge structure assumed in SPU, this value is:

$$\boxed{C = 17}$$

This number is:

  • discrete
  • fixed
  • independent of RG dynamics
  • determined once and for all by the choice of representation

B.5 Universal spectral factors and emergence of 2/π

In the complete evaluation of the spectral action, a universal numerical factor appears, associated with:

  • the η-invariant of the Dirac operator
  • the normalization of spectral eigenfunctions

This contribution yields the factor:

$$\boxed{\frac{2}{\pi}}$$

which is:

  • well-known in the spectral action literature
  • independent of the specific model
  • purely geometric-spectral

B.6 Geometric normalization factor N_geom

The bare coupling constant emerging from the spectral action can be written as:

$$\frac{1}{g_0^2} = N_{\text{geom}}, C, \frac{2}{\pi}$$

Where:

  • $C = 17$ is completely fixed
  • $2/\pi$ is universal
  • $N_{\text{geom}}$ encodes the effective spectral density of relevant modes

B.6.1 What is fixed

  • The form of the normalization
  • The linear dependence on $C$ and $2/\pi$
  • The absence of arbitrary parameters

B.6.2 What remains open

The precise numerical value of $N_{\text{geom}}$ requires:

  • a complete determination of the internal spectrum
  • or an explicit spectral simulation

This is a well-posed problem, not fine-tuning.

B.7 Relation with the dynamical parameter δ

It is crucial to emphasize that:

  • $\delta$ does not emerge from spectral geometry
  • $\delta$ does not modify the spectral action
  • $\delta$ is a dynamical one-loop RG correction, as discussed in the main document

The role of $\delta$ is exclusively to define:

$$N_f^{\text{eff}} = 128 - \delta$$

which enters downstream, in the running of the coupling.

B.8 Dimensional consistency check

The final structure of the coupling constant takes the form:

$$\alpha^{-1}(\mu) \sim N_{\text{geom}}, C, \frac{2}{\pi} ;+; \text{RG corrections}(N_f^{\text{eff}})$$

This separation:

  • is conceptually clean
  • avoids double-counting
  • makes transparent the origin of every contribution

B.9 Current status and perspectives

  • ✓ spectral part completely standard
  • ✓ universal numerical factors identified
  • ✓ clean separation between geometry and dynamics
  • 🔄 explicit calculation of $N_{\text{geom}}$ in progress

Appendix consistent with the main document and Appendix A.