Q13 -- Gauge--Gravity Spectral Synthesis

Gauge--Gravity Spectral Synthesis: A Conditional Einstein--Yang--Mills System from Projective Spectral Entropy

J. Beau, Independent Researcher, France

Abstract

The companion papers (Gravity, Q12) established that the projective spectral entropy functional $S_\Pi[g, \mathcal{A}] = \tfrac{1}{2}\log\det' A_{g,\mathcal{A}}$ produces the Einstein tensor as the horizontal $a_2$ response and the local Yang--Mills equations as the vertical $a_4$ response of the same functional.

The present paper joins the two variations within a single renormalized local action. Its robust content is a conditional Einstein--Yang--Mills system: varying the matched action with respect to the connection at fixed metric gives $D_\mu F^{a\mu\nu} = 0$, while varying the same $\mathrm{tr}_\rho(F^2)$ term with respect to the metric gives exactly the Yang--Mills stress tensor, through

$$\frac{1}{\sqrt{g}}\frac{\delta}{\delta g^{\mu\nu}}\int \sqrt{g},\mathrm{tr}_\rho(F^2),d^4x = 2\tau_{\mu\nu}, \qquad \tau_{\mu\nu} = \mathrm{tr}_\rho!\left(F_{\mu\rho}F_\nu{}^\rho - \tfrac14 g_{\mu\nu}F^2\right).$$

Since $\mathrm{tr}\rho(F^2) = I\rho F^a_{\mu\nu}F^{a,\mu\nu}$, canonical matching fixes the products $c_{\mathrm{EH}} = 1/(16\pi G_N)$ and $c_F I_\rho = 1/(4g_{\mathrm{YM}}^2)$, where $I_\rho$ is the Dynkin index of the fibre representation. The equation $G_{\mu\nu} + \Lambda_{\mathrm{eff}} g_{\mu\nu} = 8\pi G_N T^{\mathrm{YM}}{\mu\nu}$ then holds by definition of $T^{\mathrm{YM}}$, with $8\pi G_N = 1/(2c{\mathrm{EH}})$ depending on the Einstein coefficient alone. The derivation is Euclidean; identification with the Lorentzian Einstein--Yang--Mills equations requires a continuation not performed here.

Central message: one matched local action supplies both field equations; within a proper-time cutoff the spectral expansion fixes the degree of ultraviolet divergence of each sector, not the finite couplings.

Key Results

Theorem (Conditional Einstein--Yang--Mills system): joint stationarity of the matched local action yields

1. $G_{\mu\nu} + \Lambda_{\mathrm{eff}} g_{\mu\nu} = 8\pi G_N T^{\mathrm{YM}}_{\mu\nu}$,
2. $D_\mu F^{a\mu\nu} = 0$ (fixed-metric connection variation),

with $8\pi G_N = 1/(2c_{\mathrm{EH}})$, depending on the Einstein coefficient alone.

Structural (qualitative only, within a proper-time cutoff): the two sectors carry different ultraviolet divergence degrees, quadratic at $a_2$ and logarithmic at $a_4$. This is a scheme-dependent statement: the zeta-regularized determinant carries no power divergences.

Matching data, not predictions: $G_N$, $g_{\mathrm{YM}}$, $\Lambda_{\mathrm{ren}}$ (the $a_0$ term does contribute to the metric equation) and the dimension-six coefficients are independent renormalized inputs.

$a_6$: inventory only — a representative spanning list, modulo integrations by parts and Bianchi identities, not a basis. In four dimensions $a_6$ produces no ultraviolet divergence; the pure-gauge invariant at this order is $\mathrm{tr}_\rho(F^3)$; several $RF^2$-type and derivative structures occur and are interrelated by those identities, so no unique cross-term is singled out.

Status

Preprint. DOI: 10.5281/zenodo.20209859

Position in the Programme

• Depends on: Gravity paper ($S_\Pi[g]$ functional, $a_2$ Einstein sector), Q12 (Yang--Mills from $a_4$, admissible principal bundle), Q6a (gauge group $G_\Pi$), Q11 (effective Lorentzian metric)
• SU(3) status: $[H\text{-color}]{\mathrm{pointwise}}$ established in O31/O32; it enters here only through the representation $\rho$ in which $\mathrm{tr}\rho$ and $I_\rho$ are taken
• Open: non-linear completion (to be rebuilt from scratch, if at all), quantitative $a_6$ coefficients, whether admissibility constrains any combination of the matching data, Lorentzian continuation of the gauge sector

Compilation

cd q13
bash compile.sh
# or manually:
pdflatex -output-directory=out tex/q13.tex