The Gauge–Gravity Stratification Sub-Programme

Overview

The spectral gravity sub-programme (Presentation Note 4) derives the Einstein tensor as the $a_2$ infrared-dominant response of the horizontal metric variation of $S_\Pi[g]$. The gauge structure sub-programme (Presentation Note 3) identifies the gauge group $G_\Pi = \mathrm{SU}(3) \times \mathrm{SU}(2) \times \mathrm{U}(1)$ and constructs the admissible principal bundle $P_{G_\Pi}(M, G_\Pi)$. Does the same functional, extended to include the admissible gauge connection, produce Yang–Mills dynamics — and if so, at what spectral order?

The answer is yes, and the order is $a_4$. Gravity and gauge dynamics arise from the same functional by varying in orthogonal directions — horizontal (metric) and vertical (gauge connection) — at different Seeley–DeWitt orders. The conventional question "what symmetry unifies gravity and gauge?" is thereby replaced by "at what spectral order does the admissible projection respond?". This is the spectral stratification principle: $a_2 \to$ gravity, $a_4 \to$ Yang–Mills, $a_6 \to$ gauge–gravity mixing. The Yang–Mills equations $D_\mu F^{a\mu\nu} = 0$ follow from the $a_4$ vertical variation (Q12); the conditional coupled Einstein–Yang–Mills system follows from the joint variation (Q13), with the couplings entering as independent renormalization data.

Stratification, not unification. What the expansion fixes is the difference in ultraviolet divergence degree between the sectors — quadratic at $a_2$, logarithmic at $a_4$, within a proper-time cutoff — and not the finite couplings, which remain renormalization data. The difficulty of reconciling gravity with the gauge-theoretic Standard Model may reflect a difference of spectral order rather than the absence of a common symmetry group $G_{\mathrm{unif}}$.

Quick facts

Author

Jérôme Beau

Work type

Synthesis (presentation note, working paper)

License

CC BY 4.0

Scope

2 papers (Q12 — shared with Note 3 — and Q13); Yang–Mills from $a_4$, conditional coupled Einstein–Yang–Mills, $a_6$ invariant inventory, couplings as renormalization matching data

Status

Bosonic dynamical sector closed conditionally on matching; open: non-linear completion, $a_6$ coefficients, whether admissibility constrains the couplings, Lorentzian continuation of the gauge sector, fermionic matter (Note 6)

The spectral stratification chain

$\underbrace{a_2 \to G_{\mu\nu}}_{\text{horizontal } \delta_g} \;\Big|\; \underbrace{a_4 \to D_\mu F^{a\mu\nu} = 0}_{\text{vertical } \delta_A} \;\Big|\; \underbrace{a_6 \to \text{mixed invariants}}_{\text{inventory only}}$

Four conceptual stages from the single functional $S_\Pi[g, A] = \tfrac{1}{2}\log\det' A_{g, A}$: extension of the operator to the gauge sector with fixed-metric isolation of the gauge sector (Q12 Lemma 1; full horizontal–vertical decoupling does not hold, the metric variation of $F^2$ being the $a_4$ back-reaction $T^{\mathrm{YM}}$); Yang–Mills equations from the vertical $a_4$ variation (Q12 Theorem 1, structural given $G_\Pi$); conditional coupled Einstein–Yang–Mills system from the joint variation (Q13), with $G_N$, $g_{\mathrm{YM}}$ and the cosmological coefficient as independent renormalization matching data rather than predictions.

Papers of the sub-programme

Yang–Mills from the vertical $a_4$ variation.

Q12 (v1.3) — extends the Laplacian $A_g$ to $A_{g, A} = -(\nabla^A)^2 + E$ on the associated vector bundle of $P_{G_\Pi}(M, G_\Pi)$. Results: $a_4 \supset \tfrac{1}{12}\mathrm{tr}_\rho(F_{\mu\nu}F^{\mu\nu})$ (heat-kernel, proved); at fixed metric the vertical variation isolates the gauge sector (Lemma 1), but full horizontal–vertical decoupling fails via the $T^{\mathrm{YM}}$ back-reaction; Yang–Mills equations $D_\mu F^{a\mu\nu} = 0$ from $\delta_A S_\Pi = 0$ (Theorem 1, structural given $G_\Pi$). The two sectors differ in UV divergence degree ($a_2$ quadratic vs. $a_4$ logarithmic); $g_{\mathrm{YM}}$ and $G_N$ are renormalized matching data (only the log running, $\propto I_\rho$, is computed). Q12 is shared with the gauge-structure sub-programme (Presentation Note 3), which uses its gauge-group identification component.

Joint Einstein–Yang–Mills system and hierarchy.

Q13 — completes the trilogy (Gravity 3.0, Q12, Q13) by solving the joint variational problem $\delta_{g, A} S^{\mathrm{ren}} = 0$ for a single matched local action. Results: the metric variation of the gauge kinetic term is $2\tau_{\mu\nu}$ (Lemma); the conditional coupled system $G_{\mu\nu} + \Lambda_{\mathrm{eff}} g_{\mu\nu} = 8\pi G_N T^{\mathrm{YM}}_{\mu\nu}$ with $D_\mu F^{a\mu\nu} = 0$ (Theorem), where canonical matching fixes $c_{\mathrm{EH}} = 1/(16\pi G_N)$ and $c_F I_\rho = 1/(4g_{\mathrm{YM}}^2)$ so that $8\pi G_N = 1/(2c_{\mathrm{EH}})$; and an inventory of the $a_6$ invariants, modulo integrations by parts and Bianchi identities, with no unique cross-term and no $a_6$ correction asserted. The derivation is Euclidean.

Inputs and outputs

Upstream inputs. Spectral entropy functional $S_\Pi[g]$ and its $a_2$ Einstein sector from the spectral gravity sub-programme (Presentation Note 4, Gravity 3.0); the admissible principal bundle $P_{G_\Pi}(M, G_\Pi)$ and the gauge group $G_\Pi = \mathrm{SU}(3) \times \mathrm{SU}(2) \times \mathrm{U}(1)$ from the gauge structure sub-programme (Presentation Note 3, Q6a/Q12); the effective Lorentzian base manifold $(M, g^{\mu\nu} = 2\eta^{\mu\nu})$ from the emergent geometry sub-programme (Presentation Note 2, Q5b–Q11); the spectral length scale $\ell_{\mathrm{sp}}$ and BI saturation constant $c_\chi$ (Branch I); and the renormalized matching data ($G_N$, $g_{\mathrm{YM}}$, the cosmological coefficient) supplied from outside the expansion.

Outputs. Yang–Mills equations $D_\mu F^{a\mu\nu} = 0$ in the current-free sector (SM phenomenology); the conditional coupled $G_{\mu\nu} + \Lambda_{\mathrm{eff}} g_{\mu\nu} = 8\pi G_N T^{\mathrm{YM}}_{\mu\nu}$ (Q13); an inventory of the $a_6$ invariants (future precision tests, coefficients not determined); the spectral prediction that fermionic structure should appear at a dedicated Dirac-type spectral level — the direct structural motivation for the fermionic matter sub-programme (Presentation Note 6, Q14).

Status

The bosonic dynamical sector is closed conditionally on matching. The $a_4$ heat-kernel derivation of Yang–Mills (Q12) is proved, as is the metric variation giving $2\tau_{\mu\nu}$ (Q13). The fixed-metric vertical variation isolates the gauge sector (Q12 Lemma 1), but full horizontal–vertical decoupling does not hold. The coupled Einstein–Yang–Mills system (Q13) is conditional on the matched coefficients: $G_N$, $g_{\mathrm{YM}}$ and the cosmological coefficient are independent renormalization data, so no numerical value for $G_N g_{\mathrm{YM}}^2$ is claimed. The $a_6$ sector is an inventory only, and no non-linear completion of the joint functional is asserted. Per O31 Proposition 4.23, the gauge group $G_\Pi = \mathrm{SU}(3) \times \mathrm{SU}(2) \times \mathrm{U}(1)$ is an unconditional input at the pointwise level, so the $a_4$ Yang–Mills derivation holds unconditionally for the full Standard Model gauge group. Open items: the $a_6$ coefficients, whether admissibility constrains any combination of the matching data, a non-linear completion of the joint functional, the Lorentzian continuation of the gauge sector, and the coupled equations with fermionic matter currents from Note 6 (Q14).