From Spectrum to Action: Emergent Gauge Lagrangian and Topological Mass Gap
Abstract
What effective Lagrangian emerges from the spectrum of a Dirac operator in a topologically non-trivial background? We show that the heat kernel K(τ) = Tr e−τ D2 of a self-consistent chiral soliton generates, through a single proper-time integral, all structural components of the Standard Model Lagrangian: - the chiral kinetic and Skyrme terms via the local Seeley-DeWitt expansion (R2 > 0.99), with a quasi-cancellation mechanism (× 22 suppression per order); - the rotational dynamics via the Berry connection on the moduli space (99.93% of the moment of inertia); - the gauge coupling via the instanton-skyrmion correspondence (α ≈ 0.52, crossing αsQCD); - the topological Wess-Zumino-Witten term ΓWZW = Nc = 3 exactly, via the chiral heat kernel; - the gauge group SU(3) × SU(2) × U(1), the Higgs doublet, and three generations (Ngen = 3), via the spectral truncation of the Peter-Weyl algebra on the moduli space (Emergence Paper IV); - the Higgs mass MH ∈ [118, 130] GeV from the spectral invariants of the finite Dirac operator (Emergence Paper IV). A topological mass gap argument proves 0 < εval < Mq for any monotone hedgehog with B = 1. The same spectral function thus generates both the effective Lagrangian (UV) and the mass gap (IR). The total action takes the form S = ∫ℳB=1 dμ(q) ∫0∞ w(τ̂) K(τ̂; q) dτ̂ , a double integral over proper time and moduli space, where K(τ̂; q) is a universal function determined by topology (B = 1) and self-consistency (F* = G[F*]) alone. Table 2 provides a term-by-term emergence dictionary mapping each component of ℒSM to a spectral mechanism; Table 3 maps each Standard Model entity to its solitonic origin.
Related
- TOPOLOGICAL TRIVIALITY OF EXACT LAGRANGIAN COBORDISMS
- Constraints for Physical Gauge Groups Coming from the Causal Action Principle
- Gram--Wishart--Stiefel formulation of the N=2, large--d gauge theory in 1D: towards emergent Grassmannian geometry
- A Topological Formula for Potts Lattice Gauge Theory Correlations
- Spectral Emergence from Topological Solitons: A Unified Conjecture for Gauge Representations
BibTeX
@misc{dechery2026from,
title = {{From Spectrum to Action: Emergent Gauge Lagrangian and Topological Mass Gap}},
author = {Fabien Déchery},
howpublished = {HAL},
year = {2026},
month = oct,
url = {https://hal.science/hal-05793301v2},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - From Spectrum to Action: Emergent Gauge Lagrangian and Topological Mass Gap
AU - Fabien Déchery
PY - 2026
DA - 2026/10/08
PB - HAL
UR - https://hal.science/hal-05793301v2
AB - What effective Lagrangian emerges from the spectrum of a Dirac operator in a topologically non-trivial background? We show that the heat kernel K(τ) = Tr e^−τ D^2 of a self-consistent chiral soliton generates, through a single proper-time integral, all structural components of the Standard Model Lagrangian: - the chiral kinetic and Skyrme terms via the local Seeley-DeWitt expansion (R^2 > 0.99), with a quasi-cancellation mechanism (× 22 suppression per order); - the rotational dynamics via the Berry connection on the moduli space (99.93% of the moment of inertia); - the gauge coupling via the instanton-skyrmion correspondence (α ≈ 0.52, crossing α_s^QCD); - the topological Wess-Zumino-Witten term Γ_WZW = N_c = 3 exactly, via the chiral heat kernel; - the gauge group SU(3) × SU(2) × U(1), the Higgs doublet, and three generations (N_gen = 3), via the spectral truncation of the Peter-Weyl algebra on the moduli space (Emergence Paper IV); - the Higgs mass M_H ∈ [118, 130] GeV from the spectral invariants of the finite Dirac operator (Emergence Paper IV). A topological mass gap argument proves 0 < ε_val < M_q for any monotone hedgehog with B = 1. The same spectral function thus generates both the effective Lagrangian (UV) and the mass gap (IR). The total action takes the form S = ∫_ℳ_B=1 dμ(q) ∫_0^∞ w(τ̂) K(τ̂; q) dτ̂ , a double integral over proper time and moduli space, where K(τ̂; q) is a universal function determined by topology (B = 1) and self-consistency (F^* = G[F^*]) alone. Table 2 provides a term-by-term emergence dictionary mapping each component of ℒ_SM to a spectral mechanism; Table 3 maps each Standard Model entity to its solitonic origin.
ER -