A Gauge Theory of Turbulence:

Abstract

We formalize the gauge structure of the Navier--Stokes equation for incompressible fluids, interpreting $\nabla\cdot\mathbf{v}=0$ as a gauge fixing analogous to the Coulomb gauge in electromagnetism. We construct the Martin--Siggia--Rose action and its full BRST symmetry, introducing BRST doublets for the Gribov parameter $γ$ and the monodromy phase $θ$. Through a Higgs mechanism, $γ$ acquires a vacuum expectation value $γ_{0}$, generating a mass scale for the vorticity and the ghosts; we compute the one-loop effective potential and analyze vacuum stability. Intermittency --- measured by the exponents $ζ_{n}$ --- is described by fluctuations of the Higgs field $σ$ around the condensate, via a Gribov-inspired log-Poisson hierarchy that satisfies exactly $ζ_{3}=1$ (Kolmogorov's four-fifths law). Fits to DNS data favor filamentary vorticity structures, with $D_{f}\approx1$ (one parameter) or $D_{f}\approx2.1$, $Δ\approx0.44$ (two parameters). The one-loop anomalous dimension in $d=3$, $Δ_σ\approx0.50$, supports the identification $D_{f}=2Δ_σ$. The formalism unifies classical hydrodynamics with gauge theory and spontaneous symmetry breaking, opening the study of turbulence and intermittency to quantum field theory methods.