Are inverse-cascade vorticity isolines exactly conformally invariant?
In plain words
In flat turbulence, the curves where the local spin of the fluid changes sign look statistically identical to the boundaries of random clusters in a classic model of percolation, a symmetry known from equilibrium critical points. Why a driven, dissipative flow should have this symmetry is unexplained.
Precise statement
Forced 2D incompressible Navier-Stokes turbulence with large-scale friction, inverse-cascade range $\ell_f \ll r \ll L$. Zero-vorticity isolines are statistically consistent with Schramm-Loewner evolution $\mathrm{SLE}_{\kappa}$ with $\kappa = 6$ (Bernard, Boffetta, Celani, Falkovich 2006), the class of critical percolation cluster boundaries. Is this conformal invariance exact as $L/\ell_f \to \infty$, and which property of the dynamics implies it? Answer: a derivation, or a measured deviation of $\kappa$ beyond error bars.
What would settle it
An analytic derivation from the $2\mathrm{D}$ Navier-Stokes dynamics, or a high-precision numerical test of $\kappa$ and of the conformal covariance of multipoint observables.
Status in the literature
Unverified note
Evidence is numerical and experimental; a 2025 theoretical proposal models the cascade as constant-flux domains whose boundaries carry a Liouville conformal field theory (Eling, arXiv:2505.09657), without derivation from the Navier-Stokes equations.