Geometric solution of turbulence as diffusion in loop space

Strongly nonlinear dynamics, from fluid turbulence to quantum chromodynamics (QCD), has long constituted some of the most challenging problems in theoretical physics. This review describes a unified theoretical framework, the loop space calculus, which offers an analytical approach to these problems. The central idea is a shift in perspective from pointwise fields to integrated loop observables, a transformation that recasts the governing nonlinear equations into a universal linear diffusion equation in the space of loops. This framework, supported by recent mathematical analysis, is analytically solvable and yields an exact, parameter-free solution for decaying hydrodynamic turbulence-the Euler ensemble-which is shown to be dual to a solvable string theory. The theory's predictions include: (i) the unification of spatial and temporal scaling laws, governed by two related, infinite spectra of intermittency and decay exponents derived from the non-trivial zeros of the Riemann zeta function, (ii) a first-order phase transition in magnetohydrodynamic (MHD) turbulence and (iii) the formation of quantized, concentric shells in passive scalar mixing. The appearance of identical mathematical structures as solutions to the turbulent regime of Yang-Mills gradient flow points to the broad applicability of this approach. The framework also yields a new type of analytic Hodge-dual matrix surface that solves the Yang-Mills fixed-point loop equation by harmonic map, opening the way for a geometric formulation of QCD string theory. This article is part of the theme issue 'Frontiers of turbulence and statistical physics'.