Root- T T ¯ flows unify 4D duality-invariant electrodynamics and 2D integrable sigma models

We present a single, dimension-independent framework that links four-dimensional duality-invariant nonlinear electrodynamics to two-dimensional integrable sigma models. Both sectors are shown to obey the same firs-order Courant–Hilbert equation, solved by a common generating function and an auxiliary-potential formulation. Within this structure, a discrete φ parity acts as a selection rule, organizing deformation series into integer versus fractional powers. Two commuting deformations—an irrelevant parameter λ and a marginal parameter γ —admit universal flow representations that recover root- T T ¯ dynamics and extend them in a controlled way. The construction yields closed-form families (generalized Born-Infeld, logarithmic, and q deformed) and a new integrable model, all realized in 2D and 4D. These results replace case-by-case analyses with a unified route to solvable nonlinear theories, with immediate relevance to gauge dynamics, string-inspired effective actions, and integrable models.