Generalized $T\bar{T}$-like flows for scalar theories in two dimensions

Abstract: We demonstrate that the necessary condition for $SO(N) \times SO(N)$ duality invariance manifests as a partial differential equation in two-dimensional scalar theories. This condition, expressed as a partial differential equation, corresponds precisely to the integrability condition. We derive a general perturbation solution to this partial differential equation, which includes both a root $T\bar{T}$ flow equation and an irrelevant $T\bar{T}$-like flow equation. Additionally, we identify a general form for these flow equations that commute with each other.