Integrable Sigma Models and Universal Root $T\bar{T}$ Deformation via Courant-Hilbert Approach

TL;DR

This work develops a unified Courant–Hilbert approach to construct two-dimensional integrable sigma models deformed by marginal γ and irrelevant λ couplings, encoded by a PDE for invariants (P1,P2) and solvable via a generating function ℓ(τ). The CH construction yields a general Lagrangian family L = ℓ(τ) − 2Q/ℓ′(τ) with τ governed by auxiliary variables, generating PCM, ModMax-like, Born–Infeld–like, q-deformed, and logarithmic models that are consistent with dimensional reduction from 4D duality-invariant electrodynamics. A universal root-TT flow equation is established, linking the 2D models to their 4D counterparts and ensuring integrability is preserved under deformation. Perturbative analysis clarifies the γ-dependence of couplings and demonstrates how different flow forms, including single-trace variants, emerge from a common CH framework, revealing deep connections between self-duality, integrability, and deformation dynamics across dimensions.

Abstract

We develop a unified Courant--Hilbert framework for constructing two-dimensional integrable sigma models deformed by two couplings: a marginal one $γ$ and an irrelevant one $λ$. The integrability condition is encoded in a nonlinear partial differential equation (PDE) for two invariants $(P_1, P_2)$, whose general solution could be expressed through an arbitrary generating function $\ell(τ)$. This formulation encompasses and extends known models, such as ModMax and Born-Infeld, while introducing new classes of solvable models with closed-form Lagrangians, including those with logarithmic and $q$-deformations. All resulting theories obey a universal root-$T\overline{T}$ flow equation, consistent under dimensional reduction from four-dimensional duality-invariant electrodynamics. Using perturbative expansions, we recover ModMax in the free limit, determine the $γ$-dependence of the coupling functions, and show how different flow equations, including a single-trace form, naturally emerge. Our results reveal deep structural connections between self-duality, integrability, and deformation dynamics across different dimensions.