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<p>We develop a unified Courant-Hilbert framework for constructing two-dimensional integrable sigma models deformed by two couplings: a marginal one <i>γ</i> and an irrelevant one <i>λ</i>. The integrability condition is encoded in a nonlinear partial differential equation (PDE) for two invariants (<i>P</i><sub>1</sub><i>, P</i><sub>2</sub>), whose general solution could be expressed through an arbitrary generating function <i>ℓ</i>(<i>τ</i>). 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 <i>q</i>-deformations. All resulting theories obey a universal root-<InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(T\overline{T }\)</EquationSource> </InlineEquation> flow equation, consistent under dimensional reduction from four-dimensional duality-invariant electrodynamics. Using perturbative expansions, we recover ModMax in the free limit, determine the <i>γ</i>-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.</p>

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Integrable sigma models and universal root \(T\overline{T }\) deformation via Courant-Hilbert approach

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年1月16日

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  • H. Babaei-Aghbolagh,
  • Bin Chen,
  • Song He

摘要

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 (P1, P2), whose general solution could be expressed through an arbitrary generating function ℓ(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.