Courant-Hilbert deformations of Yang-Baxter sigma models

TL;DR

This work unifies Courant–Hilbert deformations with Yang–Baxter sigma models using the four-dimensional Chern–Simons framework, showing that CH deformations can be incorporated alongside YB deformations and that the master formula acquires a universal correction by the trace of the energy–momentum tensor. The authors derive a generalized Lax pair and flatness conditions for both homogeneous YB and mCYBE-based deformations, demonstrating that the resulting deformed action retains a master-form structure, $S[g]=-\int d^2x\,\text{tr}(j^{\mu}\boldsymbol{\frak J}_{\mu})$, with the deformation captured through deformed currents $\tilde{j}_{\mu}$ and $\boldsymbol{\frak J}_{\mu}$. A key contribution is the explicit CH construction: solving the PDE via Courant–Hilbert variables and inverting the relation between $\tilde{j}_{\mu}$ and $j_{\mu}$ to obtain the true Lagrangian, including two detailed examples—the root $T\overline{T}$-deformation and the standard $T\overline{T}$-deformation—where closed-form expressions for the deformed currents are derived. The results suggest a universal correction term and provide a concrete protocol for deriving CH–YB deformations, with potential implications for AdS/CFT and integrable string theories through generalized Lax structures and their boundary interpretations.

Abstract

We present integrable deformations of Yang-Baxter (YB) sigma models based on the Courant-Hilbert (CH) construction. To this end, we employ the four-dimensional Chern-Simons theory, in which the CH construction is shown in arXiv:2509.22080. As a result, the CH construction works in an intricate way alongside the YB deformations. Remarkably, the resulting deformed action can also be expressed as the sum of the master formula Lagrangian and the trace of the energy-momentum tensor. This result indicates the universality of the correction term.