$T\bar{T}$ and root-$T\bar{T}$ deformations in four-dimensional Chern-Simons theory

TL;DR

This work embeds $T\bar{T}$-type and root-$T\bar{T}$ deformations of two-dimensional integrable field theories into the four-dimensional Chern-Simons framework with disorder defects, using the degenerate $\mathcal{E}$-model as a unifying target. It develops deformed Lax pairs by implementing dynamical coordinate transformations and extends the construction from flat to curved 2D spaces via Beltrami differentials, ensuring finite actions and preserving classical integrability. The authors explicitly realize both $T\bar{T}$ and root-$T\bar{T}$ deformations for the PCM and the degenerate $\mathcal{E}$-model, derive the corresponding two-dimensional actions, and verify their flow equations, including a two-parameter deformation that commutes the two deformations. This framework connects 2D integrable deformations to a 4D bulk theory with defects, offering a geometric and gauge-theoretic perspective on integrable deformations and suggesting avenues toward higher-spin generalizations and gravity couplings.

Abstract

The four-dimensional Chern-Simons (CS) theory provides a systematic procedure for realizing two-dimensional integrable field theories. It is therefore a natural question to ask whether integrable deformations of the theories can be realized in the four-dimensional CS theory. In this work, we study $T\bar{T}$ and root-$T\bar{T}$ deformations of two-dimensional integrable field theories, formulated in terms of dynamical coordinate transformations, within the framework of four-dimensional CS theory coupled to disorder defects. We illustrate our procedure in detail for the degenerate $\mathcal{E}$-model, a specific construction that captures and unifies a broad range of integrable systems, including the principal chiral model.

Figures (2)

• Figure 1: Outline of the structure of this paper.
• Figure 2: Decomposition of $\mathfrak{d}$ into subspaces. We have a decomposition $\mathfrak{d}=\mathfrak{f} \dot{+} \mathfrak{f}' \dot{+} \mathfrak{r}_+ \dot{+} \mathfrak{r}_{-}$, with $\mathrm{dim}\,\mathfrak{f} =\mathrm{dim}\, \mathfrak{f}' = \mathrm{dim}\,\mathfrak{g}$.