$T\overline{T}$ deformation in SCFTs and integrable supersymmetric theories

We calculate the $\mathcal{S}$-multiplets for two-dimensional Euclidean $\mathcal{N}=(0,2)$ and $\mathcal{N} = (2,2)$ superconformal field theories under the $T\overline{T}$ deformation at leading order of perturbation theory in the deformation coupling. Then, from these $\mathcal{N} = (0, 2)$ deformed multiplets, we calculate two- and three-point correlators. We show the $\mathcal{N} = (0,2)$ chiral ring's elements do not flow under the $T\overline{T}$ deformation. Specializing to integrable supersymmetric seed theories, such as $\mathcal{N} = (2,2)$ Landau-Ginzburg models, we use the thermodynamic Bethe ansatz to study the S-matrices and ground state energies. From both an S-matrix perspective and Melzer's folding prescription, we show that the deformed ground state energy obeys the inviscid Burgers' equation. Finally, we show that several indices independent of $D$-term perturbations including the Witten index, Cecotti-Fendley-Intriligator-Vafa index and elliptic genus do not flow under the $T\overline{T}$ deformation.