$T\bar{T}$ Flows and (2,2) Supersymmetry

TL;DR

We construct a solvable, manifestly ${\rm N}=(2,2)$ supersymmetric deformation of two-dimensional QFTs via the supercurrent-squared operator ${\mathcal T}\overline{\mathcal T}$, which is equivalent to the conventional ${T\overline{T}}$ flow on using current conservation. The framework relies on the ${\cal S}$-multiplet and its Ferrara–Zumino refinement, and permits coupling to old-minimal supergravity to organize the deformation as a D-term in superspace. For theories with a single ${\cal N}=(2,2)$ chiral multiplet and vanishing superpotential, the finite-$\lambda$ action is on-shell equivalent to a Nambu–Goto–type action, with a precise closed-form structure in the Kähler case; with a nonzero superpotential the classical potential develops poles, signaling dramatic changes in the vacuum and soliton structure. These results illuminate how ${T\overline{T}}$-like deformations interact with extended supersymmetry, and provide a controlled setting for exploring potential IR/UV mixing and the fate of vacua under solvable deformations.

Abstract

We construct a solvable deformation of two-dimensional theories with $(2,2)$ supersymmetry using an irrelevant operator which is a bilinear in the supercurrents. This supercurrent-squared operator is manifestly supersymmetric, and equivalent to $T\bar{T}$ after using conservation laws. As illustrative examples, we deform theories involving a single $(2,2)$ chiral superfield. We show that the deformed free theory is on-shell equivalent to the $(2,2)$ Nambu-Goto action. At the classical level, models with a superpotential exhibit more surprising behavior: the deformed theory exhibits poles in the physical potential which modify the vacuum structure. This suggests that irrelevant deformations of $T\overline{T}$ type might also affect infrared physics.