$T \bar{T}$-Deformed Actions and (1,1) Supersymmetry

TL;DR

The paper develops an algebraic, topological-gravity–based method to construct $T\bar{T}$-deformed Lagrangians from seed theories, avoiding partial differential equations by solving algebraic equations for the coordinate map $X^{a}(x)$. It provides closed-form deformations for free bosons, general bosonic theories with potentials, general fermionic theories, and mixed boson–fermion systems, all satisfying the $\partial_{\lambda}\mathcal{L}_{\lambda} = T\bar{T}_{\lambda}$ relation. Special emphasis is placed on $(1,1)$ supersymmetry with $N_b=N_f$, where the complete deformed Lagrangian is shown to be off-shell SUSY to $\mathcal{O}(\lambda^{2})$ and the SUSY algebra holds to $\mathcal{O}(\lambda)$, with explicit modified transformations. The results demonstrate the viability of an algebraic, implementable framework for explicit $T\bar{T}$ constructions and point toward future extensions to SUSY nonlinear sigma models and higher-order invariances.

Abstract

We describe an algorithmic method to calculate the $T\bar{T}$ deformed Lagrangian of a given seed theory by solving an algebraic system of equations. This method is derived from the topological gravity formulation of the deformation. This algorithm is far simpler than the direct partial differential equations needed in most earlier proposals. We present several examples, including the deformed Lagrangian of (1,1) supersymmetry. We show that this Lagrangian is off-shell invariant through order $λ^2$ in the deformation parameter and verify its SUSY algebra through order $λ$.