Massive gravity generalization of $T\overline{T}$ deformations

Motivated by the two-dimensional massive gravity description of $T\overline{T}$ deformations, we propose a direct generalization in $d$ dimensions. Our methodology indicates that all terms up to order $d$ are present in the deformation. In two dimensions, $T\overline{T}$ is enhanced by a linear and a constant term, and exhibits an interesting behaviour regarding the deformed spectrum and correlators. At certain limits, this deformation can reduce to $T\overline{T}$ or $T\overline{T}+\Lambda_{2}$ consistently. Using the massive gravity method, we obtain the classically deformed action of a sigma model of bosons and fermions interacting with an arbitrary potential, extending previous results. As a consequence, a proposal regarding the deformation of higher-derivative theories is made. Moreover, a standard dimensional reduction procedure is presented, with the resulting operator matching the form of prior findings under certain assumptions. In $d\geq2$, we provide the exact structure of the quadratic terms in agreement with previous approaches, as well as the structure of the linear and constant terms. All higher order contributions are not easily evaluated, yet we derive the complete answer for all cases up to seven dimensions. Under certain conditions, these terms vanish, resulting in a quadratic operator. The trace-flow equation for this family of deformations is also derived. Finally, we investigate the class of root-$T\overline{T}$ operators in various dimensions within the scope of this formalism.