More on $T \overline{T}$-like deformations in higher dimensions

We investigate several possible generalisations of $T\overline{T}$ deformations to three- and higher-dimensional field theories. Starting from the two-dimensional $T\overline{T}$ flow, we work out its higher-dimensional uplift, which results in a non-local and non-isotropic three-dimensional theory. Starting instead from the relation between the Nambu-Goto action and $T\overline{T}$ in $d=2$, we study the flow equation obeyed by the Dirac-Nambu-Goto actions in $d>2$ dimensions, written in terms of the stress-energy tensor only. Similarly, we derive the stress-tensor flow obeyed by the Born-Infeld actions in $d$ dimensions and by the Dirac-Born-Infeld actions in $d=2$ and $d=3$.