Stress Tensor Flows, Birefringence in Non-Linear Electrodynamics, and Supersymmetry

We identify the unique stress tensor deformation which preserves zero-birefringence conditions in non-linear electrodynamics, which is a $4d$ version of the ${T\overline{T}}$ operator. We study the flows driven by this operator in the three Lagrangian theories without birefringence -- Born-Infeld, Plebanski, and reverse Born-Infeld -- all of which admit ModMax-like generalizations using a root-${T\overline{T}}$-like flow that we analyse in our paper. We demonstrate one way of making this root-${T\overline{T}}$-like flow manifestly supersymmetric by writing the deforming operator in $\mathcal{N} = 1$ superspace and exhibit two examples of superspace flows. We present scalar analogues in $d = 2$ with similar properties as these theories of electrodynamics in $d = 4$. Surprisingly, the Plebanski-type theories are fixed points of the classical ${T\overline{T}}$-like flows, while the Born-Infeld-type examples satisfy new flow equations driven by relevant operators constructed from the stress tensor. Finally, we prove that any theory obtained from a classical stress-tensor-squared deformation of a conformal field theory gives rise to a related ``subtracted'' theory for which the stress-tensor-squared operator is a constant.