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 $T\bar{T}$ 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-$T\bar{T}$-like flow that we analyse in our paper. We demonstrate one way of making this root-$T\bar{T}$-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 $T\bar{T}$-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.