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I will discuss recent work on the connection between duality-invariant theories of electrodynamics in four spacetime dimensions and deformations of the Lagrangian by functions of the energy-momentum tensor. Such deformations, which we refer to as "stress tensor flows", are inspired by  Zamolodchikov 's $T ar{T}$ deformation of two-dimensional quantum field theories. We find a surprising one-to-one correspondence between these ideas: every family of self-dual theories of electrodynamics (such as the Born-Infeld and Modified Maxwell theories) obeys a flow equation built from the energy-momentum tensor, and every stress tensor deformation gives rise to families of duality-invariant theories.

TBA