Stress Tensor flows, birefringence in non-linear electrodynamics and supersymmetry

Abstract

We identify the unique stress tensor deformation which preserves zero-birefringence conditions in non-linear electrodynamics, which is a 4d4d version of the T\overline{T}TT¯ 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}TT¯-like flow that we analyse in our paper. We demonstrate one way of making this root-T\overline{T}TT¯-like flow manifestly supersymmetric by writing the deforming operator in \mathcal{N} = 1𝒩=1 superspace and exhibit two examples of superspace flows. We present scalar analogues in d = 2d=2 with similar properties as these theories of electrodynamics in d = 4d=4. Surprisingly, the Plebanski-type theories are fixed points of the classical T\overline{T}TT¯-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.

$T \overline{T}$-like flows and $3d$ nonlinear supersymmetry

“…Comparing the final expression of (48) to that of (47), we see that the second term of R vanishes on-shell and thus R is a total derivative up to equations of motion as claimed.…”

“…Therefore, deformations of the 3d Maxwell Lagrangian -such as the stress tensor flow which produces the Born-Infeld theory that was presented in the preceding subsection -do not exhibit the usual properties for deformations of CFTs which are often used when studying perturbations of this kind. An especially useful example of such a property is the so-called trace flow equation, which plays a role in many places in the T T literature, including in cutoff AdS 3 holography [70][71][72][73][74][75] and in 4d T T -like flows [48]. We now review and extend this result.…”

“…produces the ModMax theory [44][45][46] when the initial condition is the ordinary Maxwell theory [47]. As we have mentioned in Section 1, this flow commutes with the irrelevant T T -like flow which deforms the Maxwell Lagrangian into the Born-Infeld theory, and this pair of commuting flows can be used to construct a two-parameter family of ModMax-Born-Infeld theories (and their supersymmetrix extensions) [48,49]. A similar pair of flows defines a collection of Modified-Nambu-Goto theories in two dimensions [17,34,50].…”

“…In [48], it was pointed out that one version of the four-dimensional Born-Infeld Lagrangian also satisfies a flow equation driven by a relevant operator constructed from the stress tensor, as opposed to the usual irrelevant T T -like combination which deforms the 4d Maxwell theory to Born-Infeld. This result relied upon the fact that one can construct a "subtracted" version of the theory by adding an appropriate λ-dependent constant to the Lagrangian which causes the T T -like operator, which usually drives the irrelevant flow, to become a constant.…”

“…3 A marginal flow driven by the operator R can be used to obtain the Modified Maxwell (ModMax) theory, which was introduced in [44][45][46], in four spacetime dimensions [47]. This root-T T -like flow can also be made manifestly supersymmetric [48], and along with the 4d T T -like flow, forms a 2-parameter family of commuting deformations which generate the ModMax-Born-Infeld theory (or its supersymmetric extension, in the superspace case) [49]. In two dimensions, the analogous root-T T flow can be used to construct a "Modified Scalar" theory, or combined with T T to yield a 2-parameter family of "Modified-Nambu-Goto" theories [17,34,50].…”