$ T\overline{T} $-deformed fermionic theories revisited

We revisit $$ T\overline{T} $$T T ¯ deformations of d = 2 theories with fermions with a view toward the quantization. As a simple illustration, we compute the deformed Dirac bracket for a Majorana doublet and confirm the known eigenvalue flows perturbatively. We mostly consider those $$ T\overline{T} $$T T ¯ theories that can be reconstructed from string-like theories upon integrating out the worldsheet metric. After a quick overview of how this works when we add NSR-like or GS-like fermions, we obtain a known non-supersymmetric $$ T\overline{T} $$T T ¯ deformation of a $$ \mathcal{N} $$N = (1, 1) theory from the latter, based on the Noether energy-momentum. This world- sheet reconstruction implies that the latter is actually a supersymmetric subsector of a d = 3 GS-like model, implying hidden supercharges, which we do construct explicitly. This brings us to ask about different $$ T\overline{T} $$T T ¯ deformations, such as manifestly supersymmetric $$ T\overline{T} $$T T ¯ and also more generally via the symmetric energy-momentum. We show that, for theories with fermions, such choices often lead us to doubling of degrees of freedom, with potential unitarity issues. We show that the extra sector develops a divergent gap in the “small deformation” limit and decouples in the infrared, although it remains uncertain in what sense these can be considered a deformation.