The on-shell action of supergravity & the B-side of TsT and single-trace $T\bar{T}$

We propose boundary terms for the action of the NS sector of supergravity on $M_3\times S^3\times T^4$ spacetimes – where $M_3$ is an AdS${}_3$ or a linear dilaton background – that render the Brown-York stress tensor and the on-shell action finite. For AdS${}_3$ backgrounds, we show that the on-shell action yields a free energy with chemical potentials determined by the $B$-field. TsT (T-duality + shift + T-duality) transformations of these backgrounds generate classes of linear dilaton backgrounds distinguished by their boundary conditions. Among them, there is one class of backgrounds that reproduces features of the single-trace $T\bar{T}$ deformation. These backgrounds have additional chemical potentials that can be turned off by large gauge transformations of the $B$-field. We show that the Brown-York stress tensor of these backgrounds reproduces the energies and the trace flow equation of $T\bar{T}$-deformed CFTs. We also show that the on-shell action matches the $T\bar{T}$ partition function and discuss the interpretation of these results.