  We study the UV dynamics of $\mu T \bar T$ deformed conformal field theories
formulated as a deformation of generating functions. We explore the issue of
non-perturbative completion of the $\mu$ expansion by deriving an integral
expression using the Fourier/Legendre transform technique, and show that it is
more natural to impose Neumann, as opposed to the Dirichlet, boundary
condition, for the metric at the cut-off surface recently proposed by McGough,
Mezei, and Verlinde. We also comment on interesting connection to boundary
conformal field theories.
