In this work, we investigate the partition function of 2d CFT under root-$T\bar{T}$ deformation. We demonstrate that the deformed partition function satisfies a flow equation. At large central charge sector, the deformed partition function reduces to a redefinition of the modular parameters, which preserves modular invariance under the deformed parameters. We then derive a Cardy-like formula for the asymptotic density of states using modular bootstrap trick. In the context of AdS/CFT, it was proposed the root-$T\bar{T}$ deformed CFT corresponds to the AdS$_3$ with certain deformed boundary condition. We show the deformed BTZ black hole is a quotient of hyperbolic space. In terms of Chern-Simons formulation, we compute the root-$T\bar{T}$ deformed BTZ black hole entropy and find that it obeys a Cardy-like formula, which is consistent with the modular bootstrap result. Furthermore, employing the Wilson spool technique, we compute the one-loop partition functions for the root-$T\bar{T}$ deformed AdS$_3$ geometry. Our results reveal an exact match between one-loop gravitational partition function and the large $c$ expansion of root-$T\bar{T}$ deformed CFT partition function.
Comment: 37 pages