A bstract In this work, we investigate the partition function of 2d CFT under root- TT T 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- TT T 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- TT T 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- TT T T ¯ deformed AdS 3 geometry. Our results reveal an exact agreement between one-loop gravitational partition function and the large c expansion of root- TT T T ¯ deformed CFT partition function.
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