Large N Analysis of $T\bar{T}$-deformation and Unavoidable Negative-norm States

We study non-perturbative quantum aspects of $T\bar{T}$-deformation of a free $O(N)$ vector model by employing the large $N$ limit. It is shown that bound states of the original field appear and inevitably become negative-norm states. In particular, the bound states can be regarded as the states of the conformal mode in a gravitational theory, where the Liouville action is induced with the coefficient proportional to the minus of central charge. To make the theory positive-definite, some modification is required so as to preserve diffeomorphism invariance due to the Faddeev-Popov ghosts with a negative central charge.