Encoding field theories into gravities

Abstract

We propose a method to give a d +1 geometry from a d dimensional quantum field theory in the large N expansion. We first construct a d +1 dimensional field from the d dimensional one using the gradient flow equation, whose flow time t represents the energy scale of the system such that t !0 corresponds to the ultra-violet (UV) while t →∞ to the infra-red (IR). We define the induced metric using d +1 dimensional field operators. We show that the metric defined in this way becomes classical in the large N limit: quantum fluctuations of the metric are suppressed as 1/N due to the large N factorization property. As a concrete example, we apply our method to the O(N) non-linear s model in two dimensions. We calculate the three dimensional induced metric, which describes an AdS space in the massless limit. We finally discuss several open issues for future investigations.