Geometries from field theories

Abstract

We propose a method to define a d+1d+1-dimensional geometry from a dd-dimensional quantum field theory in the 1/N1/N expansion. We first construct a d+1d+1-dimensional field theory from the dd-dimensional one via the gradient-flow equation, whose flow time tt represents the energy scale of the system such that t→0t→0 corresponds to the ultraviolet and t→∞t→∞ to the infrared. We then define the induced metric from d+1d+1-dimensional field operators. We show that the metric defined in this way becomes classical in the large-NN limit, in the sense that quantum fluctuations of the metric are suppressed as 1/N1/N due to the large-NN factorization property. As a concrete example, we apply our method to the O(N)O(N) nonlinear σσ model in two dimensions. We calculate the 3D induced metric, which is shown to describe an anti-de Sitter space in the massless limit. Finally, we discuss several open issues for future studies.