Encoding field theories into gravities
  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 $t\rightarrow 0$ corresponds to the ultra-violet (UV) while $t\rightarrow\infty$ 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 $\sigma$ 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. 
Sinya Aoki, Kengo Kikuchi, Tetsuya Onogi
hep-lat hep-th