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 $t\to 0$ corresponds to the ultra-violet (UV) while $t\to \infty$ 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.

7 pages, Proceedings of the 33rd International Symposium on Lattice Field Theory (LATTICE2015), 14 -18 July 2015, Kobe International Conference Center, Kobe, Japan

References in corpus (3)

• Perturbative analysis of the gradient flow in non-abelian gauge theories
• Renormalizability of the gradient flow in the 2D $O(N)$ non-linear sigma model
• Gradient Flow of O(N) nonlinear sigma model at large N

Cited by in corpus (1)

• Holographic geometry for non-relativistic systems emerging from generalized flow equations