Geometries from field theories

Abstract

We propose a method to define a $d+1$ dimensional geometry from a $d$ dimensional quantum field theory in the $1/N$ expansion. We first construct a $d+1$ dimensional field theory from the $d$ dimensional one via 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 then define the induced metric from $d+1$ dimensional field operators. We show that the metric defined in this way becomes classical in the large $N$ limit, in a sense that 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 is shown to describe an AdS space in the massless limit. We finally discuss several open issues in future studies.

9 pages, the title has been changed, and some contents have also been modified. This version is accepted for a publication in PTEP

References in corpus (5)

• Perturbative analysis of the gradient flow in non-abelian gauge theories
• Infinite N phase transitions in continuum Wilson loop operators
• 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
• Generalized Gradient Flow Equation and Its Application to Super Yang-Mills Theory

Cited by in corpus (10)

• Results and techniques for higher order calculations within the gradient-flow formalism
• Flow equation, conformal symmetry and AdS geometry
• Gradient flow and the Wilsonian renormalization group flow
• Holographic geometry for non-relativistic systems emerging from generalized flow equations
• Supersymmetric gradient flow in the Wess-Zumino model
• Flow equation for the scalar model in the large $N$ expansion and its applications
• Holographic computation of quantum corrections to the bulk cosmological constant
• Non-relativistic Hybrid Geometry with Gravitational Gauge-Fixing Term
• Perturbative analysis of the Wess-Zumino flow
• AdS/CFT correspondence for the $O(N)$ invariant critical ${\phi }^4$ model in 3-dimensions by the conformal smearing