We find a geometrical description from a field theoretical setup based on Wilson’s renormalization group in real space. We show that renormalization group equations of coupling parameters encode the metric structure of an emergent curved space, regarded to be an Einstein equation for the emergent gravity. Self-consistent equations of local order-parameter fields with an emergent metric turn out to describe low-energy dynamics of a strongly coupled field theory, analogous to the Maxwell equation of the Einstein-Maxwell theory in the ${\mathrm{AdS}}_{d+2}/{\mathrm{CFT}}_{d+1}$ duality conjecture. We claim that the ${\mathrm{AdS}}_3/{\mathrm{CFT}}_2$ duality may be interpreted as Landau-Ginzburg theory combined with Wilson’s renormalization group, which introduces vertex corrections into the Landau-Ginzburg theory in the large-$N_s$ limit, where $N_s$ is the number of fermion flavors.

• Received 28 April 2016

DOI:https://doi.org/10.1103/PhysRevD.93.121702