Machine learning holographic mapping by neural network renormalization group

Abstract

Exact holographic mapping (EHM) provides an explicit duality map between a conformal field theory (CFT) configuration and a massive field propagating on an emergent classical geometry. However, designing the optimal holographic mapping is challenging. Here we introduce the neural network renormalization group as a universal approach to design generic EHM for interacting field theories. Given a field theory action, we train a flow-based hierarchical deep generative neural network to reproduce the boundary field ensemble from uncorrelated bulk field fluctuations. In this way, the neural network develops the optimal renormalization-group transformations. Using the machine-designed EHM to map the CFT back to a bulk effective action, we determine the bulk geodesic distance from the residual mutual information. We have shown that the geometry measured in this way is the classical saddle-point geometry. We apply this approach to the complex φ4 theory in two-dimensional Euclidian space-time in its critical phase, and show that the emergent bulk geometry matches the three-dimensional hyperbolic geometry when geometric fluctuation is neglected.