Kähler-Einstein metrics emerging from free fermions and statistical mechanics

Abstract

We propose a statistical mechanical derivation of Kähler-Einstein metrics, i.e. solutions to Einstein's vacuum field equations in Euclidean signature (with a cosmological constant) on a compact Kähler manifold X. The microscopic theory is given by a canonical free fermion gas on X whose one-particle states are pluricanonical holomorphic sections on X (coinciding with higher spin states in the case of a Riemann surface) defined in background free manner. A heuristic, but hopefully physically illuminating, argument for the convergence in the thermodynamical (large N) limit is given, based on a recent mathematically rigorous result about exponentially small fluctuations of Slater determinants. Relations to higher-dimensional effective bosonization, the Yau-Tian-Donaldson program in Kähler geometry and quantum gravity are explored. The precise mathematical details will be investigated elsewhere. © SISSA 2011.