Kahler-Einstein metrics emerging from free fermions and statistical mechanics

Abstract

We propose a statistical mechanical derivation of Kahler-Einstein metrics, i.e. solutions to Einstein's vacuum field equations in Euclidean signature (with a cosmological constant) on a compact Kahler 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). 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 effective bosonization and the Yau-Tian-Donaldson program in Kahler geometry are pointed out. The precise mathematical details will be investigated elsewhere.

v1: 22 pages v2: 25 pages. The relation to quantum gravity has been further developed by working over the moduli space of all complex structures. Relations to Donaldson's program pointed out. References added

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