Kahler-Einstein metrics emerging from free fermions and statistical mechanics

TL;DR

This paper presents a novel approach to deriving Kahler-Einstein metrics using statistical mechanics of free fermions, linking quantum states to geometric solutions on Kahler manifolds.

Contribution

It introduces a microscopic fermionic model that leads to Kahler-Einstein metrics, connecting physics and complex geometry in a new way.

Findings

Proposes a fermionic statistical model for Kahler-Einstein metrics

Provides heuristic convergence argument in thermodynamic limit

Connects geometric analysis with quantum 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.