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Kähler-Einstein metrics emerging from free fermions and statistical mechanics

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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.

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  1. Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, Hörsalsvägen 3, SE-412 96, Gothenburg, Sweden

    Robert J. Berman

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Correspondence to Robert J. Berman.

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ArXiv ePrint: 1009.2942

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Berman, R.J. Kähler-Einstein metrics emerging from free fermions and statistical mechanics. J. High Energ. Phys. 2011, 106 (2011). https://doi.org/10.1007/JHEP10(2011)106

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