Balanced metrics and noncommutative Kaehler geometry

TL;DR

This paper explores how Einstein metrics on Kahler manifolds can be understood through quantization methods, revealing natural emergence of balanced metrics and their classical limits, with applications to Calabi-Yau geometry.

Contribution

It introduces a framework linking geometric and deformation quantization to Einstein metrics and balanced metrics, with new insights into their classical limits and applications.

Findings

Balanced metrics arise from vacuum energy conditions.

Classical limits of these metrics are Kahler-Einstein.

Applications include constructing special Lagrangian submanifolds.

Abstract

In this paper we show how Einstein metrics are naturally described using the quantization of the algebra of functions on a Kahler manifold M. In this setup one interprets M as the phase space itself, equipped with the Poisson brackets inherited from the Kahler 2-form. We compare the geometric quantization framework with several deformation quantization approaches. We find that the balanced metrics appear naturally as a result of setting the vacuum energy to be the constant function on the moduli space of semiclassical vacua. In the classical limit these metrics become Kahler-Einstein (when M admits such metrics). Finally, we sketch several applications of this formalism, such as explicit constructions of special Lagrangian submanifolds in compact Calabi-Yau manifolds.