Balanced metrics and noncommutative Kaehler geometry

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.