Balanced Metrics and Noncommutative Kähler Geometry

In this paper we show how Einstein metrics are naturally described using the quantization of the algebra of functions $C^\infty(M)$ on a Kähler manifold $M$. In this setup one interprets $M$ as the phase space itself, equipped with the Poisson brackets inherited from the Kähler 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 requiring the vacuum energy to be the constant function on the moduli space of semiclassical vacua. In the classical limit these metrics become Kähler–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.