Balanced Metrics and Noncommutative Kähler Geometry

[摘要] In this paper we show how Einstein metrics are naturally described using the quantization of the algebra of functions C ∞ ( 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.

[关键词] balanced metrics;geometric quantization;K¨ahler–Einstein