Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 6 (2010), 069, 15 pages      arXiv:0710.1304
Contribution to the Special Issue “Noncommutative Spaces and Fields”

Balanced Metrics and Noncommutative Kähler Geometry

Sergio Lukic
Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08855-0849, USA

Received March 01, 2010, in final form August 02, 2010; Published online August 27, 2010

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.

Key words: balanced metrics; geometric quantization; Kähler-Einstein.

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