SIGMA 6 (2010), 069, 15 pages      arXiv:0710.1304      https://doi.org/10.3842/SIGMA.2010.069
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

Abstract
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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References

1. Alvarez-Gaumé L., Freedman D.Z., Mukhi S., The background field method and the ultraviolet structure of the supersymmetric nonlinear σ-model, Ann. Physics 134 (1981), 85-109.
2. Axelrod S., Della Pietra S., Witten E., Geometric quantization of Chern-Simons gauge theory, J. Differential Geom. 33 (1991), 787-902.
3. Banks T., Fischler W., Shenker S.H., Susskind L., M theory as a matrix model: a conjecture, Phys. Rev. D 55 (1997), 5112-5128, hep-th/9610043.
4. Bordemann M., Meinrenken E., Schlichenmaier M., Toeplitz quantization of Kähler manifolds and gl(N), N→∞ limits, Comm. Math. Phys. 165 (1994), 281-296, hep-th/9309134.
5. Bressler P., Soibelman Y., Mirror symmetry and deformation quantization, hep-th/0202128.
6. Cattaneo A.S., Felder G., A path integral approach to the Kontsevich quantization formula, Comm. Math. Phys. 212 (2000), 591-611, math.QA/9902090.
7. Cornalba L., Taylor W., Holomorphic curves from matrices, Nuclear Phys. B 536 (1998), 513-552, hep-th/9807060.
8. Donaldson S.K., Scalar curvature and projective embeddings. I, J. Differential Geom. 59 (2001), 479-522.
9. Donaldson S.K., Some numerical results in complex differential geometry, Pure Appl. Math. Q. 5 (2009), 571-618, math.DG/0512625.
10. Douglas M.R., Karp R.L., Lukic S., Reinbacher R., Numerical solution to the hermitian Yang-Mills equation on the Fermat quintic, J. High Energy Phys. 2007 (2007), no. 12, 083, 24 pages, hep-th/0606261.
11. Douglas M.R., Karp R.L., Lukic S., Reinbacher R., Numerical Calabi-Yau metrics, J. Math. Phys. 49 (2008), 032302, 19 pages, hep-th/0612075.
12. Elitzur S., Moore G.W., Schwimmer A., Seiberg N., Remarks on the canonical quantization of the Chern-Simons-Witten theory, Nuclear Phys. B 326 (1989), 108-134.
13. Gaiotto D., Simons A., Strominger A., Yin X., D0-branes in black hole attractors, J. High Energy Phys. 2006 (2006), no. 3, 019, 24 pages, hep-th/0412179.
14. Kachru S., Lawrence A.E., Silverstein E., On the matrix description of Calabi-Yau compactifications, Phys. Rev. Lett. 80 (1998), 2996-2999, hep-th/9712223.
15. Kapustin A., Topological strings on noncommutative manifolds, Int. J. Geom. Methods Mod. Phys. 1 (2004), 49-81, hep-th/0310057.
16. Karabegov A.V., Deformation quantizations with separation of variables on a Kähler manifold, Comm. Math. Phys. 180 (1996), 745-755, hep-th/9508013.
17. Kontsevich M., Deformation quantization of Poisson manifolds. I, Lett. Math. Phys. 66 (2003), 157-216, q-alg/9709040.
18. Lu Z., On the lower order terms of the asymptotic expansion of Tian-Yau-Zelditch, Amer. J. Math. 122 (2000), 235-273, math.DG/9811126.
19. Rawnsley J.H., Coherent states and Kähler manifolds, Quart. J. Math. Oxford Ser. (2) 28 (1977), no. 112, 403-415.
20. Reshetikhin N., Takhtajan L.A., Deformation quantization of Kähler manifolds, in L.D. Faddeev's Seminar on Mathematical Physics, Amer. Math. Soc. Transl. Ser. 2, Vol. 201, Amer. Math. Soc., Providence, RI, 2000, 257-276, math.QA/9907171.
21. Seiberg N., Witten E., String theory and noncommutative geometry, J. High Energy Phys. 1999 (1999), no. 9, 032, 93 pages, hep-th/9908142.
22. Souriau J.M., Structure of dynamical systems. A symplectic view of physics, Progress in Mathematics, Vol. 149, Birkhäuser Boston, Inc., Boston, MA, 1997.
23. Zelditch S., Szegö kernels and a theorem of Tian, Internat. Math. Res. Notices 1998 (1998), no. 6, 317-331, math-ph/0002009.