Bergman Kernel from Path Integral

• Published in: Communications in mathematical physics Vol. 293; no. 1; pp. 205 - 230
• Main Authors: Douglas, Michael R., Klevtsov, Semyon
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.01.2010
• Subjects: Classical and Quantum Gravitation; Complex Systems; Mathematical and Computational Physics; Mathematical Physics; Physics; Physics and Astronomy; Quantum Physics; Relativity Theory; Theoretical; Line Bundle; Heat Kernel; Path Integral; Density Matrix; Coherent State
• ISSN: 0010-3616; 1432-0916
• DOI: 10.1007/s00220-009-0915-0

We rederive the expansion of the Bergman kernel on Kähler manifolds developed by Tian, Yau, Zelditch, Lu and Catlin, using path integral and perturbation theory, and generalize it to supersymmetric quantum mechanics. One physics interpretation of this result is as an expansion of the projector of wave functions on the lowest Landau level, in the special case that the magnetic field is proportional to the Kähler form. This is relevant for the quantum Hall effect in curved space, and for its higher dimensional generalizations. Other applications include the theory of coherent states, the study of balanced metrics, noncommutative field theory, and a conjecture on metrics in black hole backgrounds discussed in [24]. We give a short overview of these various topics. From a conceptual point of view, this expansion is noteworthy as it is a geometric expansion, somewhat similar to the DeWitt-Seeley-Gilkey et al short time expansion for the heat kernel, but in this case describing the long time limit, without depending on supersymmetry.