Bergman Kernel from Path Integral

TL;DR

This paper rederives and generalizes the Bergman kernel expansion on Kahler manifolds using path integral methods, with applications in quantum Hall effect, coherent states, and black hole metrics.

Contribution

It introduces a path integral approach to derive the Bergman kernel expansion and extends it to supersymmetric quantum mechanics, linking physics and complex geometry.

Findings

Path integral derivation of Bergman kernel expansion.

Generalization to supersymmetric quantum mechanics.

Relevance to quantum Hall effect and black hole metrics.

Abstract

We rederive the expansion of the Bergman kernel on Kahler 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 Kahler 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. 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.