Random K\"ahler Metrics

TL;DR

This paper introduces a novel approach to defining and computing path integrals over K"ahler metrics by approximating infinite-dimensional spaces with finite-dimensional Bergman metrics and analyzing their limits using large deviations theory.

Contribution

It proposes a new method leveraging Bergman metrics and large deviations to approximate and analyze path integrals over K"ahler metrics.

Findings

Demonstrates convergence of probability measures on Bergman metrics to measures on K"ahler metrics

Provides several examples illustrating the approximation method

Establishes a theoretical framework for path integrals in K"ahler geometry

Abstract

The purpose of this article is to propose a new method to define and calculate path integrals over metrics on a K\"ahler manifold. The main idea is to use finite dimensional spaces of Bergman metrics, as an approximation to the full space of K\"ahler metrics. We use the theory of large deviations to decide when a sequence of probability measures on the spaces of Bergman metrics tends to a limit measure on the space of all K\"ahler metrics. Several examples are considered.