Cartan Geometry and Infinite-Dimensional Kempf-Ness Theory

TL;DR

This paper develops an infinite-dimensional version of the Kempf-Ness theorem using Cartan bundles, enabling applications in Kähler geometry, deformation quantization, and gauge theory despite the lack of complexification.

Contribution

It introduces a novel Cartan bundle approach to extend Kempf-Ness theory to infinite dimensions, overcoming the challenge of absent complexification.

Findings

Established a convexity property of the Kempf-Ness function in infinite dimensions.

Defined a generalized Futaki character within the new framework.

Applied the theory to problems in Kähler geometry, quantization, and gauge theory.

Abstract

We pioneer the development of a rigorous infinite-dimensional framework for the Kempf-Ness theorem, addressing the significant challenge posed by the absence of a complexification for the symmetry group in infinite dimensions, e.g, the diffeomorphism group. We propose a novel approach, based on Cartan bundles, to generalize Kempf-Ness theory to infinite dimensions, invoking the fundamental role played by the Maurer-Cartan form. This approach allows us to define and study objects essential for the Kempf-Ness theorem, such as the complex model for orbits and the Kempf-Ness function, as well as establishing its convexity properties and defining a generalized Futaki character. We show how our framework can be applied to the study of various problems in K\"ahler geometry, deformation quantization, and gauge theory.