Some special Kahler metrics on SL(2,C) and their holomorphic quantization

TL;DR

This paper investigates special Kahler metrics on SL(2,C) invariant under SU(2)*SU(2) action, computes their geometric properties explicitly, and explores their holomorphic quantization, proposing a conjecture on the Hilbert space dimension in the semi-classical limit.

Contribution

It introduces a class of invariant Kahler metrics on SL(2,C), computes their geometric invariants explicitly, and analyzes their holomorphic quantization, leading to a new conjecture on Hilbert space dimensions.

Findings

Explicit formulas for volume growth and curvature quantities.

Examples include metrics with positive, negative, and zero Ricci curvature.

Results on the dimension of the quantized Hilbert space and a related conjecture.

Abstract

The group SU(2)*SU(2) acts naturally on SL(2,C) by simultaneous right and left multiplication. We study the Kahler metrics invariant under this action using global Kahler potentials. The volume growth and various curvature quantities are then explicitly computable. Examples include metrics of positive, negative and zero Ricci curvature, and the 1-lump metric of the CP^1-model on a sphere. We then look at the holomorphic quantization of these metrics, where some physically satisfactory results on the dimension of the Hilbert space can be obtained. These give rise to an interesting geometrical conjecture, regarding the dimension of this space for general Stein manifolds in the semi-classical limit.