[Paper Review] Self-dual SU(2) invariant Einstein metrics and modular dependence of theta-functions

M. V. Babich, D. Korotkin|arXiv (Cornell University)|Oct 7, 1998

Geometric Analysis and Curvature Flows4 citations

TL;DR

This paper simplifies Hitchin's construction of SU(2)-invariant self-dual Einstein metrics by employing the tau-function of a four-pole Schlesinger system, revealing a modular dependence of theta-functions on the metric parameters. The key contribution is a unified description linking integrable systems and self-dual gravity through tau-function geometry.

ABSTRACT

We simplify Hitchin's description of SU(2)-invariant self-dual Einstein metrics, making use of the tau-function of related four-pole Schlesinger system.

Motivation & Objective

• To simplify Hitchin's description of SU(2)-invariant self-dual Einstein metrics using integrable system techniques.
• To clarify the modular properties of theta-functions arising in the context of self-dual gravity.
• To establish a direct link between the tau-function of a four-pole Schlesinger system and the geometry of self-dual Einstein metrics.
• To provide a more transparent and computationally accessible formulation of these metrics by reducing dependence on complex algebraic geometry.

Proposed method

• Utilizes the tau-function associated with a four-pole Schlesinger system as a central geometric object.
• Applies the isomonodromic deformation theory to relate monodromy data to the metric structure.
• Expresses the metric components in terms of derivatives of the tau-function and associated theta-functions.
• Analyzes the modular transformation properties of the theta-functions under the action of the modular group.
• Employs the Riemann theta function with characteristics to encode the dependence on moduli parameters.
• Demonstrates that the self-dual condition is equivalent to a specific tau-function normalization and modular invariance.

Experimental results

Research questions

• RQ1 How can Hitchin’s SU(2)-invariant self-dual Einstein metrics be re-expressed using integrable system tools?
• RQ2 What is the modular behavior of the theta-functions that appear in the metric construction?
• RQ3 How does the tau-function of the four-pole Schlesinger system encode the geometry of self-dual Einstein metrics?
• RQ4 What role do monodromy data and isomonodromic deformations play in constructing these metrics?
• RQ5 Can the modular dependence of theta-functions be systematically derived from the underlying differential system?

Key findings

• The SU(2)-invariant self-dual Einstein metrics are fully reconstructed from the tau-function of a four-pole Schlesinger system.
• The metric components are expressed as rational functions of the tau-function and its derivatives.
• The theta-functions appearing in the metric exhibit a well-defined modular transformation law under the action of the modular group.
• The self-dual condition is equivalent to a specific normalization of the tau-function ensuring modular invariance.
• The construction provides a direct geometric realization of the isomonodromic tau-function in the context of Einstein gravity.
• Misprints in the original version were corrected in the final arXiv version (v3), ensuring accuracy in the final formulation.