Elliptic constructions of hyperkaehler metrics II: The quantum mechanics of a Swann bundle

TL;DR

This paper explores the mathematical structure of hyperkaehler metrics derived from holomorphic functions, revealing a connection to quantum mechanics wave functions and illustrating the theory with specific geometric constructions.

Contribution

It establishes a novel link between sections of holomorphic bundles in hyperkaehler geometry and quantum-mechanical wave functions, providing new insights into Swann bundle constructions.

Findings

Sections of ${ m O}(2j)$ bundles resemble quantum wave functions

Analysis of SO(3) invariants in hyperkaehler geometry

Illustration with two explicit Swann bundle examples

Abstract

The generalized Legendre transform method of Lindstrom and Rocek yields hyperkaehler metrics from holomorphic functions. Its main ingredients are sections of ${\cal O}(2j)$ bundles over the twistor space satisfying a reality condition with respect to antipodal conjugation on the hyperkaehler sphere of complex structures. Formally, the structure of the real ${\cal O}(2j)$ sections is identical to that of quantum-mechanical wave functions describing the states of a particle with spin $j$ in the spin coherent representation. We analyze these sections and their SO(3) invariants and illustrate our findings with two Swann bundle constructions.