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

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.