Einstein metrics, hypercomplex structures and the Toda field equation

✍ Scribed by David M.J. Calderbank; Paul Tod

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 73 KB
• Volume: 14
• Category: Article
• ISSN: 0926-2245
• DOI: 10.1016/s0926-2245(01)00037-7

✦ Synopsis

We obtain explicitly all solutions of the SU(∞) Toda field equation with the property that the associated Einstein-Weyl space admits a 2-sphere of divergence-free shear-free geodesic congruences. The solutions depend on an arbitrary holomorphic function and give rise to new hyperKähler and selfdual Einstein metrics with one-dimensional isometry group. These metrics each admit a compatible hypercomplex structure with respect to which the symmetries are triholomorphic.