Quaternionic Kahler spaces with large toric symmetry

TL;DR

This paper develops explicit formulas for quaternionic Kahler metrics with large toric symmetry, linking the geometry to reduced Higgs fields and solving associated PDEs, including special cases like self-dual Einstein manifolds.

Contribution

It introduces a new explicit construction of quaternionic Kahler metrics with toric symmetry using reduced Higgs fields and PDE systems, connecting different geometric approaches.

Findings

Explicit formulas for quaternionic Kahler metrics and connections.

Reduction of the problem to linear PDEs for Higgs fields.

Construction of solutions via generalized Legendre transform.

Abstract

We consider a general 4n-dimensional quaternionic Kahler geometry with a free action of the torus T^(n+1). The toric action lifts onto the Swann bundle of the quaternionic Kahler space to a tri-holomorphic action that commutes with the standard H* action on the bundle. By matching Pedersen and Poon's generalized Gibbons-Hawking Ansatz description of the total space with the Swann picture we extract the local geometry of the quaternionic Kahler base. Specifically, we obtain explicit expressions for the quaternionic Kahler metric and Sp(1) connection in terms of a set of reduced Higgs fields and connection 1-forms that satisfy a reduced Bogomol'nyi-type equation. We find, moreover, that these Higgs fields can be derived from a single function V satisfying a system of linear second-order partial differential constraints. In four dimensions, corresponding to the case of self-dual Einstein manifolds with two commuting Killing vector fields, our formulas coincide with those obtained through a different approach by Calderbank and Pedersen. Finally, we show how to construct explicit solutions to the reduced Bogomol'nyi and V equations by means of Lindstrom and Rocek's generalized Legendre transform construction for a large class of quaternionic Kahler manifolds related to the c-map.