Quaternionic and Hyper-K\"ahler Metrics from Generalized Sigma Models

TL;DR

This paper develops a method to derive new quaternionic and hyper-K"ahler metrics from generalized sigma models, providing explicit solutions and exploring their geometric properties relevant to supergravity.

Contribution

It introduces a generalized quaternionic sigma model approach to construct and analyze new hyper-K"ahler metrics, including explicit solutions and their geometric generalizations.

Findings

Derived new quaternionic and hyper-K"ahler metrics from sigma models.

Obtained explicit solutions analogous to Berger's sphere and Abraham-Townsend metrics.

Discussed generalizations of 4D quaternionic metrics as products of complex metrics.

Abstract

The problem of finding new metrics of interest, in the context of SUGRA, is reduced to two stages: first, solving a generalized BPS sigma model with full quaternionic structure proposed by the authors and, second, constructing the hyper-K\"{a}hler metric, or suitable deformations of this condition, taking advantage of the correspondence between the quaternionic left-regular potential and the hyper-K\"{a}hler metric of the target space. As illustration, new solutions are obtained using generalized Q-sigma model for Wess-Zumino type superpotentials. Explicit solutions analog to the Berger's sphere and Abraham-Townsend type are given and generalizations of 4-dimensional quaternionic metrics, product of complex ones, are shown and discussed.