Quaternionic K\"ahler metrics associated to special K\"ahler manifolds with mutually local variations of BPS structures

TL;DR

This paper constructs explicit quaternionic-K"ahler metrics from special K"ahler manifolds with BPS structure variations, connecting mathematical structures to string theory moduli spaces and providing formulas for their metrics.

Contribution

It introduces a new class of quaternionic-K"ahler metrics derived from special K"ahler manifolds with mutually-local BPS structures, including explicit formulas and positivity conditions.

Findings

Explicit formulas for quaternionic-K"ahler metrics

Conditions for metric positive-definiteness

Relation to supergravity c-map and string theory models

Abstract

We construct a quaternionic-K\"ahler manifold from a conical special K\"ahler manifold with a certain type of mutually-local variation of BPS structures. We give global and local explicit formulas for the quaternionic-K\"ahler metric, and specify under which conditions it is positive-definite. Locally, the metric is a deformation of the 1-loop corrected Ferrara-Sabharval metric obtained via the supergravity c-map. The type of quaternionic-K\"ahler metrics we obtain are related to work in the physics literature by S. Alexandrov and S. Banerjee, where they discuss the hypermultiplet moduli space metric of type IIA string theory, with mutually local D-instanton corrections.