Abelian Hypermultiplet Gaugings and BPS Vacua in N = 2 Supergravity

TL;DR

We develop a comprehensive framework for Abelian gaugings of hypermultiplets in ${\cal N}=2$ supergravity by exploiting symmetric quaternionic-Kähler manifolds arising from the c-map and computing the full set of Killing prepotentials and their $SU(2)$ compensators. The analysis shows that regular ${\cal N}=2$ ${\rm AdS}_4$ vacua require gauging along isometries with non-trivial compensators, and we formulate the magnetic/electric gauging structure, locality constraints, and their implications for ${\rm AdS}_4$ and ${\rm AdS}_2\times \Sigma_g$ BPS configurations. We illustrate the formalism with two M-theory inspired examples (e.g., ${\cal M}_h=G_{2(2)}/SO(4)$) yielding explicit AdS$_4$ vacua and AdS$_2$ horizons, highlighting how Killing prepotentials determine the BPS equations and entropy through the quartic invariant. The results provide a rigorous pathway to classify and construct BPS vacua and horizon geometries in ${\cal N}=2$ gauged supergravity with hypermultiplets, with future directions including fully solving horizon flows and extending to broader homogeneous spaces.

Abstract

We analyze the gauging of Abelian isometries on the hypermultiplet scalar manifolds of N = 2 supergravity in four dimensions. This involves a study of symmetric special quaternionic-Kähler manifolds, building on the work of de Wit and Van Proeyen. In particular we compute the general set of Killing prepotentials and associated compensators for these manifolds manifolds. This allows us to glean new insights about AdS 4 vacua which preserve the full N = 2 supersymmetry as well as BPS static black hole horizons.

Figures (1)

• Figure 1: Root diagram for the $G_2$ Lie algebra.