Superconformal Hypermultiplets

TL;DR

The paper develops $N=2$ hypermultiplet theories with rigid or local superconformal symmetry in four dimensions, showing that rigid theories require $(4n)$-dimensional special hyper-Kähler target spaces that are cones over tri-Sasakian manifolds and locally factor as ${f R}^4\times{\cal Q}^{4n-4}$ with a quaternionic base ${\cal Q}$. It introduces local ${\rm Sp}(n)\times{\rm Sp}(1)$ sections to formulate the Lagrangian and supersymmetry transformations in a coordinatization-free way, and derives the hyper-Kähler potential $\chi$ via a homothetic Killing vector $k_D^A$ with $D_A k_D^B=\delta^B_A$, linking to a cone structure and the associated sections $A_i^\alpha$. The work then shows how coupling to conformal supergravity via the conformal multiplet calculus yields a quaternionic target space after gauge fixing, with a precise decomposition of curvatures and a consistent Sp$(n-1)$ holonomy for the horizontal sector. These results generalize previous flat or quotient constructions, connect to the $c$-map in string compactifications, and provide a unified framework for studying hypermultiplet geometry and current superconformal couplings. The formalism offers a geometric and algebraic toolkit for exploring hypermultiplet moduli spaces in four-dimensional $N=2$ theories and their string-theoretic realizations.

Abstract

We present theories of N=2 hypermultiplets in four spacetime dimensions that are invariant under rigid or local superconformal symmetries. The target spaces of theories with rigid superconformal invariance are (4n)-dimensional {\it special} hyper-Kähler manifolds. Such manifolds can be described as cones over tri-Sasakian metrics and are locally the product of a flat four-dimensional space and a quaternionic manifold. The latter manifolds appear in the coupling of hypermultiplets to N=2 supergravity. We employ local sections of an Sp$(n)\times{\rm Sp}(1)$ bundle in the formulation of the Lagrangian and transformation rules, thus allowing for arbitrary coordinatizations of the hyper-Kähler and quaternionic manifolds.