Special geometry in hypermultiplets

TL;DR

The paper investigates how four-dimensional $N=2$ vector and hypermultiplet theories are related by the classical mirror map, focusing on how symplectic reparametrizations of the vector-multiplet special Kähler geometry induce corresponding hypermultiplet transformations. By reducing to three dimensions and employing the HKLR construction, it derives explicit $Sp(1)\times Sp(n)$ one-forms that encode hypermultiplet couplings and demonstrates covariance under symplectic reparametrizations, linking the central charges and holomorphic BPS mass across the mirror map. It provides a coordinate-independent characterization of the resulting hyper-Kähler manifolds, elucidates the role of isometries and dualities, and shows how vector-multiplet central charges map to hyper-Kähler charges through the associated two-forms, thereby clarifying the geometric content of mirror symmetry in 4D $N=2$ theories. These results contribute to understanding how special geometry constraints shape hypermultiplet moduli and offer a framework for constraining perturbative string corrections via mirror-like relations between vector and hypermultiplet sectors.

Abstract

We give a detailed analysis of pairs of vector and hypermultiplet theories with N=2 supersymmetry in four spacetime dimensions that are related by the (classical) mirror map. The symplectic reparametrizations of the special Kähler space associated with the vector multiplets induce corresponding transformations on the hypermultiplets. We construct the Sp(1)$\times$Sp($n$) one-forms in terms of which the hypermultiplet couplings are encoded and exhibit their behaviour under symplectic reparametrizations. Both vector and hypermultiplet theories allow vectorial central charges in the supersymmetry algebra associated with integrals over the Kähler and hyper-Kähler forms, respectively. We show how these charges and the holomorphic BPS mass are related by the mirror map.