Special geometry in hypermultiplets

TL;DR

This paper analyzes the relationship between vector and hypermultiplet theories with N=2 supersymmetry in four dimensions, focusing on mirror symmetry, symplectic transformations, and BPS charges.

Contribution

It provides a detailed construction of hypermultiplet couplings and their transformation properties under symplectic reparametrizations, elucidating the mirror map in N=2 theories.

Findings

Constructed Sp(1)×Sp(n) one-forms for hypermultiplet couplings

Demonstrated how mirror map relates hypermultiplet and vector multiplet charges

Analyzed the behavior of BPS masses under symplectic transformations

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 K\"ahler 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 K\"ahler and hyper-K\"ahler forms, respectively. We show how these charges and the holomorphic BPS mass are related by the mirror map.