Special Geometry of Euclidean Supersymmetry I: Vector Multiplets

Abstract

We construct two new versions of the c-map which allow us to obtain the target manifolds of hypermultiplets in Euclidean theories with rigid N = 2 supersymmetry. While the Minkowskian para-c-map is obtained by dimensional reduction of the Minkowskian vector multiplet lagrangian over time, the Euclidean para-c-map corresponds to the dimensional reduction of the Euclidean vector multiplet lagrangian. In both cases the resulting hypermultiplet target spaces are para-hyper-Kähler manifolds. We review and prove the relevant results of para-complex and para-hypercomplex geometry. In particular, we give a second, purely geometrical construction of both c-maps, by proving that the cotangent bundle N = T * M of any affine special (para-)Kähler manifold M is parahyper-Kähler.

A stringy test of flux-induced isometry gauging

“…This subtlety is present in many instanton computations, also in four dimensions. For an analysis of rigid N = 2 SUSY in Euclidean signature, see [38].…”

Membrane Instantons and de Sitter Vacua

“…Additional references on hypermultiplet moduli spaces and instantons are [11,12,13,14,15,16,17]. Furthermore a program towards formulating an instanton calculus based on N = 2 supersymmetric actions with Euclidean signature was started in [18,19]. 2 Membrane instantons were also considered in [20,21], but our analysis below differs since we do not assume the existence of a rotational symmetry between the RR scalars in the UHM scalar metric.…”

On small deformations of paracomplex manifolds

“…We recall some basic definitions of paracomplex geometry (see e.g [6], [5], [1]). Let V be a 2n-dimensional real vector space.…”