The hypermultiplet with Heisenberg isometry in N = 2 global and local supersymmetry

Abstract

The string coupling of N = 2 supersymmetric compactifications of type II string theory on a Calabi-Yau manifold belongs to the so-called universal dilaton hypermultiplet, that has four real scalars living on a quaternion-Kähler manifold. Requiring Heisenberg symmetry, which is a maximal subgroup of perturbative isometries, reduces the possible manifolds to a one-parameter family that describes the tree-level effective action deformed by the only possible perturbative correction arising at one-loop level. A similar argument can be made at the level of global supersymmetry where the scalar manifold is hyper-Kähler. In this work, the connection between global and local supersymmetry is explicitly constructed, providing a non-trivial gravity decoupled limit of type II strings already in perturbation theory.

Rigid limit for hypermultiplets and five-dimensional gauge theories

“…(4.51) 13 In fact, the vectors va, a = 2, 3, 4, may be chosen in a simpler form. Our choice instead allows to have simpler relations to the gauge theory variables, which are found below in (4.55).…”

Isometries, gaugings and N $$ \mathcal{N} $$ = 2 supergravity decoupling

“…It implies to simultaneously "zooming-in" in order to recover non-trivial hyper-Kähler geometries, as performed in ref. [3] for quaternion-Kähler spaces with Heisenberg isometry (this symmetry was discussed in ref. [4]).…”

“…This family of quaternion-Kähler spaces contains the sub-family of the Heisenberg ⋉U(1) spaces resulting from a Z gauging [3].…”

“…Firstly, only two hyper-Kähler spaces exist with this isometry group [34], and one quaternion-Kähler [16]. Secondly, the latter captures the type-II string one-loop perturbative corrections to the hypermultiplet scalar manifold [4], and can be derived through the quaternionic quotient method by gauging the Z isometry inside the Sp(2, 4) of the N = 2 hypermultiplet manifold [3]. The quaternion-Kähler space under consideration has actually extended Heisenberg ⋉ U(1) isometry and its metric reads:…”

“…It was noticed in [3] that the rigid limit of the kinetic term (3.1) with (5.1) is σ-dependent. For σ 0 two trivial, flat-space rigid limits occur around ρ 2 ∼ 0 and ρ 2 − 2σ ∼ 0.…”