Abstract
We study the emergence of Heisenberg (Bianchi II) algebra in hyper-Kähler and quaternionic spaces. This is motivated by the rôle these spaces with this symmetry play in N=2 hypermultiplet scalar manifolds. We show how to construct related pairs of hyper-Kähler and quaternionic spaces under general symmetry assumptions, the former being a zooming-in limit of the latter at vanishing scalar curvature. We further apply this method for the two hyper-Kähler spaces with Heisenberg algebra, which is reduced to U(1)×U(1) at the quaternionic level. We also show that no quaternionic spaces exist with a strict Heisenberg symmetry – as opposed to Heisenberg⋉U(1). We finally discuss the realization of the latter by gauging appropriate Sp(2,4) generators in N=2 conformal supergravity.
Highlights
As pointed out previously, the Heisenberg algebra is uniquely realized at the quaternionic level as Heisenberg ⋉ U(1), two distinct hyper-Kähler spaces exist with Bianchi II symmetry, realized either as Heisenberg ⋉ U(1), or as strict Heisenberg [11]
An important result in the present note is the obstruction for a quaternionic space to host a strict Heisenberg algebra
As a tool for such a scanning, we introduced a method which allows to uplift hyper-Kähler geometries possessing rotational Killing vectors, to quaternionic spaces
Summary
A four-dimensional hyper-Kähler space is Ricci-flat with (anti-)self-dual Riemann tensor: Rκλμν. In the presence of an isometry generated by a Killing vector ξ = ξμ∂μ, using the Bianchi identity for the Riemann tensor, it is known that. The Killing ξ is a translational vector; it is otherwise rotational. Using the Killing vector at hand, we can adapt a coordinate τ to it, ξ = ∂τ, and write the metric as a fiber along this isometry: ds. Where we note the gauge invariance δτ = f (x), δω = −∇ f (x). When ∂τ is a translational Killing vector, one is allowed to use the Gibbons–Hawking frame [17], dV = ± ⋆γ dω , γij = δij ,. The third component wZ vanishes by a gauge choice of the coordinate τ. The function Ψ generates the metric without appearing explicitly in it
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