Bianchi type A hyper-symplectic and hyper-Kähler metrics in 4D

Abstract

We present a simple explicit construction of hyper-Kähler and hyper-symplectic (also known as neutral hyper-Kähler or hyper-parakähler) metrics in 4D using the Bianchi type groups of class A. The construction underlies a correspondence between hyper-Kähler and hyper-symplectic structures in dimension four.In this section we recover some of the known hyper-Kähler metrics in dimension four. To this end, we lift the special structure on the non-Euclidean Bianchi type groups of class A to a hyper-Kähler metric on its product with (an interval in) the real line.

Supergravity solutions of two M2 branes

“…We explicitly solve the Killing spinor equation and show all the supergravity configurations preserve 1/8 of the supersymmetry in agreement with the system of three D branes after dimensional reduction. It would be interesting to construct new solutions for the system of two M2 branes on other types of Bianchi space [28] as the overall transverse space.…”

Induced hypersymplectic and hyperkähler structures on the dual of a Lie algebroid

“…which corresponds to the Heisenberg metric obtained in [2]. By Theorem 3.3, the quadruple (g, N 1 , N 2 , N 3 ) is an hyperkähler structure on the Lie algebroid T (H 3 × I).…”

“…The next example shows that the Heisenberg metric constructed in [2] can be obtained applying our construction, if we treat the problem in the Lie algebroid setting, that is, if we construct the transition tensors and get the metric from formula (6). We should stress that all the cases considered in [2] can be tackled in the same way as the one we will explain next. We define a hypersymplectic triple (ω 1 , ω 2 , ω 3 ) on the Lie algebroid T (H 3 × I), where I ⊆ R + , by setting…”

“…In [2] the authors present several (para-)hypersymplectic structures in 4-dimensional manifolds arising from the 3-dimensional Bianchi type A groups. In order to obtain, explicitly, the metrics of these structures, the authors solve a system of evolution equations and consider some particular solutions.…”

“…In order to obtain, explicitly, the metrics of these structures, the authors solve a system of evolution equations and consider some particular solutions. One of the cases considered in [2] concerns the Heisenberg group H 3 . We show that if we treat that case in the Lie algebroid setting, taking T (H 3 × I) and applying our construction, we get a hypersymplectic structure and its corresponding hyperkähler structure recovering, this way, the Heisenberg metric of [2].…”