On pseudo-hyperkähler prepotentials (Q2832720)

In this paper the authors exhibit a complete local parameterisation of pseudo-hyper-Kähler manifolds in terms of local prepotentials, i.e., holomorphic functions on a certain complex submanifold of a complex Lie group.NEWLINENEWLINEMore precisely, let \((M,g)\) be a real analytic pseudo-hyper-Kähler manifold of signature \((4p, 4q)\) where \(p,q\) are non-negative integers (hence \(\dim_{\mathbb R}M=4(p+q)\)). This means that the holonomy group of \((M,g)\) is \(\mathrm{Sp}_{p,q}\) where \(\mathrm{Sp}_{p,q}\) is the group of all symplectic transformations of \(\mathbb H^{p+q}\) endowed with a symplectic scalar product of signature \((p,q)\). Then in the article it is proved that for every real analytic pseudo-hyper-Kähler manifold \((M,g)\) the restriction of the metric to a sufficiently small open subset can be associated with an unconstrained prepotential, i.e., a holomorphic function \(L\) on a certain submanifold of the complex Lie group \(\left(\mathrm{Sp}_1(\mathbb C)\times\mathrm{Sp}_{p+q}(\mathbb C)\right)\ltimes\mathbb C^{4(p+q)}\); conversely, every isomorphism class of such functions determines a local pseudo-hyper-Kähler metric \(g\) up to isometry (see Theorems 4.4, 4.5 and 4.6). This correspondence is explicit and can be used to construct local metrics of this sort (see a five-step recipe in Section VIII).NEWLINENEWLINESince special holonomy manifolds play a crucial role in supersymmetric field theories this construction is relevant in the local classification of such theories and in the explicit construction of their supersymmetric Lagrangians.