Seminario Geometría

Complete quaternionic Kähler manifolds of cohomogeneity at least one

Ponente:  Arpan Saha (ICMAT)
Fecha:  miércoles 14 de junio de 2023 - 11:30
Lugar:  Aula Gris 2, ICMAT


Quaternionic Kähler manifolds are notable for their appearance as one of the items in Berger's list of manifolds with special Riemannian holonomy. Within this list, they have the distinction of being the only item to be generically Einstein but not Ricci-flat. This distinction is reflected in the fact that although we have locally inhomogeneous compact examples for all the other items in Berger's list, there are no known examples even of locally inhomogeneous complete quaternionic Kähler manifolds of finite volume. As a matter of fact, in the case of positive scalar curvature, it has been conjectured (and proven for sufficiently low dimension) that no nonsymmetric complete examples exist at all. However, the moduli space of quaternionic Kähler manifolds of negative scalar curvature is known to be infinite-dimensional, so there is hope of finding locally inhomogeneous complete examples of finite volume among them. Indeed, arguments from physics suggest that certain geometric constructions arising in supergravity ought to provide examples of such quaternionic Kähler metrics once the full set of quantum corrections coming from string theory are taken into account.

In this talk, I address a more modest goal within this ongoing programme, namely that of constructing explicit examples of locally inhomogeneous complete quaternionic Kähler metrics, without any condition on the volume. We shall see using Macia and Swann's realisation of the hyperkähler/ quaternionic Kähler correspondence studied by Haydys, Hitchin, and many others, in terms of Swann's twist construction, that the perturbative quantum corrections from string theory suffice for this purpose. This is based on the preprint 2210.10097 with Vicente Cortés and Alejandro Gil García, expected to appear in Communications in Mathematical Physics.