Scalar-flat Kähler metrics on non-compact symplectic toric 4-manifolds

Abstract

Abstract. In a recent paper Donaldson [D1] explains how to use an older construction of Joyce [J] to obtain four dimensional local models for scalarflat Kähler metrics with a 2-torus symmetry. In [D2], using the same idea, he recovers and generalizes the Taub-NUT metric by including it in a new family of complete scalar-flat toric Kähler metrics on R 4 . In this paper we generalize Donaldson's method and construct complete scalar-flat toric Kähler metrics on any symplectic toric 4-manifold with "strictly unbounded" moment polygon. These include the asymptotically locally Euclidean scalar-flat Kähler metrics previously constructed by Calderbank and Singer [CS], as well as new examples of complete scalar-flat toric Kähler metrics which are asymptotic to Donaldson's generalized Taub-NUT metrics. Our construction is in symplectic action-angle coordinates and determines all these metrics via their symplectic potentials. When the first Chern class is zero we obtain a new description of known Ricci-flat Kähler metrics.

Unlocking the axion-dilaton in 5D supergravity

“…This is the case for example for a multi-Taub-NUT solution where all the centers lie on an axis, and more generally any Gibbons-Hawking metric constructed from an axially symmetric harmonic function is a toric hyperkähler metric. Although this is a known result in the mathematics literature [41], we will rederive it here in a way that is completely explicit. In such solutions, the base can be written both in the Gibbons-Hawking form, where the base is fibered along the translational direction over a flat 3D base, and the Toda form (3.83), by taking the fiber to be the rotational S 1 .…”

“…In particular, the configurations we are most interested in will be governed by solutions of an ordinary nonlinear differential equation (see (3.125) below). In addition, noncompact toric Kähler manifolds are a subject of recent interest in the mathematics community, see [41] and references therein.…”

“…This definition can be relaxed however to the non-compact setting, see e.g. [41], where in return it is demanded that the moment map be proper onto its convex image, to ensure some polytopical form for that image. Here we will be a bit more loose in our nomenclature and for us a toric manifold will simply be any symplectic 2n dimensional manifold with a Hamiltonian T n action.…”

Non-supersymmetric, multi-center solutions with topological flux

“…We note that this is essentially the same construction as in [38] for scalar-flat toric Kähler 4-manifolds (which can always be written in LeBrun form). Thus the base space is defined via the functions (4.31) and (4.32) and the 2N + 2 parameters k 3 0 , k 3 i , q 0 , q i .…”

“…We also point out that the axisymmetric LeBrun metrics we consider here are toric Kähler manifolds, and there is possibly a more elegant description of what is going on with the various types of 2-cycles using the techniques of toric geometry [38].…”