ar X iv : h ep - t h / 05 05 02 7 v 1 3 M ay 2 00 5 Toric Sasaki – Einstein metrics on S 2 × S 3

We show that by taking a certain scaling limit of a Euclideanised form of the Plebanski–Demianski metrics one obtains a family of local toric Kähler–Einstein metrics. These can be used to construct local Sasaki–Einstein metrics in five dimensions, which are generalisations of the Y p,q metrics. In fact, we find that the local metrics are diffeomorphic to those recently found by Cvetic, Lu, Page and Pope. We argue that the resulting family of smooth Sasaki–Einstein manifolds all have topology S × S. We conclude by setting up the equations describing the warped version of the Calabi–Yau cones, supporting (2, 1) three–form flux. Recently Sasaki–Einstein geometry has been the focus of much attention. The interest in this subject has arisen due to the discovery in [1,2] of an infinite family Y p,q of explicit Sasaki–Einstein metrics on S×S, and the subsequent identification of the corresponding family of AdS/CFT dual quiver gauge theories in [3,4]. The construction of [2] was immediately generalised to higher dimension in reference [5] and a further generalisation subsequently appeared in [6,7]. However, dimension five is the most interesting dimension physically and the purpose of this work was to investigate if there exist other local Kähler–Einstein metrics in dimension four from which one can construct complete Sasaki–Einstein metrics in one dimension higher. Email addresses: [email protected] (Dario Martelli), [email protected] (James Sparks). Preprint submitted to Elsevier Science 2 March 2008 As we show, one can obtain a family of local toric Kähler–Einstein metrics by taking a certain scaling limit of a Euclideanised form of the Plebanski– Demianski metrics [8]. Here toric refers to the fact that the metric has two commuting holomorphic Killing vector fields. In fact the resulting metrics were found independently by Apostolov and collaborators in [9] using rather different methods. In the latter reference it is shown that this family of metrics constitute the most general local Kähler–Einstein metric which is orthotoric, a term that we define later. We also show that these Kähler–Einstein metrics are precisely those used in the recent construction of Sasaki–Einstein metrics generalising the Y p,q metrics [10]. Higher–dimensional orthotoric Kähler–Einstein metrics are given in explicit form in reference [11]. Our starting point will be the following family of local Einstein metrics in dimension four ds 4 = (p − q) [