Some Examples of Toric Sasaki—Einstein Manifolds

Abstract

Abstract. A series of examples of toric Sasaki-Einstein 5-manifolds is constructed which first appeared in the author's Ph.D. thesis [40]. These are submanifolds of the toric 3-Sasakian 7-manifolds of C. Boyer and K. Galicki. And there is a unique toric quasi-regular Sasaki-Einstein 5-manifold associated to every simply connected toric 3-Sasakian 7-manifold. Using 3-Sasakian reduction as in [8,7] an infinite series of examples is constructed of each odd second Betti number. They are all diffeomorphic to #kM∞, where M∞ ∼ = S 2 × S 3 , for k odd. We then make use of the same framework to construct positive Ricci curvature toric Sasakian metrics on the manifolds X∞#kM∞ appearing in the classification of simply connected smooth 5-manifolds due to Smale and Barden. These manifolds are not spin, thus do not admit Sasaki-Einstein metrics. They are already known to admit toric Sasakian metrics (cf. [9]) which are not of positive Ricci curvature. We then make use of the join construction of C. Boyer and K. Galicki first appearing in [6], see also [9], to construct infinitely many toric Sasaki-Einstein manifolds with arbitrarily high second Betti number of every dimension 2m + 1 ≥ 5. This is in stark contrast to the analogous case of Fano manifolds in even dimensions. IntroductionA new series of quasi-regular Sasaki-Einstein 5-manifolds is constructed. These examples first appeared in the author's Ph.D. thesis [40]. They are toric, and arise as submanifolds of toric 3-Sasakian 7-manifolds. Applying 3-Sasakian reduction to torus actions on spheres C. Boyer, K. Galicki, et al [8] produced infinitely many toric 3-Sasakian 7-manifolds. More precisely, one has a 3-Sasakian 7-manifold S Ω for each integral weight matrix Ω satisfying some conditions to ensure smoothness. This produces infinitely many examples of each b 2 (S Ω ) ≥ 1. A result of D. Calderbank and M. Singer [12] shows that, up to finite coverings, this produces all examples of toric 3-Sasakian 7-manifolds. Associated to each S Ω is its twistor space Z, a complex contact Fano 3-fold with orbifold singularities. The action of T 2 complexifies to T 2 C = C * × C * acting on Z. Furthermore, if L is the line bundle of the complex contact structure, the action defines a pencilwhere t C is the Lie algebra of T 2 C . We determine the structure of the divisors in E. The generic X ∈ E is a toric variety with orbifold structure whose orbifold anti-canonical bundle K −1 X is positive. The total space M of the associated S 1 orbifold bundle to K X has a natural Sasaki-Einstein structure. Associated to any [3]) that M is diffeomorphic to #k(S 2 × S 3 ) where b 2 (M ) = k. We have the following: Theorem 1.1. Associated to every simply connected toric 3-Sasakian 7-manifold S is a toric quasi-regular Sasaki-. This gives an invertible correspondence. That is, given either X or M in diagram 1.1 one can recover the other spaces with their respective geometries.This gives an infinite series of quasi-regular Sasaki-Einstein structures on #m(S 2 × S 3 ) for every odd m ≥ 3.In section...

Brane tilings and their applications

“…For the quasi-regular case, Boyer-Galicki has shown many existence theorems, using existence of metrics of Fano orbifolds, but the physical significance of their metrics is not clear at the moment. See the papers [161][162][163][164] and the review [165].…”

Examples of asymptotically conical Ricci-flat Kähler manifolds

“…[18,58,60]). Note that if the Sasaki link S ⊂ X is simply connected, then in the toric case H 2 (S, Z) = Z r and Smale's classification of 5-manifolds implies that S diff ∼ = #k(S 2 × S 3 ).…”

“…Thus using Theorem 1. 3 and the examples of [18,58,60] we produce Ricci-flat asymptotically conical Kähler manifolds Y asymptotic to cones over #k(S 2 × S 3 ). And in fact, this produces infinitely many examples for each k ≥ 1.…”

“…Thus every Q-Gorenstein toric Kähler cone admits a Ricciflat Kähler cone metric. See [18] and also [58,60] for the construction of infinite series of examples.…”

“…These are all diffeomorphic to S 2 × S 3 and include the first examples of irregular Sasaki-Einstein manifolds. In [18,58,60] infinite series of toric Sasaki-Einstein manifolds are constructed. Together these give infinitely many examples for each b 2 (S) ≥ 1.…”