Sasaki-Einstein manifolds and volume minimisation
The authors study the following objects: Given a \((2n-1)\)-dimensional Riemannian manifold \((L, g_{L})\), we say that it is Sasakian if the metric cone \(X=\mathbb{R}^{+}\times L\) is a Kähler manifold, where the cone metric \(g_{X}\) is defined to be \(g_{X}=dr^{2}+ r^{2}g_{L}\), \(r>0\) is the coordinate on \(\mathbb{R}^{+}\). Hence \(X\) admits a parallel complex structure \(J\), and then, the Reeb vector vector field \(\xi\) can be defined as \(\xi = J(r\frac{\partial}{\partial r})\), where \( r\frac{\partial}{\partial r}\) is the vector field on \(X\) which is the infinitesimal generator of the one-parameter family of dilation diffeomorphisms \((r, p) \longmapsto (e^{t}r, p)\). Restricted to a vector field on the link \(L\) of the cone \(X\), the orbits of \(\xi\) define a foliation of \(L\). So, if the Kähler metric \(g_{X}\) on \(X\) is actually Ricci-flat, then the Riemannian manifold \((L, g_{L})\) is called Sasakian-Einstein. The authors show that the Einstein-Hilbert action \(\mathcal{S}(g_{L})=\int_{L}[s(g_{L}) + 2(n-1)(3-2n)]d\mu\), restricted to the space of Sasakian metrics on \(L\), where \(s(g_{L})\) is the scalar curvature of \(g_{L}\), and \(d\mu\) is the associated Riemannian measure, is just the volume functional for the link: \(S(g_{L})= 4(n-1) vol(L)\). In the remainder of the paper, by reducing their problem to studying the volume of the link, they prove that the functional \(S (g_{L}) \) depends only on the Reeb vector field. Using the Duistermaat-Heckman formula, the authors derive a very explicit formula for the volume of \(L\) at a critical Reeb vector field \(\xi\) and they prove later that the volume of a Sasaki-Einstein manifold, relative to that of the round sphere, is always an algebraic number. Since physical study of the so-called AdS/CFT correspondence has placed the importance on the understanding of the geometry of Sasaki-Einstein manifolds, the authors also show that their results provide, in complex dimension \(n=3\) and via AdS/CFT, the geometric counterpart of \(a\)-maximisation in four dimensional surperconformal field theories. They give several examples which are toric Sasaki-Einstein in order to support their results.
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