Sasaki-Einstein Manifolds and Volume Minimisation

TL;DR

The paper realises a general, non-toric geometric framework in which the Einstein--Hilbert action on Sasakian metrics on the link $L$ of a Calabi--Yau cone $X$ reduces to a volume functional depending solely on the Reeb vector field $\xi$, enabling a convex variation with a unique critical Reeb direction when a Sasaki--Einstein metric exists. It connects this geometric extremisation to Duistermaat--Heckman localisation and to an equivariant index limit that counts holomorphic functions, thereby expressing vol[$L$] as a fixed-point topological sum and proving the algebraicity of the normalized volume. In the toric case, the results specialise to explicit sums over fixed points and lattice data, reproducing the known $Z$-minimisation/$a$-maximisation correspondence and yielding concrete examples such as the conifold and del Pezzo exemplars, as well as the $Y^{p,q}$ family. The work also clarifies the role of the Futaki invariant as the obstruction to KE metrics on transverse bases and situates the Reeb vector as a central element of the isometry/algebraic structure, with implications for AdS/CFT and the structure of dual SCFTs. Overall, the paper provides a unified, non-toric generalisation of volume minimisation in Sasakian geometry with broad geometric and physical implications.

Abstract

We study a variational problem whose critical point determines the Reeb vector field for a Sasaki-Einstein manifold. This extends our previous work on Sasakian geometry by lifting the condition that the manifolds are toric. We show that the Einstein-Hilbert action, restricted to a space of Sasakian metrics on a link L in a Calabi-Yau cone X, is the volume functional, which in fact is a function on the space of Reeb vector fields. We relate this function both to the Duistermaat-Heckman formula and also to a limit of a certain equivariant index on X that counts holomorphic functions. Both formulae may be evaluated by localisation. This leads to a general formula for the volume function in terms of topological fixed point data. As a result we prove that the volume of a Sasaki-Einstein manifold, relative to that of the round sphere, is always an algebraic number. In complex dimension n=3 these results provide, via AdS/CFT, the geometric counterpart of a-maximisation in four dimensional superconformal field theories. We also show that our variational problem dynamically sets to zero the Futaki invariant of the transverse space, the latter being an obstruction to the existence of a Kahler-Einstein metric.

Figures (9)

• Figure 1: Toric diagram for the conifold.
• Figure 2: A small resolution of the conifold.
• Figure 3: The other small resolution of the conifold.
• Figure 4: Toric diagram for the complex cone over $dP_1$.
• Figure 5: Canonical bundle over $dP_1$.