A New Infinite Class of Sasaki-Einstein Manifolds

TL;DR

This work constructs, from any positive-curvature Kähler–Einstein base $B_{2n}$, a countable family of compact Sasaki–Einstein manifolds $X_{2n+3}$ of dimension $2n+3$, each admitting a Killing spinor and with a Calabi–Yau cone having holonomy in $SU(n+2)$. The approach combines a local $(2n+2)$-dimensional Kähler–Einstein base $Y_{2n+2}$ (an $S^2$-bundle over $B_{2n}$) with a U(1) Sasaki direction, and imposes integrality conditions on a connection to yield a global, complete Sasaki–Einstein metric for countably many parameter values $\kappa$ (with $-1<\kappa<0$). The resulting $X_{2n+3}$ are simply-connected and spin, and their Sasaki structure is regular or irregular depending on whether $f(\rho_2)$ is rational; when specialized to $n=1$ or $n=2$ they reproduce known AdS solutions (e.g., $AdS_5\times X_5$ and $AdS_4\times X_7$) and encompass new inhomogeneous cohomogeneity-one examples. The construction thus broadens explicit Sasaki–Einstein geometries, connecting to established homogeneous spaces like $T^{1,1}$, $Q^{1,1,1}$ and $M^{3,2}$ and providing new holographic duals for superconformal field theories.

Abstract

We show that for every positive curvature Kahler-Einstein manifold in dimension 2n there is a countably infinite class of associated Sasaki-Einstein manifolds X_{2n+3} in dimension 2n+3. When n=1 we recover a recently discovered family of supersymmetric AdS_5 x X_5 solutions of type IIB string theory, while when n=2 we obtain new supersymmetric AdS_4 x X_7 solutions of D=11 supergravity. Both are expected to provide new supergravity duals of superconformal field theories.