Construction of Einstein-Sasaki metrics in $D\ge 7$

We construct explicit Einstein-Kähler metrics in all even dimensions $D=2n+4\ge 6$, in terms of a $2n$-dimensional Einstein-Kähler base metric. These are cohomogeneity 2 metrics which have the new feature of including a NUT-type parameter, or gravomagnetic charge, in addition to.." in addition to mass and rotation parameters. Using a canonical construction, these metrics all yield Einstein-Sasaki metrics in dimensions $D=2n+5\ge 7$. As is commonly the case in this type of construction, for suitable choices of the free parameters the Einstein-Sasaki metrics can extend smoothly onto complete and nonsingular manifolds, even though the underlying Einstein-Kähler metric has conical singularities. We discuss some explicit examples in the case of seven-dimensional Einstein-Sasaki spaces. These new spaces can provide supersymmetric backgrounds in M theory, which play a rôle in the ${\mathrm{AdS}}_4/{\mathrm{CFT}}_3$ correspondence.

Figure 1

The functions $X(x)$ (curve from top left to bottom right) and $Y(y)$ (curve from bottom left to top right), showing the nonoverlapping closed intervals in which they are greater than or equal to zero. The horizontal axis is $x$ and $y$, respectively, for the two functions. [$X(x)$ is heading to $-\infty$ at large positive $x$, while $Y(y)$ is heading to $+\infty$ at large positive $y$.]

Figure 2

The functions $X(x)$ and $Y(y)$, showing the nonoverlapping closed intervals in which they are less than or equal to zero. The horizontal axis is respectively $x$ and $y$ for the two functions. [$X(x)$ is heading to $-\infty$ at large positive $x$ and to $+\infty$ at large negative $x$, while $Y(y)$ is heading to $+\infty$ at large positive $y$ and to $-\infty$ at large negative $y$.]