Eguchi–Hanson space

In mathematics and theoretical physics, the Eguchi–Hanson space is a non-compact, self-dual, asymptotically locally Euclidean (ALE) metric on the cotangent bundle of the 2-sphere T^*S². The holonomy group of this 4-real-dimensional manifold is SU(2). The metric is generally attributed to the physicists Tohru Eguchi and Andrew J. Hanson; it was discovered independently by the mathematician Eugenio Calabi around the same time in 1979. The Eguchi-Hanson metric has Ricci tensor equal to zero, making it a solution to the vacuum Einstein equations of general relativity, albeit with Riemannian rather than Lorentzian metric signature. It may be regarded as a resolution of the A₁ singularity according to the ADE classification which is the singularity at the fixed point of the C²/Z₂ orbifold where the Z₂ group inverts the signs of both complex coordinates in C². The even dimensional space C^d/2/Z_d/2 of (real-)dimension $d$ can be described using complex coordinates $w_i\in {\mathbb{ℂ}}^{d/2}$ with a metric

$g_{i\overline{j}}=(1+\frac{{\rho }^d}{r^d}{)}^{2/d}[{\delta }_{i\overline{j}}-\frac{{\rho }^d{w}_i{\overline{w}}_{\overline{j}}}{r^2({\rho }^d+r^d)}],$

where $\rho$ is a scale setting constant and $r^2=|w{|}_{{\mathbb{ℂ}}^{d/2}}^2$. Aside from its inherent importance in pure geometry, the space is important in string theory. Certain types of K3 surfaces can be approximated as a combination of several Eguchi–Hanson metrics since both have the same holonomy group. Similarly, the space can also be used to construct Calabi–Yau manifolds by replacing the orbifold singularities of $T^6/{\mathbb{\mathbb{Z}}}_3$ with Eguchi–Hanson spaces. The Eguchi–Hanson metric is the prototypical example of a gravitational instanton; detailed expressions for the metric are given in that article. It is then an example of a hyperkähler manifold.