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 Cd/2/Zd/2 of (real-)dimension $$d$$ can be described using complex coordinates $$w_i \in \mathbb C^{d/2}$$ with a metric



g_{i \bar j} = \bigg(1+\frac{\rho^d}{r^{d}}\bigg)^{2/d}\bigg[\delta_{i\bar j}-\frac{\rho^d w_i \bar w_{\bar j}}{r^2(\rho^d+r^{d})}\bigg], $$

where $$\rho$$ is a scale setting constant and $$r^2 = |w|^2_{\mathbb C^{d/2}}$$.

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 $$T^6/\mathbb Z_3$$ 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.