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.[1] [2]
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
wi\inCd/2
gi=(1+
| \rhod | |
| rd |
)2/d[\deltai\bar-
| \rhodwi\barw\bar | |
| r2(\rhod+rd) |
],
where
\rho
r2=
| 2 | |
| |w| | |
| Cd/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
| 6/Z | |
| T | |
| 3 |
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.