It will be convenient to write the gravitational instanton solutions below using left-invariant 1-forms on the three-sphere S3 (viewed as the group Sp(1) or SU(2)). These can be defined in terms of Euler angles by
Eguchi–Hanson metric
The Eguchi–Hanson space is defined by a metric the cotangent bundle of the 2-sphere . This metric is
where . This metric is smooth everywhere if it has no conical singularity at , . For this happens if has a period of , which gives a flat metric on R4; However, for this happens if has a period of .
Asymptotically (i.e., in the limit ) the metric looks like
which naively seems as the flat metric on R4. However, for , has only half the usual periodicity, as we have seen. Thus the metric is asymptotically R4 with the identification , which is a Z2 subgroup of SO(4), the rotation group of R4. Therefore, the metric is said to be asymptotically
R4/Z2.
There is a transformation to another coordinate system, in which the metric looks like
where
- (For a = 0, , and the new coordinates are defined as follows: one first defines and then parametrizes , and by the R3 coordinates , i.e. ).
In the new coordinates, has the usual periodicity
One may replace V by
For some n points , i = 1, 2..., n.
This gives a multi-center Eguchi–Hanson gravitational instanton, which is again smooth everywhere if the angular coordinates have the usual periodicities (to avoid conical singularities). The asymptotic limit () is equivalent to taking all to zero, and by changing coordinates back to r, and , and redefining , we get the asymptotic metric
This is R4/Zn = C2/Zn, because it is R4 with the angular coordinate replaced by , which has the wrong periodicity ( instead of ). In other words, it is R4 identified under , or, equivalently, C2 identified under zi ~ zi for i = 1, 2.
To conclude, the multi-center Eguchi–Hanson geometry is a Kähler Ricci flat geometry which is asymptotically C2/Zn. According to Yau's theorem this is the only geometry satisfying these properties. Therefore, this is also the geometry of a C2/Zn orbifold in string theory after its conical singularity has been smoothed away by its "blow up" (i.e., deformation).[3]