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# 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), as it is for a Calabi-Yau K3 surface. While the metric is generally attributed to the physicists Eguchi and Hanson,[1] it was actually discovered independently by the mathematician Eugenio Calabi[2] around the same time.

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.

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.

The Eguchi–Hanson metric is the prototypical example of a gravitational instanton.

References

Eguchi, Tohru; Hanson, Andrew J. (1979). "Selfdual solutions to Euclidean gravity" (PDF). Annals of Physics 120: 82–105. Bibcode:1979AnPhy.120...82E. doi:10.1016/0003-4916(79)90282-3.

Calabi, Eugenio (1979). "Métriques kählériennes et fibrés holomorphes". Annales Scientifiques de l'École Normale Supérieure. Quatrième Série, 12 (2): 269–294.

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