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Eguchi–Hanson space

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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 can be described using complex coordinates with a metric

where is a scale setting constant and .

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 with Eguchi–Hanson spaces.[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.[2]

References

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  1. ^ Eguchi, Tohru; Hanson, Andrew J. (1979). "Selfdual solutions to Euclidean gravity" (PDF). Annals of Physics. 120 (1): 82–105. Bibcode:1979AnPhy.120...82E. doi:10.1016/0003-4916(79)90282-3. OSTI 1447072.
  2. ^ a b Calabi, Eugenio (1979). "Métriques kählériennes et fibrés holomorphes". Annales Scientifiques de l'École Normale Supérieure. Quatrième Série, 12 (2): 269–294. doi:10.24033/asens.1367.
  3. ^ Polchinski, J. (1998). "17". String Theory Volume II: Superstring Theory and Beyond. Cambridge University Press. p. 309-310. ISBN 978-1551439761.