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Eguchi–Hanson space

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Concept in mathematics and theoretical physics

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 TS. 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 C/Z2 orbifold where the Z2 group inverts the signs of both complex coordinates in C. The even dimensional space C/Zd/2 of (real-)dimension d {\displaystyle d} can be described using complex coordinates w i ∈ C d / 2 {\displaystyle w_{i}\in \mathbb {C} ^{d/2}} with a metric

g i j ¯ = ( 1 + ρ d r d ) 2 / d [ δ i j ¯ − ρ d w i w ¯ j ¯ r 2 ( ρ d + r d ) ] , {\displaystyle g_{i{\bar {j}}}={\bigg (}1+{\frac {\rho ^{d}}{r^{d}}}{\bigg )}^{2/d}{\bigg },}

where ρ {\displaystyle \rho } is a scale setting constant and r 2 = | w | C d / 2 2 {\displaystyle r^{2}=|w|_{\mathbb {C} ^{d/2}}^{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 / Z 3 {\displaystyle 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.

References

  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. ^ 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.


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