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Just as the title says:

  1. What is the easiest definition of a Hyperkahler Manifold?

  2. Could you give some examples of Hyperkahler manifolds, and manifolds which fail to be hyperkahler?

  3. Why are such manifolds considered to be interesting in physics, and how do they arise in the study of supersymmetric gauge theories?

  4. Apart from SUSY, are there any other branches of physics in which Hyperkahler manifolds appear?

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  • $\begingroup$ Related post for Kahler manifolds: physics.stackexchange.com/q/4972/2451 $\endgroup$ – Qmechanic Oct 26 '13 at 20:04
  • $\begingroup$ Thanks for the interesting link, however there Calabi-Yau's are mainly discussed, while here I ask for Hyperkahler, which I think is quite different... $\endgroup$ – Federico Carta Oct 26 '13 at 20:12
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Partial answer :

2) Examples of hyperkahler manifolds (Ref1, Ref2, Ref3)

  • Even-dimensional complex vector space
  • Even-dimensional complex torus ($T^4$)
  • $K3$
  • moduli spaces (instantons, monopoles)
  • quiver varieties
  • Resolution of singularities

3) Supersymmetry and hyperkahler manifolds (Ref4)

  • N = 4 supersymmetric nonlinear $\sigma$-models
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