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hyperkähler manifold

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Contents

Definition

For nn \in \mathbb{N} a natural number, a 4n4n-dimensional Riemannian manifold is a hyperkähler manifold if its holonomy group is (a subgroup of) the quaternionic unitary group Sp(n). Regarded as a subgroup of the central product group Sp(n).Sp(1) this means that hyper-Kähler manifolds are special cases of quaternion-Kähler manifolds, though the latter are often taken to be only those Riemannian manifolds with full Sp(n).Sp(1)-holonomy.

Equivalently, a hyperkähler manifold is a Riemannian manifold (M,g)(M,g) with three complex structures I,J,KI, J, K which are Kähler with respect to the metric gg and satisfy the quaternionic identities

I 2=J 2=K 2=IJK=1.I^2=J^2=K^2=I J K=-1.

Properties

As part of the Berger classification

classification of special holonomy manifolds by Berger's theorem:

\,G-structure\,\,special holonomy\,\,dimension\,\,preserved differential form\,
\,\mathbb{C}\,\,Kähler manifold\,\,U(n)\,2n\,2n\,\,Kähler forms ω 2\omega_2\,
\,Calabi-Yau manifold\,\,SU(n)\,2n\,2n\,
\,\mathbb{H}\,\,quaternionic Kähler manifold\,\,Sp(n).Sp(1)\,4n\,4n\,ω 4=ω 1ω 1+ω 2ω 2+ω 3ω 3\,\omega_4 = \omega_1\wedge \omega_1+ \omega_2\wedge \omega_2 + \omega_3\wedge \omega_3\,
\,hyper-Kähler manifold\,\,Sp(n)\,4n\,4n\,ω=aω 2 (1)+bω 2 (2)+cω 2 (3)\,\omega = a \omega^{(1)}_2+ b \omega^{(2)}_2 + c \omega^{(3)}_2\, (a 2+b 2+c 2=1a^2 + b^2 + c^2 = 1)
𝕆\,\mathbb{O}\,\,Spin(7) manifold\,\,Spin(7)\,\,8\,\,Cayley form\,
\,G2 manifold\,\,G2\,7\,7\,\,associative 3-form\,

As special \mathbb{H}-Riemannian manifolds

\;normed division algebra\;𝔸\;\mathbb{A}\;\;Riemannian 𝔸\mathbb{A}-manifolds\;\;special Riemannian 𝔸\mathbb{A}-manifolds\;
\;real numbers\;\;\mathbb{R}\;\;Riemannian manifold\;\;oriented Riemannian manifold\;
\;complex numbers\;\;\mathbb{C}\;\;Kähler manifold\;\;Calabi-Yau manifold\;
\;quaternions\;\;\mathbb{H}\;\;quaternion-Kähler manifold\;\;hyperkähler manifold\;
\;octonions\;𝕆\;\mathbb{O}\;\;Spin(7)-manifold\;\;G2-manifold\;

(Leung 02)

Rozansky-Witten weight systems

Every hyperkähler manifold induces a Rozansky-Witten weight system with coefficients in certain Dolbeault cohomology-groups. For compact hyperkähler manifolds there are induced Rozansky-Witten weight system with values in the ground field, hence actual weight systems.

Examples

Compact hyperkähler manifolds

The only known examples of compact hyperkähler manifolds are Hilbert schemes of points X [n+1]X^{[n+1]} (for nn \in \mathbb{N}) for XX either

  1. a K3-surface

  2. a 4-torus (in which case the compact hyperkähler manifolds is really the fiber of (𝕋 4) [n]𝕋 4(\mathbb{T}^4)^{[n]} \to \mathbb{T}^4)

(Beauville 83) and two exceptional examples (O’Grady 99, O’Grady 03 ), see Sawon 04, Sec. 5.3.

Coulomb- and Higgs-branches of D=3D=3 𝒩=4\mathcal{N} =4 SYM

Both the Coulomb branch and the Higgs branch of D=3 N=4 super Yang-Mills theories are hyperkähler manifolds (Seiberg-Witten 96, see e.g. dBHOO 96). In special cases they are compact hyperkähler manifolds (Intriligator 99).

References

General

See also

Compact hyperkähler manifolds

Rozansky-Wutten invariants

With an eye towards Rozansky-Witten theory (ground field-valued Rozansky-Witten weight systems):

Examples

The example of Coulomb branches of D=3 N=4 super Yang-Mills theory originates with

Last revised on December 31, 2019 at 11:01:25. See the history of this page for a list of all contributions to it.