# nLab hyperkähler manifold

Contents

complex geometry

### Examples

#### Riemannian geometry

Riemannian geometry

# Contents

## Definition

For $n \in \mathbb{N}$ a natural number, a $4n$-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 $(M,g)$ with three complex structures $I, J, K$ which are Kähler with respect to the metric $g$ and satisfy the quaternionic identities

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

## 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\,$$\,$Kähler forms $\omega_2\,$
$\,$Calabi-Yau manifold$\,$$\,$SU(n)$\,$$\,2n\,$
$\,\mathbb{H}\,$$\,$quaternionic Kähler manifold$\,$$\,$Sp(n).Sp(1)$\,$$\,4n\,$$\,\omega_4 = \omega_1\wedge \omega_1+ \omega_2\wedge \omega_2 + \omega_3\wedge \omega_3\,$
$\,$hyper-Kähler manifold$\,$$\,$Sp(n)$\,$$\,4n\,$$\,\omega = a \omega^{(1)}_2+ b \omega^{(2)}_2 + c \omega^{(3)}_2\,$ ($a^2 + b^2 + c^2 = 1$)
$\,\mathbb{O}\,$$\,$Spin(7) manifold$\,$$\,$Spin(7)$\,$$\,$8$\,$$\,$Cayley form$\,$
$\,$G2 manifold$\,$$\,$G2$\,$$\,7\,$$\,$associative 3-form$\,$

### As special $\mathbb{H}$-Riemannian manifolds

$\;$normed division algebra$\;$$\;\mathbb{A}\;$$\;$Riemannian $\mathbb{A}$-manifolds$\;$$\;$special Riemannian $\mathbb{A}$-manifolds$\;$
$\;$real numbers$\;$$\;\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$\;$$\;\mathbb{O}\;$$\;$Spin(7)-manifold$\;$$\;$G2-manifold$\;$

(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 $X^{[n+1]}$ (for $n \in \mathbb{N}$) for $X$ either

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

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

### Coulomb- and Higgs-branches of $D=3$$\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).