# nLab compact hyperkähler manifold

Contents

complex geometry

### Examples

#### Riemannian geometry

Riemannian geometry

# Contents

## Examples

### Hilbert schemes on $K3$ and $\mathbb{T}^4$

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).

## Properties

### Relation to Rozansky-Witten weight systems

In order for Rozansky-Witten weight systems to take values in the ground field, hence to be actual weight systems, the hyperkähler manifold has to be compact (i.e. closed).

## References

### In Rozansky-Witten theory

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

Last revised on January 2, 2020 at 07:32:45. See the history of this page for a list of all contributions to it.