nLab
compact hyperkähler manifold

Contents

Contents

Idea

A compact (i.e. closed) hyperkähler manifold.

Examples

Hilbert schemes on K3K3 and 𝕋 4\mathbb{T}^4

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).

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

General

Examples from Coulomb branches

On D=3 N=4 super Yang-Mills theories with compact hyperkähler manifold Coulomb branches obtained by KK-compactification of little string theories:

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.