nLab compact hyperkähler manifold




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


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).


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).



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 12:32:45. See the history of this page for a list of all contributions to it.