nLab compact hyperkähler manifold

Idea

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

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 K3-surface

2. 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

General

• Arnaud Beauville, Variétés Kähleriennes dont la premiere classe de Chern est nulle, Jour.

  Diff. Geom. 18 (1983), 755–782 (euclid.jdg/1214438181)

• Kieran O’Grady, Desingularized moduli spaces of sheaves on a K3, J. Reine Angew. Math. 512 (1999), 49–117 (arXiv:alg-geom/9708009, arXiv:math/9805099)

• Kieran O’Grady, A new six dimensional irreducible symplectic variety, J. Algebraic Geom. 12 (2003), 435-505 (arXiv:math/0010187)

• Daniel Huybrechts, Compact Hyperkähler Manifolds, In: Ellingsrud G., Ranestad K., Olson L., Strømme S.A. (eds.) Calabi-Yau Manifolds and Related Geometries, Universitext. Springer, Berlin, Heidelberg 2003 (doi:10.1007/978-3-642-19004-9_3,

  arXiv:alg-geom/9705025)

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:

• Kenneth Intriligator, Compactified Little String Theories and Compact Moduli Spaces of Vacua, Phys. Rev. D61:106005, 2000 (arXiv:hep-th/9909219)

In Rozansky-Witten theory

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

• Justin Roberts, Simon Willerton, p. 17 of: On the Rozansky-Witten weight systems, Algebr. Geom. Topol. 10 (2010) 1455-1519 (arXiv:math/0602653)

• Justin Sawon, Section 5.3 of: Rozansky-Witten invariants of hyperkähler manifolds, Cambridge 2000 (arXiv:math/0404360)