nLab hyperkähler manifold

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 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).

Related concepts

• Kähler manifold, hyper-Kähler manifold, quaternionic Kähler manifold

• compact hyperkähler manifold

• ADE singularity

• Rozansky-Witten theory

References

General

See also

• Wikipedia, Hyperkähler manifold

Compact hyperkähler manifolds

• 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)

Rozansky-Witten invariants

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 manifold, Cambridge 2000 (arXiv:math/0404360)

Examples

The example of Coulomb branches of D=3 N=4 super Yang-Mills theory originates with

• Nathan Seiberg, Edward Witten, Gauge Dynamics And Compactification To Three Dimensions, In: J.M. Drouffe, J.B. Zuber (eds.) The mathematical beauty of physics: A memorial volume for Claude Itzykson Proceedings, Conference, Saclay, France, June 5-7, 1996 (arXiv:hep-th/9607163, spire:420925)