Hyperkähler manifold

Hyperkähler manifold

In differential geometry, a hyperkähler manifold is a Riemannian manifold of dimension 4k and holonomy group contained in Sp(k) (here Sp(k) denotes a compact form of a symplectic group, identified with the group of quaternionic-linear unitary endomorphisms of a k-dimensional quaternionic Hermitian space). Hyperkähler manifolds are special classes of Kähler manifolds. They can be thought of as quaternionic analogues of Kähler manifolds. All hyperkähler manifolds are Ricci-flat and are thus Calabi–Yau manifolds (this can be easily seen by noting that Sp(k) is a subgroup of SU(2k)).

Hyperkähler manifolds were defined by E. Calabi in 1978.

Contents

• Quaternionic structure 1
• Holomorphic symplectic form 2
• Examples 3

Quaternionic structure

Every hyperkähler manifold M has a 2-sphere of complex structures (i.e. integrable almost complex structures) with respect to which the metric is Kähler.

In particular, it is an almost quaternionic manifold, meaning that there are three distinct complex structures, I, J, and K, which satisfy the quaternion relations

I^2 = J^2 = K^2 = IJK = -1.\,

Any linear combination

aI + bJ + cK \,

with a, b, c real numbers such that

a^2 + b^2 + c^2 = 1 \,

is also a complex structure on M. In particular, the tangent space TxM is a quaternionic vector space for each point x of M. Sp(k) can be considered as the group of orthogonal transformations of \mathbb{R}^{4k}=\mathbb{H}^k which are linear with respect to I, J and K. From this it follows that the holonomy of the manifold is contained in Sp(k). Conversely, if the holonomy group of the Riemannian manifold M is contained in Sp(k), choose complex structures Ix, Jx and Kx on TxM which make TxM into a quaternionic vector space. Parallel transport of these complex structures gives the required quaternionic structure on M.

Holomorphic symplectic form

A hyperkähler manifold (M,I,J,K), considered as a complex manifold (M,I), is holomorphically symplectic (equipped with a holomorphic, non-degenerate 2-form). The converse is also true in the case of compact manifolds, due to Yau's proof of the Calabi conjecture: Given a compact, Kähler, holomorphically symplectic manifold (M,I), it is always equipped with a compatible hyperkähler metric. Such a metric is unique in a given Kähler class. Compact hyperkähler manifolds have been extensively studied using techniques from algebraic geometry, sometimes under a name holomorphically symplectic manifolds. Due to Bogomolov's decomposition theorem (1974), the holonomy group of a compact holomorphically symplectic manifold M is exactly Sp(k) if and only if M is simply connected and any pair of holomorphic symplectic forms on M are scalar multiples of each other.

Examples

Due to Kodaira's classification of complex surfaces, we know that any compact hyperkähler 4-manifold is either a K3 surface or a compact torus T^4. (Every Calabi–Yau manifold in 4 (real) dimensions is a hyperkähler manifold, because SU(2) is isomorphic to Sp(1).)

A Hilbert scheme of points on a compact hyperkähler 4-manifold is again hyperkähler. This gives rise to two series of compact examples: Hilbert schemes of points on a K3 surface and generalized Kummer varieties.

Non-compact, complete, hyperkähler 4-manifolds which are asymptotic to H/G, where H denotes the quaternions and G is a finite subgroup of Sp(1), are known as asymptotically locally Euclidean, or ALE, spaces. These spaces, and various generalizations involving different asymptotic behaviors, are studied in physics under the name gravitational instantons. The Gibbons–Hawking ansatz gives examples invariant under a circle action.

Many examples of noncompact hyperkähler manifolds arise as moduli spaces of solutions to certain gauge theory equations which arise from the dimensional reduction of the anti-self dual Yang–Mills equations: instanton moduli spaces, monopole moduli spaces, spaces of solutions to Hitchin's self-duality equations on Riemann surfaces, space of solutions to Nahm's equations. Another class of examples are the Nakajima quiver varieties, which are of great importance in representation theory.