Hyperkähler manifold
Encyclopedia
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 an
-dimensional quaternionic Hermitian space). Hyperkähler manifolds are special classes of Kähler manifold
s. They can be thought of as quaternion
ic 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.
of complex structures (i.e. integrable almost complex structures) with respect to which the metric
is Kähler.
In particular, there are three distinct complex structures, I, J, and K, which satisfy the quaternion relations
Any linear combination
with
real numbers such that
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
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.
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.
that any compact
hyperkähler 4-manifold is either a K3 surface
or a compact torus
.
(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 quaternion
s 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 instanton
s.
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.
Riemannian manifold
In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...
of dimension 4k and holonomy group contained in Sp(k)
Symplectic group
In mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups, denoted Sp and Sp. The latter is sometimes called the compact symplectic group to distinguish it from the former. Many authors prefer slightly different notations, usually...
(here Sp(k) denotes a compact form of a symplectic group
Symplectic group
In mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups, denoted Sp and Sp. The latter is sometimes called the compact symplectic group to distinguish it from the former. Many authors prefer slightly different notations, usually...
, identified
with the group of quaternionic-linear unitary endomorphisms
of an
-dimensional quaternionic Hermitian space). Hyperkähler manifolds are special classes of Kähler manifoldKähler manifold
In mathematics, a Kähler manifold is a manifold with unitary structure satisfying an integrability condition.In particular, it is a Riemannian manifold, a complex manifold, and a symplectic manifold, with these three structures all mutually compatible.This threefold structure corresponds to the...
s. They can be thought of as quaternion
Quaternion
In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...
ic 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)
Symplectic group
In mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups, denoted Sp and Sp. The latter is sometimes called the compact symplectic group to distinguish it from the former. Many authors prefer slightly different notations, usually...
is a subgroup
Subgroup
In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
of SU(2k)
Special unitary group
The special unitary group of degree n, denoted SU, is the group of n×n unitary matrices with determinant 1. The group operation is that of matrix multiplication...
).
Hyperkähler manifolds were defined by E. Calabi
Eugenio Calabi
Eugenio Calabi is a Italian American mathematician and professor emeritus at the University of Pennsylvania, specializing in differential geometry, partial differential equations and their applications....
in 1978.
Quaternionic structure
Every hyperkähler manifold M has a 2-sphereSphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
of complex structures (i.e. integrable almost complex structures) with respect to which the metric
Metric tensor
In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...
is Kähler.
In particular, there are three distinct complex structures, I, J, and K, which satisfy the quaternion relations
Any linear combination
with
real numbers such thatis 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
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 transportParallel transport
In geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection , then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the...
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 converseis also true in the case of compact manifolds, due to Yau
Shing-Tung Yau
Shing-Tung Yau is a Chinese American mathematician working in differential geometry. He was born in Shantou, Guangdong Province, China into a family of scholars from Jiaoling, Guangdong Province....
's proof of the Calabi
Eugenio Calabi
Eugenio Calabi is a Italian American mathematician and professor emeritus at the University of Pennsylvania, specializing in differential geometry, partial differential equations and their applications....
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
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, sometimes under a name holomorphically symplectic manifolds. Due to Bogomolov
Fedor Bogomolov
Fedor Alekseyevich Bogomolov is a Russian and American mathematician, known for his research in algebraic geometry and number theory. Bogomolov worked at Steklov Institute in Moscow before he became a professor at Courant Institute...
'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 knowthat any compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
hyperkähler 4-manifold is either a K3 surface
K3 surface
In mathematics, a K3 surface is a complex or algebraic smooth minimal complete surface that is regular and has trivial canonical bundle.In the Enriques-Kodaira classification of surfaces they form one of the 5 classes of surfaces of Kodaira dimension 0....
or a compact torus
.(Every Calabi-Yau manifold in 4 (real) dimensions is a hyperkähler manifold, because SU(2)
Special unitary group
The special unitary group of degree n, denoted SU, is the group of n×n unitary matrices with determinant 1. The group operation is that of matrix multiplication...
is isomorphic to Sp(1)
Symplectic group
In mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups, denoted Sp and Sp. The latter is sometimes called the compact symplectic group to distinguish it from the former. Many authors prefer slightly different notations, usually...
.)
A Hilbert scheme
Hilbert scheme
In algebraic geometry, a branch of mathematics, a Hilbert scheme is a scheme that is the parameter space for the closed subschemes of some projective space , refining the Chow variety. The Hilbert scheme is a disjoint union of projective subschemes corresponding to Hilbert polynomials...
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
Hilbert scheme
In algebraic geometry, a branch of mathematics, a Hilbert scheme is a scheme that is the parameter space for the closed subschemes of some projective space , refining the Chow variety. The Hilbert scheme is a disjoint union of projective subschemes corresponding to Hilbert polynomials...
Non-compact, complete, hyperkähler 4-manifolds which are asymptotic to H/G, where H denotes the quaternion
Quaternion
In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...
s and G is a finite subgroup
Subgroup
In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
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
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
under the name gravitational instanton
Gravitational instanton
In mathematical physics and differential geometry, a gravitational instanton is a four-dimensional complete Riemannian manifold satisfying the vacuum Einstein equations. They are so named because they are analogues in quantum theories of gravity of instantons in Yang–Mills theory...
s.
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.
External links
- Maciej Dunajski and Lionel J. Mason(2000)"[ftp://ftp.esi.ac.at/pub/Preprints/esi821.pdf Hyper Kahler Hierarchies and their Twistor Theory]"