Eguchi–Hanson space
Encyclopedia
In mathematics
and theoretical physics
, the Eguchi–Hanson space is a non-compact, self-dual, asymptotically locally Euclidean (ALE) metric on the cotangent bundle
of the 2-sphere T*S2. The holonomy group of this 4-real-dimensional manifold
is SU(2), much like for the K3 surface
s.
The geometrical shape has the Ricci tensor equal to zero; this fact makes it relevant as a solution to Einstein's equations of general relativity. And it may be regarded as a resolution
of the A1 singularity according to the ADE classification
which is the singularity at the fixed point of the C2/Z2 orbifold
where the Z2 group inverts the signs of both complex coordinates in C2.
Aside from its inherent importance in pure geometry
, the space is important in string theory
. Certain types of K3 surface
s can be approximated as a combination of several Eguchi–Hanson metrics.
The Eguchi–Hanson metric is also relevant for gravitational instanton
s.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
and theoretical physics
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
, the Eguchi–Hanson space is a non-compact, self-dual, asymptotically locally Euclidean (ALE) metric on the cotangent bundle
Cotangent bundle
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold...
of the 2-sphere T*S2. The holonomy group of this 4-real-dimensional manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
is SU(2), much like for the 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....
s.
The geometrical shape has the Ricci tensor equal to zero; this fact makes it relevant as a solution to Einstein's equations of general relativity. And it may be regarded as a resolution
Resolution of singularities
In algebraic geometry, the problem of resolution of singularities asks whether every algebraic variety V has a resolution, a non-singular variety W with a proper birational map W→V...
of the A1 singularity according to the ADE classification
ADE classification
In mathematics, the ADE classification is the complete list of simply laced Dynkin diagrams or other mathematical objects satisfying analogous axioms; "simply laced" means that there are no multiple edges, which corresponds to all simple roots in the root system forming angles of \pi/2 = 90^\circ ...
which is the singularity at the fixed point of the C2/Z2 orbifold
Orbifold
In the mathematical disciplines of topology, geometry, and geometric group theory, an orbifold is a generalization of a manifold...
where the Z2 group inverts the signs of both complex coordinates in C2.
Aside from its inherent importance in pure geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, the space is important in string theory
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
. Certain types of 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....
s can be approximated as a combination of several Eguchi–Hanson metrics.
The Eguchi–Hanson metric is also relevant for 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.