In mathematics and

theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...

, the Eguchi–Hanson space is a non-compact,

self-dual In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a Injective function, one-to-one fashion, often (but not always) by means of an Involution (mathematics), involutio ...

, asymptotically locally Euclidean (ALE) metric on the cotangent bundle of the 2-sphere ''T''^*''S''². The

holonomy group In differential geometry, the holonomy of a connection on a smooth manifold is a general geometrical consequence of the curvature of the connection measuring the extent to which parallel transport around closed loops fails to preserve the geomet ...

of this 4-real-dimensional manifold is SU(2). The metric is generally attributed to the physicists Tohru Eguchi and

Andrew J. Hanson Andrew J. Hanson (born 1944) is an American theoretical physicist and computer scientist. Hanson is best known in theoretical physics as the co-discoverer of the Eguchi-Hanson Metric, the first Gravitational instanton. This Einstein metric is ...

; it was discovered independently by the mathematician

Eugenio Calabi Eugenio Calabi (born 11 May 1923) is an Italian-born American mathematician and the Thomas A. Scott Professor of Mathematics, Emeritus, at the University of Pennsylvania, specializing in differential geometry, partial differential equations and ...

around the same time in 1979. The Eguchi-Hanson metric has

Ricci tensor In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measur ...

equal to zero, making it a solution to the vacuum

Einstein equations In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it. The equations were published by Einstein in 1915 in the form ...

of general relativity, albeit with Riemannian rather than Lorentzian

metric signature In mathematics, the signature of a metric tensor ''g'' (or equivalently, a real quadratic form thought of as a real symmetric bilinear form on a finite-dimensional vector space) is the number (counted with multiplicity) of positive, negative and ...

. It may be regarded as a

resolution Resolution(s) may refer to: Common meanings * Resolution (debate), the statement which is debated in policy debate * Resolution (law), a written motion adopted by a deliberative body * New Year's resolution, a commitment that an individual mak ...

of the ''A''₁ singularity according to the

ADE classification In mathematics, the ADE classification (originally ''A-D-E'' classifications) is a situation where certain kinds of objects are in correspondence with simply laced Dynkin diagrams. The question of giving a common origin to these classifications, r ...

which is the singularity at the fixed point of the ''C''²/''Z''₂

orbifold In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space which is locally a finite group quotient of a Euclidean space. D ...

where the ''Z''₂ group inverts the signs of both complex coordinates in ''C''². The even dimensional space of dimension

d

can be described using complex coordinates

w_i \in \mathbb C^

with a metric :

g_ = \bigg(1+\frac\bigg)^\bigg delta_-\frac\bigg

where

\rho

is a scale setting constant and

r^2 = , w, ^2_

. Aside from its inherent importance in pure

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

, the space is important in string theory. Certain types of

K3 surface In mathematics, a complex analytic K3 surface is a compact connected complex manifold of dimension 2 with trivial canonical bundle and irregularity zero. An (algebraic) K3 surface over any field means a smooth proper geometrically connected al ...

s can be approximated as a combination of several Eguchi–Hanson metrics since both have the same holonomy group. Similarly, the space can also be used to construct

Calabi–Yau manifold In algebraic geometry, a Calabi–Yau manifold, also known as a Calabi–Yau space, is a particular type of manifold which has properties, such as Ricci flatness, yielding applications in theoretical physics. Particularly in superstri ...

s by replacing the orbifold singularities of

T^6/\mathbb Z_3

with Eguchi–Hanson spaces. The Eguchi–Hanson metric is the prototypical example of a

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 o ...

; detailed expressions for the metric are given in that article. It is then an example of a

hyperkähler manifold In differential geometry, a hyperkähler manifold is a Riemannian manifold (M, g) endowed with three integrable almost complex structures I, J, K that are Kähler with respect to the Riemannian metric g and satisfy the quaternionic relations I^2 ...

References

Differential geometry String theory {{string-theory-stub