mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

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 List of natural phenomena, natural phenomena. This is in contrast to experimental p ...

, the Eguchi–Hanson space is a non-compact, self-dual, asymptotically locally Euclidean (ALE) metric on the

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. It may be described also as the dual bundle to the tangent bundle. This m ...

of the 2-sphere ''T''^*''S''². The

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

of this 4-real-dimensional

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...

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 physics, theoretical physicist and computer scientist. Hanson is best known in theoretical physics as the co-discoverer of the Eguchi–Hanson space, Eguchi–Hanson metric, the first gravi ...

; it was discovered independently by the mathematician

Eugenio Calabi Eugenio Calabi (May 11, 1923 – September 25, 2023) was an Italian-born American mathematician and the Thomas A. Scott Professor of Mathematics at the University of Pennsylvania, specializing in differential geometry, partial differential equa ...

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 measure ...

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 Albert Einstein in 1915 in the ...

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 z ...

. It may be regarded as a resolution 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 that 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 ''C''^d/2/''Z''_d/2 of (real-)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 a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, the space is important in

string theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...

. Certain types of

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

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 and differential geometry, a Calabi–Yau manifold, also known as a Calabi–Yau space, is a particular type of manifold which has certain properties, such as Ricci flatness, yielding applications in theoretical physics. P ...

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 ...

; 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