In topology and group theory, an orbifold (for "orbit-manifold") is a generalization of a manifold. It is a topological space (called an underlying space) with an orbifold structure (see below). The underlying space locally looks like a quotient space of a Euclidean space under the action of a finite group of isometries. Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ... Group theory is that branch of mathematics concerned with the study of groups. ... On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or gluing together certain points of a given space. ... In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... In mathematics, a finite group is a group which has finitely many elements. ... In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces. ...

In string theory, the word "orbifold" has additional meaning, discussed below. Interaction in the subatomic world: world lines of pointlike particles in the Standard Model or a world sheet swept up by closed strings in string theory String theory is a model of fundamental physics whose building blocks are one-dimensional extended objects (strings) rather than the zero-dimensional points (particles...

The main example of underlying space is a quotient space of a manifold under the action of a finite group of diffeomorphisms, in particular a manifold with boundary carries natural orbifold structure, since it is the Z₂-factor of its double. A factor space of a manifold along a smooth S¹-action without fixed points carries the structure of an orbifold (this is not a partial case of the main example). In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ... This is a glossary of terms specific to differential geometry and differential topology. ...

Orbifold structure gives a natural stratification by open manifolds on its underlying space, where one stratum corresponds to a set of singular points of the same type. Stratification in mathematical logic In mathematical logic, stratification is any consistent assignment of numbers to predicate symbols guaranteeing that a unique formal interpretation of a logical theory exists. ...

It should be noted that one topological space can carry many different orbifold structures. For example, consider the orbifold O associated with a factor space of the 2-sphere along a rotation by ; it is homeomorphic to the 2-sphere, but the natural orbifold structure is different. It is possible to adopt most of the characteristics of manifolds to orbifolds and these characteristics are usually different from correspondent characteristics of underlying space. In the above example, the orbifold fundamental group of O is Z₂ and its orbifold Euler characteristic is 1.

Formal definition

Like a manifold, an orbifold is specified by local conditions; however, whereas a manifold locally looks like , an orbifold locally looks like a quotient of . Hence an orbifold need not be a manifold.

A (topological) orbifold O, is a Hausdorff topological space X with countable base, called the underlying space, with an orbifold structure, which is defined by an orbifold atlas (see below). In mathematics, a base (or basis) B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B. We say that the base generates the topology T. Bases...

An orbifold chart is an open subset together with open set Rⁿ and a continuous map which satisfy the following property: there is a finite group Γ acting by linear transformations on V and a homeomorphism such that , where denotes the projection . In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...

A collection of orbifold charts is called an orbifold atlas if it satisfies the following properties:

1. ,
2. if then there is a neighborhood and and a homeomorphism such that .

The orbifold atlas defines the orbifold structure completely and we regard two orbifold atlases of X to give the same orbifold structure if they can be combined to give a larger orbifold atlas.

One can add differentiability conditions on the gluing map in the above definition and get a definition of differentiable orbifolds in the same way as it was done for manifolds.

Orbifolds in string theory

In string theory, the word "orbifold" has a slightly new meaning. For mathematicians, an orbifold is a generalization of the notion of manifold that allows the presence of the points whose neighborhood is diffeomorphic to a coset of Rⁿ, i.e. Rⁿ / Γ. In physics, the notion of an orbifold usually describes an object that can be globally written as a coset M / G where M is a manifold (or a theory), and G is a group of its isometries (or symmetries) - not necessarily all of them. In string theory, these symmetries do not have to have a geometric interpretation. Interaction in the subatomic world: world lines of pointlike particles in the Standard Model or a world sheet swept up by closed strings in string theory String theory is a model of fundamental physics whose building blocks are one-dimensional extended objects (strings) rather than the zero-dimensional points (particles... On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ...

A quantum field theory defined on an orbifold becomes singular near the fixed points of G. However string theory requires us to add new parts of the closed string Hilbert space - namely the twisted sectors where the fields defined on the closed strings are periodic up to an action from G. Orbifolding is therefore a general procedure of string theory to derive a new string theory from an old string theory in which the elements of G have been identified with the identity. Such a procedure reduces the number of states because the states must be invariant under G, but it also increases the number of states because of the extra twisted sectors. The result is usually a perfectly smooth, new string theory. Quantum field theory (QFT) is the application of quantum mechanics to fields. ... A closed string is a one-dimensional fundamental object in string theory that has no end-points, and therefore is topologically equivalent to a circle. ... In mathematics, a Hilbert space is a generalization of Euclidean space which is not restricted to finite dimensions. ...

D-branes propagating on the orbifolds are described, at low energies, by gauge theories defined by the quiver diagrams. In theoretical physics, D-branes are a special class of p-branes, named for the physicist Johann Dirichlet. ... == Headline text In physics, a quiver diagram is a graph representing the matter content of a gauge theory that describes D-branes on orbifolds. ...

History

Orbifolds and related concepts are implicit in the work of pioneers such as Henri Poincare. The first formal definition of an orbifold-like object was given by Ichiro Satake in 1956; he defined the V-manifold, which had a codimension 2 singular locus, in the context of Riemannian geometry. William Thurston, who was unaware of Satake's work, later in the mid 1970s defined and named the more general notion of orbifold as part of his study of hyperbolic structures. Henri Poincaré, photograph from the frontispiece of the 1913 edition of Last Thoughts Jules Henri Poincaré (April 29, 1854 – July 17, 1912) was one of Frances greatest mathematicians, theoretical scientists and a philosopher of science. ... William Thurston William Paul Thurston (born October 30, 1946) is an American mathematician. ...

Further reading

William Thurston, The Geometry and Topology of Three-Manifolds (Chapter 13), Princeton University lecture notes (1978-1981).