Orbifold

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.

In string theory, the word "orbifold" has additional meaning, discussed below.

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 $S^{1}$-action without fixed points carries the structure of an orbifold (this is not a partial case of the main example).

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.

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 $\pi _{}^{}$; 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 $\mathbb {R} ^{n}$, an orbifold locally looks like a quotient of $\mathbb {R} ^{n}$. 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).

An orbifold chart is an open subset $U\subset X$ together with open set $V\subset$Rⁿ and a continuous map $\phi :U\to V$ which satisfy the following property: there is a finite group $\Gamma$ acting by linear transformations on $V$ and a homeomorphism $\vartheta :V/\Gamma \to U$ such that $\phi =\vartheta \circ \pi$, where $\pi \,$ denotes the projection $V\to V/\Gamma$.

A collection of orbifold charts $\{\phi _{\alpha }:U_{\alpha }\to V_{\alpha }\}$ is called an orbifold atlas if it satisfies the following properties:

1. $\displaystyle \bigcup _{\alpha }U_{\alpha }=X$,
2. if $\phi _{\alpha }(x)=\phi _{\beta }(y)\,$ then there is a neighborhood $x\in V_{x}\subset V_{\alpha }$ and $y\in V_{y}\subset V_{\beta }$ and a homeomorphism $\psi :V_{x}\to V_{y}$ such that $\phi _{\alpha }=\phi _{\beta }\circ \psi$.

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 $\psi _{}^{}$ 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 $R^{n}$, i.e. $R^{n}/\Gamma$. 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.

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.

D-branes propagating on the orbifolds are described, at low energies, by gauge theories defined by the quiver diagrams.

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.

Further reading

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