Definition using orbifold atlas

Like a manifold, an orbifold is specified by local conditions; however, instead of being locally modelled on open subsets of ${\displaystyle \mathbb {R} ^{n}}$, an orbifold is locally modelled on quotients of open subsets of ${\displaystyle \mathbb {R} ^{n}}$ by finite group actions. The structure of an orbifold encodes not only that of the underlying quotient space, which need not be a manifold, but also that of the isotropy subgroups.

An n-dimensional orbifold is a Hausdorff topological space X, called the underlying space, with a covering by a collection of open sets ${\displaystyle U_{i}}$, closed under finite intersection. For each ${\displaystyle U_{i}}$, there is

• an open subset ${\displaystyle V_{i}}$ of ${\displaystyle \mathbb {R} ^{n}}$, invariant under a faithful linear action of a finite group ${\displaystyle \Gamma _{i}}$;
• a continuous map ${\displaystyle \varphi _{i}}$ of ${\displaystyle V_{i}}$ onto ${\displaystyle U_{i}}$ invariant under ${\displaystyle \Gamma _{i}}$, called an orbifold chart, which defines a homeomorphism between ${\displaystyle V_{i}/\Gamma _{i}}$ and ${\displaystyle U_{i}}$.

The collection of orbifold charts is called an orbifold atlas if the following properties are satisfied:

• for each inclusion U_i ${\displaystyle \subset }$ U_j there is an injective group homomorphism f_ij : Γ_i ${\displaystyle \rightarrow }$ Γ_j
• for each inclusion U_i ${\displaystyle \subset }$ U_j there is a Γ_i -equivariant homeomorphism ψ_ij, called a gluing map, of V_i onto an open subset of V_j
• the gluing maps are compatible with the charts, i.e. φ_j·ψ_ij = φ_i
• the gluing maps are unique up to composition with group elements, i.e. any other possible gluing map from V_i to V_j has the form g·ψ_ij for a unique g in Γ_j

As for atlases on manifolds, two orbifold atlases of X are equivalent if they can be consistently combined to give a larger orbifold atlas. An orbifold structure is therefore an equivalence class of orbifold atlases.

Note that the orbifold structure determines the isotropy subgroup of any point of the orbifold up to isomorphism: it can be computed as the stabilizer of the point in any orbifold chart. If U_i ${\displaystyle \subset }$ U_j ${\displaystyle \subset }$ U_k, then there is a unique transition element g_ijk in Γ_k such that

g_ijk·ψ_ik = ψ_jk·ψ_ij

These transition elements satisfy

(Ad g_ijk)·f_ik = f_jk·f_ij

as well as the cocycle relation (guaranteeing associativity)

f_km(g_ijk)·g_ikm = g_ijm·g_jkm.

More generally, attached to an open covering of an orbifold by orbifold charts, there is the combinatorial data of a so-called complex of groups (see below).

Exactly as in the case of manifolds, differentiability conditions can be imposed on the gluing maps to give a definition of a differentiable orbifold. It will be a Riemannian orbifold if in addition there are invariant Riemannian metrics on the orbifold charts and the gluing maps are isometries.

Definition using Lie groupoids

Recall that a groupoid consists of a set of objects ${\displaystyle G_{0}}$, a set of arrows ${\displaystyle G_{1}}$, and structural maps including the source and the target maps ${\displaystyle s,t:G_{1}\to G_{0}}$ and other maps allowing arrows to be composed and inverted. It is called a Lie groupoid if both ${\displaystyle G_{0}}$ and ${\displaystyle G_{1}}$ are smooth manifolds, all structural maps are smooth, and both the source and the target maps are submersions. The intersection of the source and the target fiber at a given point ${\displaystyle x\in G_{0}}$, i.e. the set ${\displaystyle (G_{1})_{x}:=s^{-1}(x)\cap t^{-1}(x)}$, is the Lie group called the isotropy group of ${\displaystyle G_{1}}$ at ${\displaystyle x}$. A Lie groupoid is called proper if the map ${\displaystyle (s,t):G_{1}\to G_{0}\times G_{0}}$ is a proper map, and étale if both the source and the target maps are local diffeomorphisms.

An orbifold groupoid is given by one of the following equivalent definitions:

• a proper étale Lie groupoid;
• a proper Lie groupoid whose isotropies are discrete spaces.

Since the isotropy groups of proper groupoids are automatically compact, the discreteness condition implies that the isotropies must be actually finite groups.^[10]

Orbifold groupoids play the same role as orbifold atlases in the definition above. Indeed, an orbifold structure on a Hausdorff topological space ${\displaystyle X}$ is defined as the Morita equivalence class of an orbifold groupoid ${\displaystyle G\rightrightarrows M}$ together with a homeomorphism ${\displaystyle |M/G|\simeq X}$, where ${\displaystyle |M/G|}$ is the orbit space of the Lie groupoid ${\displaystyle G}$ (i.e. the quotient of ${\displaystyle M}$ by the equivalent relation when ${\displaystyle x\sim y}$ if there is a ${\displaystyle g\in G}$ with ${\displaystyle s(g)=x}$ and ${\displaystyle t(g)=y}$). This definition shows that orbifolds are a particular kind of differentiable stack.

Orbifolds as diffeologies

Orbifolds can be defined in the general framework of diffeology^[12] and have been proved to be equivalent^[13] to Ichirô Satake's original definition:^[1]

Definition. An orbifold is a diffeological space locally diffeomorphic at each point to some ${\displaystyle \mathbb {R} ^{n}/G}$, where ${\displaystyle n}$ is an integer and ${\displaystyle G}$ is a finite linear group which may change from point to point.

This definition calls a few remarks:

• This definition mimics the definition of a manifold in diffeology, which is a diffeological space locally diffeomorphic at each point to ${\displaystyle \mathbb {R} ^{n}}$.
• An orbifold is regarded first as a diffeological space, a set equipped with a diffeology. Then, the diffeology is tested to be locally diffeomorphic at each point to a quotient ${\displaystyle \mathbb {R} ^{n}/G}$ with ${\displaystyle G}$ a finite linear group.
• This definition is equivalent^[14] with Haefliger orbifolds.^[15]
• {Orbifolds} makes a subcategory of the category {Diffeology} whose objects are diffeological spaces and morphisms smooth maps. A smooth map between orbifolds is any map which is smooth for their diffeologies. This resolves, in the context of Satake's definition, his remark:^[16] "The notion of ${\displaystyle C^{\infty }}$-map thus defined is inconvenient in the point that a composite of two ${\displaystyle C^{\infty }}$-map defined in a different choice of defining families is not always a ${\displaystyle C^{\infty }}$-map." Indeed, there are smooth maps between orbifolds that do not lift locally as equivariant maps.^[17]

Note that the fundamental group of an orbifold as a diffeological space is not the same as the fundamental group as defined above. That last one is related to the structure groupoid^[18] and its isotropy groups.

Definition

A complex of groups (Y,f,g) on an abstract simplicial complex Y is given by

• a finite group Γ_σ for each simplex σ of Y
• an injective homomorphism f_στ : Γ_τ ${\displaystyle \rightarrow }$ Γ_σ whenever σ ${\displaystyle \subset }$ τ
• for every inclusion ρ ${\displaystyle \subset }$ σ ${\displaystyle \subset }$ τ, a group element g_ρστ in Γ_ρ such that (Ad g_ρστ)·f_ρτ = f_ρσ·f_στ (here Ad denotes the adjoint action by conjugation)

The group elements must in addition satisfy the cocycle condition

f_πρ(g_ρστ) g_πρτ = g_πστ g_πρσ

for every chain of simplices ${\displaystyle \pi \subset \rho \subset \sigma \subset \tau .}$ (This condition is vacuous if Y has dimension 2 or less.)

Any choice of elements h_στ in Γ_σ yields an equivalent complex of groups by defining

• f'_στ = (Ad h_στ)·f_στ
• g'_ρστ = h_ρσ·f_ρσ(h_στ)·g_ρστ·h_ρτ⁻¹

A complex of groups is called simple whenever g_ρστ = 1 everywhere.

• An easy inductive argument shows that every complex of groups on a simplex is equivalent to a complex of groups with g_ρστ = 1 everywhere.

It is often more convenient and conceptually appealing to pass to the barycentric subdivision of Y. The vertices of this subdivision correspond to the simplices of Y, so that each vertex has a group attached to it. The edges of the barycentric subdivision are naturally oriented (corresponding to inclusions of simplices) and each directed edge gives an inclusion of groups. Each triangle has a transition element attached to it belonging to the group of exactly one vertex; and the tetrahedra, if there are any, give cocycle relations for the transition elements. Thus a complex of groups involves only the 3-skeleton of the barycentric subdivision; and only the 2-skeleton if it is simple.

Example

If X is an orbifold (or orbispace), choose a covering by open subsets from amongst the orbifold charts f_i : V_i ${\displaystyle \rightarrow }$ U_i. Let Y be the abstract simplicial complex given by the nerve of the covering: its vertices are the sets of the cover and its n-simplices correspond to non-empty intersections U_α = U_i1 ${\displaystyle \cap }$ ··· ${\displaystyle \cap }$ U_in. For each such simplex there is an associated group Γ_α and the homomorphisms f_ij become the homomorphisms f_στ. For every triple ρ ${\displaystyle \subset }$ σ ${\displaystyle \subset }$ τ corresponding to intersections

${\displaystyle U_{i}\supset U_{i}\cap U_{j}\supset U_{i}\cap U_{j}\cap U_{k}}$

there are charts φ_i : V_i ${\displaystyle \rightarrow }$ U_i, φ_ij : V_ij ${\displaystyle \rightarrow }$ U_i ${\displaystyle \cap }$ U_j and φ_ijk : V_ijk ${\displaystyle \rightarrow }$ U_i ${\displaystyle \cap }$ U_j ${\displaystyle \cap }$ U_k and gluing maps ψ : V _ij ${\displaystyle \rightarrow }$ V_i, ψ' : V _ijk ${\displaystyle \rightarrow }$ V_ij and ψ" : V _ijk ${\displaystyle \rightarrow }$ V_i.

There is a unique transition element g_ρστ in Γ_i such that g_ρστ·ψ" = ψ·ψ′. The relations satisfied by the transition elements of an orbifold imply those required for a complex of groups. In this way a complex of groups can be canonically associated to the nerve of an open covering by orbifold (or orbispace) charts. In the language of non-commutative sheaf theory and gerbes, the complex of groups in this case arises as a sheaf of groups associated to the covering U_i; the data g_ρστ is a 2-cocycle in non-commutative sheaf cohomology and the data h_στ gives a 2-coboundary perturbation.

Edge-path group

The edge-path group of a complex of groups can be defined as a natural generalisation of the edge path group of a simplicial complex. In the barycentric subdivision of Y, take generators e_ij corresponding to edges from i to j where i ${\displaystyle \rightarrow }$ j, so that there is an injection ψ_ij : Γ_i ${\displaystyle \rightarrow }$ Γ_j. Let Γ be the group generated by the e_ij and Γ_k with relations

e_ij ⁻¹ · g · e_ij = ψ_ij(g)

for g in Γ_i and

e_ik = e_jk·e_ij·g_ijk

if i ${\displaystyle \rightarrow }$ j ${\displaystyle \rightarrow }$ k.

For a fixed vertex i₀, the edge-path group Γ(i₀) is defined to be the subgroup of Γ generated by all products

g₀ · e_{i0 i1} · g₁ · e_{i1 i2} · ··· · g_n · e_{ini 0}

where i₀, i₁, ..., i_n, i₀ is an edge-path, g_k lies in Γ_ik and e_ji=e_ij⁻¹ if i ${\displaystyle \rightarrow }$ j.

Developable complexes

A simplicial proper action of a discrete group Γ on a simplicial complex X with finite quotient is said to be regular if it satisfies one of the following equivalent conditions:^[9]

• X admits a finite subcomplex as fundamental domain;
• the quotient Y = X/Γ has a natural simplicial structure;
• the quotient simplicial structure on orbit-representatives of vertices is consistent;
• if (v₀, ..., v_k) and (g₀·v₀, ..., g_k·v_k) are simplices, then g·v_i = g_i·v_i for some g in Γ.

The fundamental domain and quotient Y = X / Γ can naturally be identified as simplicial complexes in this case, given by the stabilisers of the simplices in the fundamental domain. A complex of groups Y is said to be developable if it arises in this way.

• A complex of groups is developable if and only if the homomorphisms of Γ_σ into the edge-path group are injective.
• A complex of groups is developable if and only if for each simplex σ there is an injective homomorphism θ_σ from Γ_σ into a fixed discrete group Γ such that θ_τ·f_στ = θ_σ. In this case the simplicial complex X is canonically defined: it has k-simplices (σ, xΓ_σ) where σ is a k-simplex of Y and x runs over Γ / Γ_σ. Consistency can be checked using the fact that the restriction of the complex of groups to a simplex is equivalent to one with trivial cocycle g_ρστ.

The action of Γ on the barycentric subdivision X ' of X always satisfies the following condition, weaker than regularity:

• whenever σ and g·σ are subsimplices of some simplex τ, they are equal, i.e. σ = g·σ

Indeed, simplices in X ' correspond to chains of simplices in X, so that a subsimplices, given by subchains of simplices, is uniquely determined by the sizes of the simplices in the subchain. When an action satisfies this condition, then g necessarily fixes all the vertices of σ. A straightforward inductive argument shows that such an action becomes regular on the barycentric subdivision; in particular

• the action on the second barycentric subdivision X" is regular;
• Γ is naturally isomorphic to the edge-path group defined using edge-paths and vertex stabilisers for the barycentric subdivision of the fundamental domain in X".

There is in fact no need to pass to a third barycentric subdivision: as Haefliger observes using the language of category theory, in this case the 3-skeleton of the fundamental domain of X" already carries all the necessary data – including transition elements for triangles – to define an edge-path group isomorphic to Γ.

In two dimensions this is particularly simple to describe. The fundamental domain of X" has the same structure as the barycentric subdivision Y ' of a complex of groups Y, namely:

• a finite 2-dimensional simplicial complex Z;
• an orientation for all edges i ${\displaystyle \rightarrow }$ j;
• if i ${\displaystyle \rightarrow }$ j and j ${\displaystyle \rightarrow }$ k are edges, then i ${\displaystyle \rightarrow }$ k is an edge and (i, j, k) is a triangle;
• finite groups attached to vertices, inclusions to edges and transition elements, describing compatibility, to triangles.

An edge-path group can then be defined. A similar structure is inherited by the barycentric subdivision Z ' and its edge-path group is isomorphic to that of Z.

Definition

Let X be a finite simplicial complex with barycentric subdivision X '. An orbihedron structure consists of:

• for each vertex i of X ', a simplicial complex L_i' endowed with a rigid simplicial action of a finite group Γ_i.
• a simplicial map φ_i of L_i' onto the link L_i of i in X ', identifying the quotient L_i' / Γ_i with L_i.

This action of Γ_i on L_i' extends to a simplicial action on the simplicial cone C_i over L_i' (the simplicial join of i and L_i'), fixing the centre i of the cone. The map φ_i extends to a simplicial map of C_i onto the star St(i) of i, carrying the centre onto i; thus φ_i identifies C_i / Γ_i, the quotient of the star of i in C_i, with St(i) and gives an orbihedron chart at i.

• for each directed edge i ${\displaystyle \rightarrow }$ j of X ', an injective homomorphism f_ij of Γ_i into Γ_j.
• for each directed edge i ${\displaystyle \rightarrow }$ j, a Γ_i equivariant simplicial gluing map ψ_ij of C_i into C_j.
• the gluing maps are compatible with the charts, i.e. φ_j·ψ_ij = φ_i.
• the gluing maps are unique up to composition with group elements, i.e. any other possible gluing map from V_i to V_j has the form g·ψ_ij for a unique g in Γ_j.

If i${\displaystyle \rightarrow }$ j ${\displaystyle \rightarrow }$ k, then there is a unique transition element g_ijk in Γ_k such that

g_ijk·ψ_ik = ψ_jk·ψ_ij

These transition elements satisfy

(Ad g_ijk)·f_ik = f_jk·f_ij

as well as the cocycle relation

ψ_km(g_ijk)·g_ikm = g_ijm·g_jkm.

Main properties

• The group theoretic data of an orbihedron gives a complex of groups on X, because the vertices i of the barycentric subdivision X ' correspond to the simplices in X.
• Every complex of groups on X is associated with an essentially unique orbihedron structure on X. This key fact follows by noting that the star and link of a vertex i of X ', corresponding to a simplex σ of X, have natural decompositions: the star is isomorphic to the abstract simplicial complex given by the join of σ and the barycentric subdivision σ' of σ; and the link is isomorphic to join of the link of σ in X and the link of the barycentre of σ in σ'. Restricting the complex of groups to the link of σ in X, all the groups Γ_τ come with injective homomorphisms into Γ_σ. Since the link of i in X ' is canonically covered by a simplicial complex on which Γ_σ acts, this defines an orbihedron structure on X.
• The orbihedron fundamental group is (tautologically) just the edge-path group of the associated complex of groups.
• Every orbihedron is also naturally an orbispace: indeed in the geometric realization of the simplicial complex, orbispace charts can be defined using the interiors of stars.
• The orbihedron fundamental group can be naturally identified with the orbispace fundamental group of the associated orbispace. This follows by applying the simplicial approximation theorem to segments of an orbispace path lying in an orbispace chart: it is a straightforward variant of the classical proof that the fundamental group of a polyhedron can be identified with its edge-path group.
• The orbispace associated to an orbihedron has a canonical metric structure, coming locally from the length metric in the standard geometric realization in Euclidean space, with vertices mapped to an orthonormal basis. Other metric structures are also used, involving length metrics obtained by realizing the simplices in hyperbolic space, with simplices identified isometrically along common boundaries.
• The orbispace associated to an orbihedron is non-positively curved if and only if the link in each orbihedron chart has girth greater than or equal to 6, i.e. any closed circuit in the link has length at least 6. This condition, well known from the theory of Hadamard spaces, depends only on the underlying complex of groups.
• When the universal covering orbihedron is non-positively curved the fundamental group is infinite and is generated by isomorphic copies of the isotropy groups. This follows from the corresponding result for orbispaces.

Mumford's example

The Fano plane

Let α = ${\displaystyle {\sqrt {-7}}}$ be given by the binomial expansion of (1 − 8)^1/2 in Q₂ and set K = Q(α) ${\displaystyle \subset }$ Q₂. Let

ζ = exp 2πi/7

λ = (α − 1)/2 = ζ + ζ² + ζ⁴

μ = λ/λ*.

Let E = Q(ζ), a 3-dimensional vector space over K with basis 1, ζ, and ζ². Define K-linear operators on E as follows:

• σ is the generator of the Galois group of E over K, an element of order 3 given by σ(ζ) = ζ²
• τ is the operator of multiplication by ζ on E, an element of order 7
• ρ is the operator given by ρ(ζ) = 1, ρ(ζ²) = ζ and ρ(1) = μ·ζ², so that ρ³ is scalar multiplication by μ.

The elements ρ, σ, and τ generate a discrete subgroup of GL₃(K) which acts properly on the affine Bruhat–Tits building corresponding to SL₃(Q₂). This group acts transitively on all vertices, edges and triangles in the building. Let

σ₁ = σ, σ₂ = ρσρ⁻¹, σ₃ = ρ²σρ⁻².

Then

• σ₁, σ₂ and σ₃ generate a subgroup Γ of SL₃(K).
• Γ is the smallest subgroup generated by σ and τ, invariant under conjugation by ρ.
• Γ acts simply transitively on the triangles in the building.
• There is a triangle Δ such that the stabiliser of its edges are the subgroups of order 3 generated by the σ_i's.
• The stabiliser of a vertices of Δ is the Frobenius group of order 21 generated by the two order 3 elements stabilising the edges meeting at the vertex.
• The stabiliser of Δ is trivial.

The elements σ and τ generate the stabiliser of a vertex. The link of this vertex can be identified with the spherical building of SL₃(F₂) and the stabiliser can be identified with the collineation group of the Fano plane generated by a 3-fold symmetry σ fixing a point and a cyclic permutation τ of all 7 points, satisfying στ = τ²σ. Identifying F₈* with the Fano plane, σ can be taken to be the restriction of the Frobenius automorphism σ(x) = x²² of F₈ and τ to be multiplication by any element not in the prime field F₂, i.e. an order 7 generator of the cyclic multiplicative group of F₈. This Frobenius group acts simply transitively on the 21 flags in the Fano plane, i.e. lines with marked points. The formulas for σ and τ on E thus "lift" the formulas on F₈.

Mumford also obtains an action simply transitive on the vertices of the building by passing to a subgroup of Γ₁ = <ρ, σ, τ, −I>. The group Γ₁ preserves the Q(α)-valued Hermitian form

f(x,y) = xy* + σ(xy*) + σ²(xy*)

on Q(ζ) and can be identified with U₃(f) ${\displaystyle \cap }$ GL₃(S) where S = Z[α,½]. Since S/(α) = F₇, there is a homomorphism of the group Γ₁ into GL₃(F₇). This action leaves invariant a 2-dimensional subspace in F₇³ and hence gives rise to a homomorphism Ψ of Γ₁ into SL₂(F₇), a group of order 16·3·7. On the other hand, the stabiliser of a vertex is a subgroup of order 21 and Ψ is injective on this subgroup. Thus if the congruence subgroup Γ₀ is defined as the inverse image under Ψ of the 2-Sylow subgroup of SL₂(F₇), the action of Γ₀ on vertices must be simply transitive.

Generalizations

Other examples of triangles or 2-dimensional complexes of groups can be constructed by variations of the above example.

Cartwright et al. consider actions on buildings that are simply transitive on vertices. Each such action produces a bijection (or modified duality) between the points x and lines x* in the flag complex of a finite projective plane and a collection of oriented triangles of points (x,y,z), invariant under cyclic permutation, such that x lies on z*, y lies on x* and z lies on y* and any two points uniquely determine the third. The groups produced have generators x, labelled by points, and relations xyz = 1 for each triangle. Generically this construction will not correspond to an action on a classical affine building.

More generally, as shown by Ballmann and Brin, similar algebraic data encodes all actions that are simply transitively on the vertices of a non-positively curved 2-dimensional simplicial complex, provided the link of each vertex has girth at least 6. This data consists of:

• a generating set S containing inverses, but not the identity;
• a set of relations g h k = 1, invariant under cyclic permutation.

The elements g in S label the vertices g·v in the link of a fixed vertex v; and the relations correspond to edges (g⁻¹·v, h·v) in that link. The graph with vertices S and edges (g, h), for g⁻¹h in S, must have girth at least 6. The original simplicial complex can be reconstructed using complexes of groups and the second barycentric subdivision.

The bipartite Heawood graph

Further examples of non-positively curved 2-dimensional complexes of groups have been constructed by Swiatkowski based on actions simply transitive on oriented edges and inducing a 3-fold symmetry on each triangle; in this case too the complex of groups is obtained from the regular action on the second barycentric subdivision. The simplest example, discovered earlier with Ballmann, starts from a finite group H with a symmetric set of generators S, not containing the identity, such that the corresponding Cayley graph has girth at least 6. The associated group is generated by H and an involution τ subject to (τg)³ = 1 for each g in S.

In fact, if Γ acts in this way, fixing an edge (v, w), there is an involution τ interchanging v and w. The link of v is made up of vertices g·w for g in a symmetric subset S of H = Γ_v, generating H if the link is connected. The assumption on triangles implies that

τ·(g·w) = g⁻¹·w

for g in S. Thus, if σ = τg and u = g⁻¹·w, then

σ(v) = w, σ(w) = u, σ(u) = w.

By simple transitivity on the triangle (v, w, u), it follows that σ³ = 1.

The second barycentric subdivision gives a complex of groups consisting of singletons or pairs of barycentrically subdivided triangles joined along their large sides: these pairs are indexed by the quotient space S/~ obtained by identifying inverses in S. The single or "coupled" triangles are in turn joined along one common "spine". All stabilisers of simplices are trivial except for the two vertices at the ends of the spine, with stabilisers H and <τ>, and the remaining vertices of the large triangles, with stabiliser generated by an appropriate σ. Three of the smaller triangles in each large triangle contain transition elements.

When all the elements of S are involutions, none of the triangles need to be doubled. If H is taken to be the dihedral group D₇ of order 14, generated by an involution a and an element b of order 7 such that

ab= b⁻¹a,

then H is generated by the 3 involutions a, ab and ab⁵. The link of each vertex is given by the corresponding Cayley graph, so is just the bipartite Heawood graph, i.e. exactly the same as in the affine building for SL₃(Q₂). This link structure implies that the corresponding simplicial complex is necessarily a Euclidean building. At present, however, it seems to be unknown whether any of these types of action can in fact be realised on a classical affine building: Mumford's group Γ₁ (modulo scalars) is only simply transitive on edges, not on oriented edges.

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 quotient of Rⁿ by a finite group, i.e. Rⁿ/Γ. In physics, the notion of an orbifold usually describes an object that can be globally written as an orbit space 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. Open strings attached to these D-branes have no twisted sector, and so the number of open string states is reduced by the orbifolding procedure.

More specifically, when the orbifold group G is a discrete subgroup of spacetime isometries, then if it has no fixed point, the result is usually a compact smooth space; the twisted sector consists of closed strings wound around the compact dimension, which are called winding states.

When the orbifold group G is a discrete subgroup of spacetime isometries, and it has fixed points, then these usually have conical singularities, because Rⁿ/Z_k has such a singularity at the fixed point of Z_k. In string theory, gravitational singularities are usually a sign of extra degrees of freedom which are located at a locus point in spacetime. In the case of the orbifold these degrees of freedom are the twisted states, which are strings "stuck" at the fixed points. When the fields related with these twisted states acquire a non-zero vacuum expectation value, the singularity is deformed, i.e. the metric is changed and becomes regular at this point and around it. An example for a resulting geometry is the Eguchi–Hanson spacetime.

From the point of view of D-branes in the vicinity of the fixed points, the effective theory of the open strings attached to these D-branes is a supersymmetric field theory, whose space of vacua has a singular point, where additional massless degrees of freedom exist. The fields related with the closed string twisted sector couple to the open strings in such a way as to add a Fayet–Iliopoulos term to the supersymmetric field theory Lagrangian, so that when such a field acquires a non-zero vacuum expectation value, the Fayet–Iliopoulos term is non-zero, and thereby deforms the theory (i.e. changes it) so that the singularity no longer exists , .

Music theory

Beyond their manifold and various applications in mathematics and physics, orbifolds have been applied to music theory at least as early as 1985 in the work of Guerino Mazzola^[26][27] and later by Dmitri Tymoczko and collaborators (Tymoczko 2006) and (Callender & Tymoczko 2008) harv error: no target: CITEREFCallenderTymoczko2008 (help).^[28][29] One of the papers of Tymoczko was the first music theory paper published by the journal Science.^[30][31][32] Mazzola and Tymoczko have participated in debate regarding their theories documented in a series of commentaries available at their respective web sites.^[33][34]

Animated slices of the three-dimensional orbifold ${\displaystyle T^{3}/S_{3}}$. Slices of cubes standing on end (with their long diagonals perpendicular to the plane of the image) form colored Voronoi regions (colored by chord type) which represent the three-note chords at their centers, with augmented triads at the very center, surrounded by major and minor triads (lime green and navy blue). The white regions are degenerate trichords (one-note repeated three times), with the three lines (representing two note chords) connecting their centers forming the walls of the twisted triangular prism, 2D planes perpendicular to plane of the image acting as mirrors.

Tymoczko models musical chords consisting of n notes, which are not necessarily distinct, as points in the orbifold ${\displaystyle T^{n}/S_{n}}$ – the space of n unordered points (not necessarily distinct) in the circle, realized as the quotient of the n-torus ${\displaystyle T^{n}}$ (the space of n ordered points on the circle) by the symmetric group ${\displaystyle S_{n}}$ (corresponding from moving from an ordered set to an unordered set).

Musically, this is explained as follows:

• Musical tones depend on the frequency (pitch) of their fundamental, and thus are parametrized by the positive real numbers, R⁺.
• Musical tones that differ by an octave (a doubling of frequency) are considered the same tone – this corresponds to taking the logarithm base 2 of frequencies (yielding the real numbers, as ${\displaystyle \mathbf {R} =\log _{2}\mathbf {R} ^{+}}$), then quotienting by the integers (corresponding to differing by some number of octaves), yielding a circle (as ${\displaystyle S^{1}=\mathbf {R} /\mathbf {Z} }$).
• Chords correspond to multiple tones without respect to order – thus t notes (with order) correspond to t ordered points on the circle, or equivalently a single point on the t-torus ${\displaystyle T^{t}:=S^{1}\times \cdots \times S^{1},}$ and omitting order corresponds to taking the quotient by ${\displaystyle S_{t},}$ yielding an orbifold.

For dyads (two tones), this yields the closed Möbius strip; for triads (three tones), this yields an orbifold that can be described as a triangular prism with the top and bottom triangular faces identified with a 120° twist (a ⅓ twist) – equivalently, as a solid torus in 3 dimensions with a cross-section an equilateral triangle and such a twist.

The resulting orbifold is naturally stratified by repeated tones (properly, by integer partitions of t) – the open set consists of distinct tones (the partition ${\displaystyle t=1+1+\cdots +1}$), while there is a 1-dimensional singular set consisting of all tones being the same (the partition ${\displaystyle t=t}$), which topologically is a circle, and various intermediate partitions. There is also a notable circle which runs through the center of the open set consisting of equally spaced points. In the case of triads, the three side faces of the prism correspond to two tones being the same and the third different (the partition ${\displaystyle 3=2+1}$), while the three edges of the prism correspond to the 1-dimensional singular set. The top and bottom faces are part of the open set, and only appear because the orbifold has been cut – if viewed as a triangular torus with a twist, these artifacts disappear.

Tymoczko argues that chords close to the center (with tones equally or almost equally spaced) form the basis of much of traditional Western harmony, and that visualizing them in this way assists in analysis. There are 4 chords on the center (equally spaced under equal temperament – spacing of 4/4/4 between tones), corresponding to the augmented triads (thought of as musical sets) C♯FA, DF♯A♯, D♯GB, and EG♯C (then they cycle: FAC♯ = C♯FA), with the 12 major chords and 12 minor chords being the points next to but not on the center – almost evenly spaced but not quite. Major chords correspond to 4/3/5 (or equivalently, 5/4/3) spacing, while minor chords correspond to 3/4/5 spacing. Key changes then correspond to movement between these points in the orbifold, with smoother changes effected by movement between nearby points.