In topology and geometric group theory, an orbifold (for "orbit-manifold") is a generalization of a manifold. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of Geometric group theory is an area in Mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be It is a topological space (called the underlying space) with an orbifold structure (see below). The underlying space locally looks like the quotient space of a Euclidean space under the linear action of a finite group. In Topology and related areas of Mathematics, a quotient space (also called an identification space) is intuitively speaking the result of identifying In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that In Mathematics, a finite group is a group which has finitely many elements In mathematics definitions of orbifold have been given several times: by Satake in the context of automorphic forms in the 1950's under the name V-manifold^[1] ; by Thurston in the context of the geometry of 3-manifolds in the 1970's^[2] when he coined the name orbifold, after a vote by his students; and by Haefliger in the 1980's in the context of Gromov's programme on CAT(k) spaces under the name orbihedron. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, the general notion of automorphic form is the extension to Analytic functions perhaps of Several complex variables, of the theory of William Paul Thurston (born October 30, 1946) is an American Mathematician. In Mathematics, a 3-manifold is a 3-dimensional Manifold. The topological Piecewise-linear, and smooth categories are all equivalent in three dimensions Mikhail Leonidovich Gromov Russian Михаил Леонидович Громов (born December 23, 1943, also known In Mathematics, a CAT( k) space is a specific type of Metric space. ^[3] The definition of Thurston will be described here: it is the most widely used and is applicable in all cases.

Mathematically, orbifolds arose first as surfaces with singular points long before they were formally defined. ^[4] One of the first classical examples arose in the theory of modular forms^[5] with the action of the modular group SL(2,Z) on the upper half-plane: a version of the Riemann-Roch theorem holds after the quotient is compactified by the addition of two orbifold cusp points. In Mathematics, a modular form is a (complex Analytic function on the Upper half-plane satisfying a certain kind of Functional equation and In Mathematics, the modular group Γ is a fundamental object of study in Number theory, Geometry, algebra, and many other areas of advanced In Mathematics, the upper half-plane H is the set of Complex numbers \mathbb{H} = \{x + iy \| y > 0 x y \in \mathbb{R} \} In Mathematics, specifically in Complex analysis and Algebraic geometry, the Riemann–Roch theorem is an important tool in the computation of the dimension In 3-manifold theory, the theory of Seifert fiber spaces, initiated by Seifert, can be phrased in terms of 2-dimensional orbifolds. In Mathematics, a 3-manifold is a 3-dimensional Manifold. The topological Piecewise-linear, and smooth categories are all equivalent in three dimensions A Seifert fiber space is a 3-manifold together with a "nice" decomposition as a disjoint union of circles Herbert Karl Johannes Seifert ( May 27, 1907 – October 1, 1996) was a German Mathematician known for his work in Topology ^[6] In geometric group theory, post-Gromov, discrete groups have been studied in terms of the local curvature properties of orbihedra and their covering spaces. Geometric group theory is an area in Mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and Mikhail Leonidovich Gromov Russian Михаил Леонидович Громов (born December 23, 1943, also known ^[7]

In string theory, the word "orbifold" has a slightly different meaning,^[8] discussed in detail below. String theory is a still-developing scientific approach to Theoretical physics, whose original building blocks are one-dimensional extended objects called strings In conformal field theory, a mathematical part of string theory, it is often used to refer to the theory attached to the fixed point subalgebra of a vertex algebra under the action of a finite group of automorphisms. A conformal field theory (CFT is a Quantum field theory (or Statistical mechanics model at the Critical point) that is Invariant under

The main example of underlying space is a quotient space of a manifold under the properly discontinuous action of a possibly infinite group of diffeomorphisms with finite isotropy subgroups. In Topology and related branches of Mathematics, an action of a group G on a Topological space X is called properly In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, a diffeomorphism is an Isomorphism of Smooth manifolds It is an Invertible function that maps one Differentiable In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. ^[9] In particular this applies to any action of a finite group; thus a manifold with boundary carries a natural orbifold structure, since it is the quotient of its double by an action of Z₂. In Mathematics, a finite group is a group which has finitely many elements A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be This is a Glossary of terms specific to Differential geometry and Differential topology. Similarly the quotient space of a manifold by a smooth proper action of S¹ carries the structure of an orbifold.

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 has several usages in mathematics In mathematical logic In Mathematical logic, stratification is any consistent assignment of numbers

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. In Mathematics, the fundamental group is one of the basic concepts of Algebraic topology. In Mathematics, and more specifically in Algebraic topology and Polyhedral combinatorics, the Euler characteristic is a Topological invariant

Contents 1 Formal definitions 1.1 Examples 2 Orbifold fundamental group 3 Non-positively curved orbispaces 4 Complexes of groups 4.1 Definition 4.2 Example 4.3 Edge-path group 4.4 Developable complexes 5 Orbihedra 5.1 Definition 5.2 Main properties 6 Triangles of groups 6.1 Mumford's example 6.2 Generalizations 7 2-dimensional orbifolds 8 3-dimensional orbifolds 9 Orbifolds in string theory 9.1 Calabi-Yau manifolds 10 Notes 11 References

Formal definitions

Like a manifold, an orbifold is specified by local conditions; however, instead of being locally modelled on open subsets of Rⁿ, an orbifold is locally modelled on quotients of open subsets of Rⁿ 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. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups.

An n-dimensional orbifold is a Hausdorff topological space X, called the underlying space, with a covering by a collection of open sets U_i, closed under finite intersection. In Topology and related branches of Mathematics, a Hausdorff space, separated space or T2 space is a Topological space For each U_i, there is

• an open subset V_i of Rⁿ, invariant under a faithful linear action of a finite group Γ_i
• a continuous map φ_i of V_i onto U_i invariant under Γ_i, called an orbifold chart, which defines a homeomorphism between V_i / Γ_i and U_i.

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

• for each inclusion U_i U_j there is an injective group homomorphism f_ij : Γ_i Γ_j
• for each inclusion U_i 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. In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function 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

The orbifold atlas defines the orbifold structure completely: two orbifold atlases of X give the same orbifold structure if they can be consistently combined to give a larger orbifold atlas. 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. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. If U_i U_j 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)

ψ_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. In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M

For applications in geometric group theory, it often convenient to have a slightly more general notion of orbifold, due to Haefliger. Geometric group theory is an area in Mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and An orbispace is defined by replacing the model for the orbifold charts by a locally compact space with a rigid action of a finite group, i. In Topology and related branches of Mathematics, a Topological space is called locally compact if roughly speaking each small portion of the space looks e. one for which points with trivial isotropy are dense. (This condition is automatically satisfied by faithful linear actions, because the points fixed by any non-trivial group element form a proper linear subspace. The concept of a linear subspace (or vector subspace) is important in Linear algebra and related fields of Mathematics. ) It is also useful to consider metric space structures on an orbispace, given by invariant metrics on the orbispace charts for which the gluing maps preserve distance. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined In this case each orbispace chart is usually required to be a length space with unique geodesics connecting any two points. In the mathematical study of Metric spaces, one can consider the Arclength of paths in the space In the mathematical study of Metric spaces, one can consider the Arclength of paths in the space

Examples

• If M is a Riemannian n-manifold with a cocompact proper isometric action of a discrete group Γ, then the orbit space X = M/Γ is naturally an orbifold: for each x in X take a representative m in M and an open neighbourhood V_m of m invariant under the stabiliser Γ_m, identified equivariantly with a Γ_m-subset of T_mM under the exponential map at m; finitely many neighbourhoods cover X and each of their finite intersections, if non-empty, is covered by an intersection of Γ-translates g_m·V_m with corresponding group g_m Γ g_m^-1. In Mathematics, an action of a group G on a Topological space X is cocompact if the Quotient space X In Mathematics, a Continuous function between Topological spaces is called proper if Inverse images of compact subsets are compact Orbifolds that arise in this way are called developable or good.

• If N is a compact manifold with boundary, its double M can formed by gluing together a copy of N and its mirror image along their common boundary. There is natural reflection action of Z₂ on the manifold M fixing the common boundary; the quotient space can be identified with N, so that N has a natural orbifold structure.

• A classical theorem of Henri Poincaré constructs Fuchsian groups as hyperbolic reflection groups generated by reflections in the edges of a geodesic triangle in the hyperbolic plane for the Poincaré metric. Jules Henri Poincaré ( 29 April 1854 &ndash 17 July 1912) (ˈʒyl ɑ̃ˈʁi pwɛ̃kaˈʁe was a French Mathematician In Mathematics, a Fuchsian group is a particular type of group of isometries of the Hyperbolic plane. A reflection group is a Group action, acting on a finite dimensional Vector space, which is generated by reflections elements that fix a Hyperplane in In In Mathematics, the Poincaré metric, named after Henri Poincaré, is the Metric tensor describing a two-dimensional surface of constant negative Curvature If the triangle has angles π / n_i for positive integers n_i, the triangle is a fundamental domain and naturally a 2-dimensional orbifold. In Geometry, the fundamental domain of a Symmetry group of an object or pattern is a part of the pattern as small as possible which based on the Symmetry The corresponding group is an example of a hyperbolic triangle group. In Mathematics, a triangle group is a group that can be realized geometrically by sequences of reflections across the sides of a Triangle. Poincaré also gave a 3-dimensional version of this result for Kleinian groups: in this case the Kleinian group Γ is generated by hyperbolic reflections and the orbifold is H³ / Γ. In Mathematics, a Kleinian group, named after Felix Klein, is a finitely generated Discrete group &Gamma of orientation preserving conformal

• If M is closed 2-manifold, new orbifold structures can be defined on Mi by removing finitely many disjoint closed discs from M and gluing back copies of discs D/ Γ_i where D is the closed unit disc and Γ_i is a finite cyclic group of rotations. In Mathematics, the open unit disk around P (where P is a given point in the plane) is the set of points whose distance from P is This generalises Poincaré's construction.

Orbifold fundamental group

There are several ways to define the orbifold fundamental group. More sophisticated approaches use orbifold covering spaces or classifying spaces of groupoids. In Mathematics, a covering space is a Topological space C which "covers" another space X by a Surjective Local homeomorphism In Mathematics, a classifying space BG in Homotopy theory of a Topological group G is the quotient of a Weakly contractible In Mathematics, especially in Category theory and Homotopy theory The simplest approach (adopted by Haefliger and known also to Thurston) extends the usual notion of loop used in the standard definition of the fundamental group. In Mathematics, the fundamental group is one of the basic concepts of Algebraic topology. In Mathematics, the fundamental group is one of the basic concepts of Algebraic topology.

An orbifold path is a path in the underlying space provided with an explicit piecewise lift of path segments to orbifold charts and explicit group elements identifying paths in overlapping charts; if the underlying path is a loop, it is called an orbifold loop. Two orbifold paths are identified if they are related through multiplication by group elements in orbifold charts. The orbifold fundamental group is the group formed by homotopy classes of orbifold loops. In Topology, two continuous functions from one Topological space to another are called homotopic ( Greek homos = identical

If the orbifold arises as the quotient of a simply connected manifold M by a proper rigid action of a discrete group Γ, the orbifold fundamental group can be identified with Γ. In Topology, a geometrical object or space is called simply connected (or 1-connected) if it is Path-connected and every path between two points can be In general it is an extension of Γ by π₁ M. In Mathematics, a group extension is a general means of describing a group in terms of a particular Normal subgroup and Quotient group.

The orbifold is said to be developable or good if it arises as the quotient by a group action; otherwise it is called bad. A universal covering orbifold can be constructed for an orbifold by direct analogy with the construction of the universal covering space of a topological space, namely as the space of pairs consisting of points of the orbifold and homotopy classes of orbifold paths joining them to the basepoint. In Mathematics, the fundamental group is one of the basic concepts of Algebraic topology. This space is naturally an orbifold.

Note that if an orbifold chart on a contractible open subset corresponds to a group Γ, then there is a natural local homomorphism of Γ into the orbifold fundamental group. In Mathematics, a Topological space X is contractible if the Identity map on X is Null-homotopic, i

In fact the following conditions are equivalent:

• The orbifold is developable.
• The orbifold structure on the universal covering orbifold is trivial.
• The local homomorphisms are all injective for a covering by contractible open sets.

Non-positively curved orbispaces

Let X be an orbispace endowed with a metric space structure for which the charts are geodesic length spaces. The preceding definitions and results for orbifolds can be generalized to give definitions of orbispace fundamental group and universal covering orbispace, with analogous criteria for developability. The distance functions on the orbispace charts can be used to define the length of an orbispace path in the universal covering orbispace. If the distance function in each chart is non-positively curved, then the Birkhoff curve shortening argument can be used to prove that any orbispace path with fixed endpoints is homotopic to a unique geodesic. In Mathematics, a CAT( k) space is a specific type of Metric space. In Mathematics, specifically in Topology, a surface is a Two-dimensional Manifold. Applying this to constant paths in an orbispace chart, it follows that each local homomorphism is injective and hence:

• every non-positively curved orbispace is developable (i. e. good).

Complexes of groups

Every orbifold has associated with it an additional combinatorial structure given by a complex of 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 Γ_σ whenever σ τ
• for every inclusion ρ σ 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 π ρ σ τ. In Mathematics, an abstract simplicial complex is a purely combinatorial description of the geometric notion of a Simplicial complex, consisting of a family of In Mathematics, the adjoint representation (or adjoint action) of a Lie group G is the natural representation of G on its (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. In Geometry, the barycentric subdivision is a standard way of dividing an arbitrary convex Polygon into Triangles, a convex Polyhedron 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 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 ··· U_in. In Mathematics, the nerve of an open covering is a construction in Topology, of an Abstract simplicial complex from an Open covering of a 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 σ τ corresponding to intersections

U_i U_i U_j U_i U_j U_k

there are charts φ_i : V_i U_i, φ_ij : V_ij U_i U_j and φ_ijk : V_ijk U_i U_j U_k and gluing maps ψ : V _ij V_i, ψ' : V _ijk V_ij and ψ" : V _ijk 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. In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space. In Mathematics, a gerbe is a construct in Homological algebra and Topology. In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space. In Mathematics, sheaf cohomology is the aspect of Sheaf theory, concerned with sheaves of Abelian groups that applies Homological algebra to

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 Mathematics, the fundamental group is one of the basic concepts of Algebraic topology. In the barycentric subdivision of Y, take generators e_ij corresponding to edges from i to j where i j, so that there is an injection ψ_ij : Γ_i 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 j 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 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 (see Bredon 1972):

• 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, . In Topology and related branches of Mathematics, an action of a group G on a Topological space X is called properly In Mathematics, a simplicial complex is a Topological space of a particular kind constructed by "gluing together" points Line segments In Geometry, the fundamental domain of a Symmetry group of an object or pattern is a part of the pattern as small as possible which based on the Symmetry . . , 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 ths 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 subdivison 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 Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets

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 j;
• if i j and j k are edges, then i 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.

Orbihedra

If a countable discrete group acts by a regular simplicial proper action on a simplicial complex, the quotient can be given not only the structure of a complex of groups, but also that of an orbispace. In Topology and related branches of Mathematics, an action of a group G on a Topological space X is called properly In Mathematics, a simplicial complex is a Topological space of a particular kind constructed by "gluing together" points Line segments This leads more generally to the definition of "orbihedron", the simplicial analogue of an orbifold.

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. In Geometry, the link of a vertex of a 2- Dimensional Simplicial complex is a graph that encodes information about the local structure

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. In Mathematics, a simplicial complex is a Topological space of a particular kind constructed by "gluing together" points Line segments

• for each directed edge i j of X ', an injective homomorphism f_ij of Γ_i into Γ_j.
• for each directed edge i 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 j 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 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. In Mathematics, the simplicial approximation theorem is a foundational result for Algebraic topology, guaranteeing that Continuous mappings can be (by a In Mathematics, the fundamental group is one of the basic concepts of Algebraic topology. In Mathematics, a simplicial complex is a Topological space of a particular kind constructed by "gluing together" points Line segments In Mathematics, the fundamental group is one of the basic concepts of Algebraic topology.

• 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. In Mathematics, hyperbolic n -space, denoted H n, is the maximally symmetric Simply connected, n -dimensional

• 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. In Mathematics, spaces of non-positive curvature occur in many contexts and form a generalization of Hyperbolic geometry. 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. In Mathematics, a CAT( k) space is a specific type of Metric space.

• 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.

Triangles of groups

Historically one of the most important applications of orbifolds in geometric group theory has been to triangles of groups. Geometric group theory is an area in Mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and This is the simplest 2-dimensional example generalising the 1-dimensional "interval of groups" discussed in Serre's lectures on trees, where amalgamated free products are studied in terms of actions on trees. In Abstract algebra, the free product of groups constructs a group from two or more given ones Such triangles of groups arise any time a discrete group acts simply transitively on the triangles in the affine Bruhat-Tits building for SL₃(Q_p); in 1979 Mumford discovered the first example for p = 2 (see below) as a step in producing an algebraic surface not isomorphic to projective space, but having the same Betti numbers. In Mathematics, a building (also Tits building, Bruhat–Tits building) is a combinatorial and geometric structure which simultaneously generalizes certain David Bryant Mumford (born 11 June 1937) is a Mathematician known for distinguished work in Algebraic geometry, and then for research into In Mathematics, an algebraic surface is an Algebraic variety of dimension two In Mathematics a projective space is a set of elements constructed from a vector space such that a distinct element of the projective space consists of all non-zero vectors which In Algebraic topology, the Betti number of a Topological space is in intuitive terms a way of counting the maximum number of cuts that can be made without dividing Triangles of groups were worked out in detail by Gersten and Stallings, while the more general case of complexes of groups, described above, was developed independently by Haefliger. The underlying geometric method of analysing finitely presented groups in terms of metric spaces of non-positive curvature is due to Gromov. In this context triangles of groups correspond to non-positively curved 2-dimensional simplicial complexes with the regular action of a group, transitive on triangles.

A triangle of groups is a simple complex of groups consisting of a triangle with vertices A, B, C. There are groups

• Γ_A, Γ_B, Γ_C at each vertex
• Γ_BC, Γ_CA, Γ_AB for each edge
• Γ_ABC for the triangle itself

There is an injective homomorphisms of Γ_ABC into all the other groups and of an edge group Γ_XY into Γ_X and Γ_Y. The three ways of mapping Γ_ABC into a vertex group all agree. (Often Γ_ABC is the trivial group. ) The Euclidean metric structure on the corresponding orbispace is non-positively curved if and only if the link of each of the vertices in the orbihedron chart has girth at least 6.

This girth at each vertex is always even and, as observed by Stallings, can be described at a vertex A, say, as the length of the smallest word in the kernel of the natural homomorphism into Γ_A of the amalgamated free product over Γ_ABC of the edge groups Γ_AB and Γ_AC:

The result using the Euclidean metric structure is not optimal. In Abstract algebra, the free product of groups constructs a group from two or more given ones Angles α, β, γ at the vertices A, B and C were defined by Stallings as 2π divided by the girth. In the Euclidean case α, β, γ ≤ π/3. However, if it is only required that α + β + γ ≤ π, it is possible to identify the triangle with the corresponding geodesic triangle in the hyperbolic plane with the Poincaré metric (or the Euclidean plane if equality holds). In In Mathematics, the Poincaré metric, named after Henri Poincaré, is the Metric tensor describing a two-dimensional surface of constant negative Curvature It is a classical result from hyperbolic geometry that the hyperbolic medians intersect in the hyperbolic barycentre, ^[10] just as in the familiar Euclidean case. The barycentric subdivision and metric from this model yield a non-positively curved metric structure on the corresponding orbispace. Thus, if α+β+γ≤π,

• the orbispace of the triangle of groups is developable;
• the corresponding edge-path group, which can also be described as the colimit of the triangle of groups, is infinite;
• the homomorphisms of the vertex groups into the edge-path group are injections. In Category theory, a branch of Mathematics, the abstract notion of a limit captures the essential properties of universal constructions that are used in various parts

Mumford's example

Let α = be given by the binomial expansion of (1-8)^1/2 in Q₂ and set K = Q(α) Q₂. In Finite geometry, the Fano plane (after Gino Fano) is the Projective plane with the least number of points and lines 7 each In Mathematics, the binomial theorem is an important Formula giving the expansion of powers of Sums Its simplest version says 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. In Mathematics, a Galois group is a group associated with a certain type of Field extension.

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). In Topology and related branches of Mathematics, an action of a group G on a Topological space X is called properly In Mathematics, a building (also Tits building, Bruhat–Tits building) is a combinatorial and geometric structure which simultaneously generalizes certain 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. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups.
• 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. In Mathematics, a Frobenius group is a transitive Permutation group on a Finite set, such that no non-trivial elementfixes more than one point
• 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. In Geometry, the link of a vertex of a 2- Dimensional Simplicial complex is a graph that encodes information about the local structure A collineation is a one-to-one map from one Projective space to another or from a Projective plane onto itself such that the images of collinear points are themselves In Finite geometry, the Fano plane (after Gino Fano) is the Projective plane with the least number of points and lines 7 each Identifying F₈* with the Fano plane, σ can be taken to be the restriction of the Frobenius automorphism σ(x)=x² of F₈ and τ to be multiplication by any element not in the prime field F₂, i. In Commutative algebra and field theory, which are branches of Mathematics, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a In Mathematics, the characteristic of a ring R, often denoted char( R) is defined to be the smallest number of times one must add the ring's e. an order 7 generator of the cyclic multiplicative group of F₈. In Abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements 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>. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. 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) 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. In Mathematics, a congruence subgroup of a Matrix group with Integer entries is a Subgroup defined by congruence conditions on the entries In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage In Mathematics, specifically Group theory, the Sylow theorems, named after Ludwig Sylow, form a partial converse to Lagrange's theorem, which

Generalizations

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

Cartwright et all 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. In Mathematics, particularly in Linear algebra, a flag is an increasing sequence of Subspaces of a Vector space V. See Real projective plane and Complex projective plane, for the cases met as manifolds of respective dimension 2 and 4 In Mathematics 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.

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. In the mathematical field of Graph theory, the Heawood graph is an Undirected graph with 14 vertices and 21 edges 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. In Mathematics, the Cayley graph, also known as the Cayley colour graph, is the graph that encodes the structure of a Discrete group. 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. In Mathematics, a dihedral group is the group of symmetries of a Regular polygon, including both rotations and reflections The link of each vertex is given by the corresponding Cayley graph, so is just the bipartite Heawood graph, i. In the mathematical field of Graph theory, the Heawood graph is an Undirected graph with 14 vertices and 21 edges 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. In Mathematics, a building (also Tits building, Bruhat–Tits building) is a combinatorial and geometric structure which simultaneously generalizes certain 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.

2-dimensional orbifolds

In two dimensions, there are three singular point types of an orbifold:

• A boundary point
• An elliptic point of order n, such as the origin of R² quotiented out by a cyclic group of order n of rotations.
• A corner reflector of order n: the origin of R² quotiented out by a dihedral group of order 2n.

A compact 2-dimensional orbifold has an Euler characteristic Χ given by

Χ = Χ(X₀) − Σ(1 − 1/n_i )/2 − Σ(1 − 1/m_i )

where Χ(X₀) is the Euler characteristic of the underlying topological manifold X₀, and n_i are the orders of the corner reflectors, and m_i are the orders of the elliptic points. In Mathematics, and more specifically in Algebraic topology and Polyhedral combinatorics, the Euler characteristic is a Topological invariant

A 2-dimensional compact connected orbifold has a hyperbolic structure if its Euler characteristic is less than 0, a Euclidean structure if it is 0, and if its Euler characteristic is positive it is either bad or has an elliptic structure (an orbifold is called bad if it does not have a manifold as a covering space). In other words, its universal covering space has a hyperbolic, Euclidean, or spherical structure.

The compact 2-dimensional connected orbifolds that are not hyperbolic are listed in the table below. The 17 parabolic orbifolds are the quotients of the plane by the 17 wallpaper groups. A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern based on the

Type	Euler characteristic	Underlying 2-manifold	Orders of elliptic points	Orders of corner reflectors
Bad	1 + 1/n	Sphere	n > 1
Bad	1/m + 1/n	Sphere	n > m > 1
Bad	1/2 + 1/2n	Disk		n > 1
Bad	1/2m + 1/2n	Disk		n > m > 1
Elliptic	2	Sphere
Elliptic	2/n	Sphere	n,n
Elliptic	1/n	Sphere	2, 2, n
Elliptic	1/6	Sphere	2, 3, 3
Elliptic	1/12	Sphere	2, 3, 4
Elliptic	1/30	Sphere	2, 3, 5
Elliptic	1	Disc
Elliptic	1/n	Disc		n, n
Elliptic	1/2n	Disc		2, 2, n
Elliptic	1/12	Disc		2, 3, 3
Elliptic	1/24	Disc		2, 3, 4
Elliptic	1/60	Disc		2, 3, 5
Elliptic	1/n	Disc	n
Elliptic	1/2n	Disc	2	n
Elliptic	1/12	Disc	3	2
Elliptic	1	Projective plane
Elliptic	1/n	Projective plane	n
Parabolic	0	Sphere	2, 3, 6
Parabolic	0	Sphere	2, 4, 4
Parabolic	0	Sphere	3, 3, 3
Parabolic	0	Sphere	2, 2, 2, 2
Parabolic	0	Disk		2, 3, 6
Parabolic	0	Disk		2, 4, 4
Parabolic	0	Disk		3, 3, 3
Parabolic	0	Disk		2, 2, 2, 2
Parabolic	0	Disk	2	2, 2
Parabolic	0	Disk	3	3
Parabolic	0	Disk	4	2
Parabolic	0	Disk	2, 2
Parabolic	0	Projective plane	2, 2
Parabolic	0	Torus
Parabolic	0	Klein bottle
Parabolic	0	Annulus
Parabolic	0	Moebius band

3-dimensional orbifolds

A 3-manifold is said to be small if it is closed, irreducible and does not contain any incompressible surfaces.

Orbifold Theorem. Let M be a small 3-manifold. Let φ be a non-trivial periodic orientation-preserving diffeomorphism of M. Then M admits a φ-invariant hyperbolic or Seifert fibered structure.

This theorem is a special case of Thurston's geometrization theorem for 3-dimensional orbifolds, announced without proof in 1981; it forms part of his geometrization conjecture for 3-manifolds. Thurston's geometrization conjecture states that compact 3-manifolds can be decomposed into Submanifolds that have geometric structures In particular it implies that if X is a compact, connected, orientable, irreducible, atoroidal 3-orbifold with non-empty singular locus, then M has a geometric structure (in the sense of orbifolds). Complete proofs of the theorem have been given by Boileau, Leeb & Porti and Cooper, Hodgson & Kerckhoff.

Orbifolds in string theory

In string theory, the word "orbifold" has a slightly new meaning. String theory is a still-developing scientific approach to Theoretical physics, whose original building blocks are one-dimensional extended objects called strings 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. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be In Mathematics, a diffeomorphism is an Isomorphism of Smooth manifolds It is an Invertible function that maps one Differentiable 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. In quantum field theory (QFT the forces between particles are mediated by other particles 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. A string is one of the main objects of study in String theory, a branch of Theoretical physics. This article assumes some familiarity with Analytic geometry and the concept of a limit. 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. In String theory, D-branes are a class of extended objects upon which open strings can end with Dirichlet boundary conditions after which they are named In Physics, a quiver diagram is a graph representing the Matter content of a Gauge theory that describes D-branes on orbifolds. 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. In String theory, D-branes are a class of extended objects upon which open strings can end with Dirichlet boundary conditions after which they are named

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 windingstates.

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. A gravitational singularity (sometimes spacetime singularity) is approximately a place where quantities which are used to measure the Gravitational field become In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an In string theory, gravitational singularities are usually a sign of extra degrees of freedom which are located at 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. In Quantum field theory the vacuum expectation value (also called condensate) of an operator is its average Expected value in the vacuum e. the metric is changed and become regular at this point and around it. An example for a resulting geometry is the is Eguchi-Hanson spacetime. In Mathematical physics and Differential geometry, a gravitational instanton is a four-dimensional complete Riemannian manifold satisfying

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-Illiopolys term to the supersymmetric field theory Lagrangian, so that when such a field acquires a non-zero vacuum expectation value, the Fayet-Illiopolys term is non-zero, and thereby deforms the theory (i. In Quantum field theory the vacuum expectation value (also called condensate) of an operator is its average Expected value in the vacuum e. changes it) so that the singularity no longer exists [1], [2].

Calabi-Yau manifolds

Main article: Calabi-Yau manifold

In superstring theory,^[11][12] the construction of realistic phenomenological models requires dimensional reduction because the strings naturally propagate in a 10-dimensional space whilst the observed dimension of space-time of the universe is 4. In mathematics Calabi&ndashYau manifolds are compact Kähler manifolds whose Canonical bundle is trivial See also String theory Superstring theory is an attempt to explain all of the particles and Fundamental forces of nature in one theory by modelling This article is on dimensional reduction in physics For the statistics concept see Dimensionality reduction. SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS Formal constraints on the theories nevertheless place restrictions on the compactified space in which the extra "hidden" variables live: when looking for realistic 4-dimensional models with supersymmetry, the auxiliary compactified space must be a 6-dimensional Calabi-Yau manifold. In Particle physics, supersymmetry (often abbreviated SUSY) is a Symmetry that relates elementary particles of one spin to another particle that In mathematics Calabi&ndashYau manifolds are compact Kähler manifolds whose Canonical bundle is trivial

There are a large number of possible Calabi-Yau manifolds (tens of thousands), whence the use of the term "swampland" in the current theoretical physics literature to describe the baffling choice. The general study of Calabi-Yau manifolds is mathematically complex and for a long time examples have been hard to construct explicitly. Orbifolds have therefore proved very useful since they automatically satisfy the constraints imposed by supersymmetry. They provide degenerate examples of Calabi-Yau manifolds due to their singular points, but this is completely acceptable from the point of view of theoretical physics. In Mathematics, a singularity is in general a point at which a given mathematical object is not defined or a point of an exceptional set where it fails to be Such orbifolds are called "supersymmetric": they are technicaly easier to study than general Calabi-yau manifolds. It is very often possible to associate a continuous family of non-singular Calabi-Yau manifolds to a singular supersymmetric orbifold. In 4 dimensions this can be illustrated using complex K3 surfaces:

• Every K3 surface admits 16 cycles of dimension 2 that are topologically equivalent to usual 2-spheres. In Mathematics, in the field of Complex manifolds a K3 surface is an important and interesting example of a compact complex surface ( Complex dimension Making the surface of these spheres tend to zero, the K3 surface develops 16 singularities. This limit represents a point on the boundary of the moduli space of K3 surfaces and corresponds to the orbifold obtained by taking the quotient of the torus by the symmetry of inversion. In Algebraic geometry, a moduli space is a geometric space (usually a scheme or an Algebraic stack) whose points represent algebro-geometric objects of

The study of Calabi-Yau manifolds in string theory and the duality between different models of string theory (type IIA and IIB) led to the idea of mirror symmetry in 1988. Mirror symmetry may refer to Mirror symmetry (string theory, a relation between two Calabi-Yau manifolds in string theory Homological mirror The role of orbifolds was first pointed out by Dixon, Harvey, Vafa and Witten around the same time.

Notes

References

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