Riemannian orbifold



Riemannian geometry

Higher geometry



The concept of Riemannian orbifolds is the joint generalization of the concepts of Riemannian manifolds and orbifolds:

A Riemannian orbifold is an orbifold equipped with an orbifold atlas where each chart (U^ i,G)(\widehat{U}_i, G) is equipped with a Riemannian metric such that the action of GG is by isometries, and such that the transition functions from one chart to the other are isometries.

A key aspect is that the orbifold singularities behave like carrying singular curvature, notably there are flat orbifolds (also “Euclidean orbifolds”, i.e. Riemannian orbifolds with vanishing Riemann curvature away from the singularities) whose underlying topological spaces are n-spheres (see below).

Key examples of flat orbifolds are global homotopy quotients 𝕋 nG\mathbb{T}^n \sslash G of the n-torus 𝕋 n\mathbb{T}^n equipped with its canonical flat Riemannian metric. These flat orbifolds are called toroidal orbifolds.


under construction

Every flat orbifold whose underlying metric space is connected and complete) is a global quotient of Euclidean space/Cartesian space n\mathbb{R}^n

(Ratcliffe 06, 13.3.10)


Non-compact orbifolds

Basic examples of non-compact Riemannian orbifolds are conical singularities.

In the flat case these are homotopy quotients of the form VGV\sslash G for GG a finite group and VRO(G)V \in RO(G) a finite-dimensional orthogonal linear representation of GG.

graphics grabbed from Blumenhagen-Lüst-Theisen 13

For V=V = \mathbb{H} equipped with the canonical action of finite subgroups of SU(2) these are the ADE-singularities.

Compact flat orbifolds from crystallographic groups


(compact flat orbifolds from crystallographic groups)

Let EE be a Euclidean space and SIso(E)S \subset Iso(E) a crystallographic group acting on it, with translational normal subgroup lattice NSN \subset S and corresponding point group G=S/NG = S/N.

1 1 normal subgrouplattice of translations N E translationgroup crystallographicgroup S Iso(E) Euclideanisometry group pointgroup G O(E) orthogonalgroup 1 1 \array{ & 1 && 1 \\ & \downarrow && \downarrow \\ {\text{normal subgroup} \atop \text{lattice of translations}} & N &\subset& E & {\text{translation} \atop \text{group}} \\ & \big\downarrow && \big\downarrow \\ {\text{crystallographic} \atop \text{group}} & S &\subset& Iso(E) & {\text{Euclidean} \atop \text{isometry group}} \\ & \big\downarrow && \big\downarrow \\ {\text{point} \atop \text{group}} & G &\subset& O(E) & {\text{orthogonal} \atop \text{group}} \\ & \downarrow && \downarrow \\ & 1 && 1 }

Then the action of GG on EE descends to the quotient space torus E/NE/N (this Prop.)

E g E E/N g E/N \array{ E &\overset{g}{\longrightarrow}& E \\ \big\downarrow && \big\downarrow \\ E/N &\underset{g}{\longrightarrow}& E/N }

The resulting homotopy quotient (E/N)G(E/N)\sslash G is a compact flat orbifold.

The following is the class of special cases of Example for point group being the involution-action by reflection at a point:


(coordinate reflection on n-torus)

Let 𝕋 d d/ d\mathbb{T}^d \coloneqq \mathbb{R}^d / \mathbb{Z}^d be the d-torus and consider the action of the cyclic group 2\mathbb{Z}_2 by canonical coordinate reflection

2×𝕋 d 𝕋 d (σ,x) x. \array{ \mathbb{Z}_2 \times \mathbb{T}^d &\longrightarrow& \mathbb{T}^d \\ (\sigma, \vec x) &\mapsto& - \vec x } \,.

The resulting homotopy quotient orbifold 𝕋 d 2\mathbb{T}^d\sslash\mathbb{Z}_2 has 2 d2^d singularities/fixed points, namely the points with all coordinates in {0,1/2mod}\{0\,,\, 1/2\, \mathrm{mod} \mathbb{Z}\}.

In applications to string theory orbifolds of the form p,1×𝕋 d 2\mathbb{R}^{p,1} \times \mathbb{T}^d\sslash \mathbb{Z}_2 play the role of orientifold spacetimes with 2 d2^d Op-planes.

Flat compact 2-dimensional orbifolds

In 2 dimensions the crystallographic groups are the “wallpaper groups”. Hence, as a special case of Example , the flat compact 2-dimensional orbifolds may be classified as homotopy quotients of the 2-torus by wallpaper groups (for review see e.g. Guerreiro 09):

graphics grabbed from Bettiol-Derdzinski-Piccione 18

Flat compact 4-dimensional orbifolds

The orbifold quotient of the 4-torus by the sign involution on all four canonical coordinates is the flat compact 4-dimensional orbifold known as the Kummer surface T 4 2T^4 \sslash \mathbb{Z}_2 – the special case of Example for d=4d = 4. This is a singular K3-surface (e.g. Bettiol-Derdzinski-Piccione 18, 5.5)

graphics grabbed from Snowden 11

Also 𝕋 4 4\mathbb{T}^4\sslash\mathbb{Z}_4 gives a toroidal orbifold. As orientifolds with D5-branes in type IIB string theory these are discussed in Buchel-Shiu-Tye 99, Sec. II.

Flat compact 6-dimensional orbifolds

see FRTV 12

7-Dimensional G 2G_2-orbifolds

see G2-orbifold



Discussion of gravity and maybe quantum gravity on orbifolds:

  • Helio V. Fagundes, Teofilo Vargas, Orbifolds, Quantum Cosmology, and Nontrivial Topology (arXiv:gr-qc/0611048)

Discussion of perturbative string theory on toroidal orbifolds

For more see the references at orbifold.

Flat orbifolds

Of dimension 2

In 2 dimensions

2d toroidal orientifolds:

Of dimension 4

Flat (toroidal) orbifolds of dimension 4:

In the the context of black holes in string theory:

  • Justin R. David, Gautam Mandal, Spenta R. Wadia, Microscopic Formulation of Black Holes in String Theory, Phys.Rept.369:549-686,2002 (arXiv:hep-th/0203048)

In the context of RR-field tadpole cancellation for intersecting D-brane models on toroidal orientifolds:

Specifically K3 orientifolds (𝕋 4/G ADE\mathbb{T}^4/G_{ADE}) in type IIB string theory, hence for D9-branes and D5-branes:

Specifically K3 orientifolds (𝕋 4/G ADE\mathbb{T}^4/G_{ADE}) in type IIA string theory, hence for D8-branes and D4-branes:

In the context of Mathieu moonshine from string sigma models on K3s:

Of dimension 6

In 6 dimensions (mostly motivated as singular Calabi-Yau compactifications of heterotic string theory to 4d)

  • Jens Erler, Albrecht Klemm, Comment on the Generation Number in Orbifold Compactifications, Commun. Math. Phys. 153:579-604, 1993 (arXiv:hep-th/9207111)

  • Dieter Lüst, S. Reffert, E. Scheidegger, S. Stieberger, Resolved Toroidal Orbifolds and their Orientifolds, Adv.Theor.Math.Phys.12:67-183, 2008 (arXiv:hep-th/0609014)

  • S. Reffert, Toroidal Orbifolds: Resolutions, Orientifolds and Applications in String Phenomenology (arXiv:hep-th/0609040)

  • Ron Donagi, Katrin Wendland, On orbifolds and free fermion constructions, J. Geom. Phys. 59:942-968, 2009 (arXiv:0809.0330)

  • Maximilian Fischer, Michael Ratz, Jesus Torrado, Patrick K.S. Vaudrevange, Classification of symmetric toroidal orbifolds, JHEP 1301 (2013) 084 (arXiv:1209.3906)

Last revised on September 15, 2019 at 15:52:37. See the history of this page for a list of all contributions to it.