higher geometry / derived geometry
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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 $(\widehat{U}_i, G)$ is equipped with a Riemannian metric such that the action of $G$ 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 $\mathbb{T}^n \sslash G$ of the n-torus $\mathbb{T}^n$ 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 $\mathbb{R}^n$
Basic examples of non-compact Riemannian orbifolds are conical singularities.
In the flat case these are homotopy quotients of the form $V\sslash G$ for $G$ a finite group and $V \in RO(G)$ a finite-dimensional orthogonal linear representation of $G$.
graphics grabbed from Blumenhagen-Lüst-Theisen 13
For $V = \mathbb{H}$ equipped with the canonical action of finite subgroups of SU(2) these are the ADE-singularities.
(compact flat orbifolds from crystallographic groups)
Let $E$ be a Euclidean space and $S \subset Iso(E)$ a crystallographic group acting on it, with translational normal subgroup lattice $N \subset S$ and corresponding point group $G = S/N$.
Then the action of $G$ on $E$ descends to the quotient space torus $E/N$ (this Prop.)
The resulting homotopy quotient $(E/N)\sslash G$ 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 $\mathbb{T}^d \coloneqq \mathbb{R}^d / \mathbb{Z}^d$ be the d-torus and consider the action of the cyclic group $\mathbb{Z}_2$ by canonical coordinate reflection
The resulting homotopy quotient orbifold $\mathbb{T}^d\sslash\mathbb{Z}_2$ has $2^d$ singularities/fixed points, namely the points with all coordinates in $\{0\,,\, 1/2\, \mathrm{mod} \mathbb{Z}\}$.
In applications to string theory orbifolds of the form $\mathbb{R}^{p,1} \times \mathbb{T}^d\sslash \mathbb{Z}_2$ play the role of orientifold spacetimes with $2^d$ Op-planes.
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
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 $T^4 \sslash \mathbb{Z}_2$ – the special case of Example for $d = 4$. This is a singular K3-surface (e.g. Bettiol-Derdzinski-Piccione 18, 5.5)
graphics grabbed from Snowden 11
see FRTV 12
see G2-orbifold
Joseph Ernest Borzellino, Riemannian Geometry of Orbifolds, 1992 (pdf 1, pdf 2)
John Ratcliffe, Geometric Orbifolds, chapter 13 in Foundations of Hyperbolic Manifolds, Graduate Texts in Mathematics 149, Springer 2006 (doi:10.1007/978-0-387-47322-2, pdf)
Christian Lange, Orbifolds from a metric viewpoint (arXiv:1801.03472)
Renato G. Bettiol, Andrzej Derdzinski, Paolo Piccione, Teichmüller theory and collapse of flat manifolds, Annali di Matematica (2018) 197: 1247 (arXiv:1705.08431, doi:10.1007/s10231-017-0723-7)
Discussion of gravity and maybe quantum gravity on orbifolds:
Discussion of perturbative string theory on toroidal orbifolds
For more see the references at orbifold.
In 2 dimensions
John Milnor, On Lattès Maps (arXiv:math/0402147)
João Guerreiro, Orbifolds and Wallpaper Patterns (pdf)
Flat (toroidal) orbifolds of dimension 4 are discussed in
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 black holes in string theory)
In the context of Mathieu moonshine from string sigma models on K3s:
Matthias Gaberdiel, Stefan Hohenegger, Roberto Volpato, Symmetries of K3 sigma models, Commun.Num.Theor.Phys. 6 (2012) 1-50 (arXiv:1106.4315)
Matthias Gaberdiel, Roberto Volpato, Mathieu Moonshine and Orbifold K3s (arXiv:1206.5143)
In 6 dimensions (mostly motivated as singular Calabi-Yau compactifications of heterotic string theory to 4d)
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 May 19, 2019 at 07:43:36. See the history of this page for a list of all contributions to it.