higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
∞-Lie theory (higher geometry)
Where an ordinary manifold is a space locally modeled on Cartesian spaces, an orbifold is, more generally, a space that is locally modeled on smooth action groupoids (=“stack quotients”) of a finite group acting on a Cartesian space.
This turns out to be equivalent (Moerdijk, Moerdijk-Pronk) to saying that an orbifold is a proper étale Lie groupoid. (Morita equivalent Lie groupoids correspond to the same orbifolds.)
The word orbifold was invented in (Thurston 1992), while the original name was $V$-manifold (Satake), and was taken in a more restrictive sense, assuming that the actions of finite groups on the charts are always effective. Nowdays we call such orbifolds effective and those which are global quotients by a finite group global quotient orbifolds.
There is also a notion of finite stabilizers in algebraic geometry. A singular variety is called an (algebraic) orbifold if it has only so-called orbifold singularities.
An orbifold is a stack presented by an orbifold groupoid.
One can consider a bicategory of proper étale Lie groupoids and the orbifolds will be the objects of certain bicategorical localization of this bicategory (a result of Moerdijk and Pronk).
Equivalently, every orbifold is globally a quotient of a smooth manifold by an action of finite-dimensional Lie group with finite stabilizers in each point. (eg (Adem-Leida-Ruan 2007), Corollary 1.24)
In (ALR 07, theorem 1.23) is asserted that every effective orbifold $X$ (paracompact, Hausdorff) is isomorphic to a global quotient orbifold, specifically to a global quotient of $O(n)$ (where $n$ is the dimension of $X$) acting on the frame bundle of $X$ (which is a manifold).
It has been noticed that the topological invariants of the underlying topological space of an orbifold as a topological space with an orbifold structure are not appropriate, but have to be corrected leading to orbifold Euler characteristics, orbifold cohomology etc. One of the constructions which is useful in this respect is the inertia orbifold (the inertia stack of the original orbifold) which gives rise to “twisted sectors” in Hilbert space of a quantum field theory on the orbifold, and also to twisted sectors in the appropriate cohomology spaces. A further generalization gives multitwisted sectors.
Some basic building blocks of orbifolds:
The quotient of a ball by a discrete subgroup of the special orthogonal group of rotations. Is an orbifold, and orbifolds may be obtained by cutting out balls from ordinary smooth manifolds and gluing in these orbifold quotients.
The moduli stack of elliptic curves over the complex numbers is an orbifold, being the homotopy quotient of the upper half plane by the special linear group acting by Möbius transformations.
For $\mathcal{G}$ any orbifold, then the mapping space $\mathcal{G}^{\Pi(S^1)} = \mathcal{G}^{B\mathbb{Z}}$ is again an orbifold, called the inertia orbifold.
Orbifolds are in differential geometry what Deligne-Mumford stacks are in algebraic geometry. See also at geometric invariant theory and GIT-stable point.
If the finiteness condition is dropped one also speaks of orbispaces and generally of stacks.
Orbifolds may be regarded as a kind of stratified spaces.
See also
Orbifolds in string theory:
Original sources on orbifolds include
I. Satake, The Gauss–Bonnet theorem for $V$-manifolds, J. Math. Soc. Japan 9 (1957), 464–492.
William Thurston, Three-dimensional geometry and topology, preliminary draft, University of Minnesota, Minnesota, (1992) which in completed and revised form is available as his book: The Geometry and Topology of Three-Manifolds; in particular the orbifold discussion is in chapter 13.
Discussion of orbifold as Lie groupoids/differentiable stacks is in
Ieke Moerdijk, Orbifolds as Groupoids: an Introduction (arXiv:math.DG/0203100)
Ieke Moerdijk, Dorette Pronk, Orbifolds, sheaves and groupoids, K-theory 12 3-21 (1997) (pdf)
Eugene Lerman, Orbifolds as stacks? (arXiv)
The mapping stacks of orbifolds are discussed in
Orbifolds often appear as moduli spaces in differential geometric setting:
The generalization of orbifolds to weighted branched manifolds is discussed in
A reference dealing with the string topology of orbifolds is
See also
(which is mainly tailored toward Thurston’s approach).
Orbifold cobordisms are discussed in
K. S. Druschel, Oriented Orbifold Cobordism, Pacific J. Math., 164(2) (1994), 299-319.
K. S. Druschel, The Cobordism of Oriented Three Dimensional Orbifolds, Pacific J. Math., bf 193(1) (2000), 45-55.
Andres Angel, Orbifold cobordism (pdf)
See also at orbifold cobordism.
Orbifolds as target spaces for a string sigma-model were first considered in
and then further developed notably in
For topological strings the path integral as a pull-push transform for target orbifolds – in analogy to what Gromov-Witten theory is for Deligne-Mumford stacks – has first been considered in
A review with further pointers is in