nLab orbifold



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An orbifold is much like a smooth manifold but possibly with singularities of the form of fixed points of finite group-actions.

Where a smooth manifold is a space locally modeled on Cartesian spaces/Euclidean spaces n\mathbb{R}^n, an orbifold is, more generally, a space that is locally modeled on smooth action groupoids (homotopy quotients) nG\mathbb{R}^n\sslash G of a finite group GG acting on a Cartesian space.

This turns out to be broadly captured (Moerdijk-Pronk 97, Moerdijk 02) by saying that an orbifold is a proper étale Lie groupoid. (Morita equivalent Lie groupoids correspond to the same orbifolds.)

The word orbifold was introduced in (Thurston 1992), while the original name was VV-manifold (Satake), and was taken in a more restrictive sense, assuming that the actions of finite groups on the charts are always effective. Nowadays these are called effective orbifolds and those which are global quotients by a finite group are 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.



Global quotient orbifolds

In (ALR 07, theorem 1.23) it is asserted that every effective orbifold XX (paracompact, Hausdorff) is isomorphic to a global quotient orbifold, specifically to a global quotient of O(n)O(n) (where nn is the dimension of XX) acting on the frame bundle of XX (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.


Orbifolds are in differential geometry what Deligne-Mumford stacks are in algebraic geometry. See also at geometric invariant theory and GIT-stable point.

Orbifolds may be regarded as a kind of stratified spaces.

See also

Orbifolds in string theory:



The original articles:

and specifically for orbifolds in complex geometry:

For careful comparative review of the definitions in these original articles see IKZ 10.

Survey of basic orbifold theory:

See also

On good orbifolds:

Textbook account:

Application to moduli spaces of curves and moduli spaces of Riemann surfaces:

On Riemannian orbifolds:

  • 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)

  • S. T. Hyde, S. J. Ramsden and V. Robins, Unification and classification of two-dimensional crystalline patterns using orbifolds, Acta Cryst. (2014). A70, 319-337 (doi:10.1107/S205327331400549X)

Survey of applications in mathematical physics and notably in string theory:

Orbifolds often appear as moduli spaces in differential geometric setting:

The generalization of orbifolds to weighted branched manifolds is discussed in

  • Dusa McDuff, Groupoids, branched manifolds and multisections, J. Symplectic Geom. Volume 4, Number 3 (2006), 259-315 (project euclid).

On orbifolds, orbifold cohomology and specifically on Chen-Ruan cohomology and orbifold K-theory:

As Lie groupoids

Discussion of orbifolds as Lie groupoids/differentiable stacks:


  • Alexander Amenta, The Geometry of Orbifolds via Lie Groupoids, ANU 2012 (arXiv:1309.6367)

Analogous discussion for topological orbifolds as topological stacks:

  • Vesta Coufal, Dorette Pronk, Carmen Rovi, Laura Scull, Courtney Thatcher, Orbispaces and their Mapping Spaces via Groupoids: A Categorical Approach, Contemporary Mathematics 641 (2015): 135-166 (arXiv:1401.4772)

Discussion of the corresponding perspective in algebraic geometry, via Deligne-Mumford stacks:

  • Andrew Kresch, On the geometry of Deligne-Mumford stacks (doi:10.5167/uzh-21342, pdf), in: D. Abramovich, A. Bertram, L. Katzarkov, R. Pandharipande, M. Thaddeus (eds.) Algebraic Geometry: Seattle 2005, Proceedings of Symposia in Pure Mathematics 80, Providence, Rhode Island: American Mathematical Society 2009, 259-271 (pspum-80-1)

The mapping stacks of orbifolds are discussed in

Discussion of principal bundles and fiber bundles over orbifolds:

An expected relation of orbifolds (orbispaces) to global equivariant homotopy theory:

As diffeological spaces

On orbifolds regarded as naive local quotient spaces (instead of homotopy quotients/Lie groupoids/differentiable stacks) but as such formed in diffeological spaces:

and as stratified diffeological spaces:

On this approach seen in the broader context of cohesive higher differential geometry:

Orbifold cobordism

Orbifold cobordisms are discussed in

  • K. S. Druschel, Oriented Orbifold Cobordism, Pacific J. Math., 164(2) (1994), 299-319 (doi:10.2140/pjm.1994.164.299, pdf)

  • 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.

tangential structure on orbifolds (in the context of factorization homology):

In string theory

(from Green 86)

In perturbative string theory, orbifolds as target spaces for a string sigma-model were first considered in

and then further developed notably in

See also:

For more references on orbifolds in string theory see also at

Review of heterotic string phenomenology on orbifolds:

Discussion of blow-up of orbifold singularities in string theory:

In terms of vertex operator algebras:

Review of orbifolds in the context of string KK-compactifications and intersecting D-brane models:

and for orbifolds of G₂-manifolds for M-theory on G₂-manifolds:

and for heterotic string phenomenology:

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

Review with further pointers:

On non-supersymmetric flat orbifolds of supergravity theories:

  • Anamaria Font, Alexis Hernandez, Non-Supersymmetric Orbifolds, Nucl. Phys. B 634 (2002) 51-70 [arXiv:hep-th/0202057]

and specifically fluxed KK-compactification of D=6 supergravity on the pillowcase orbifold:

On supergravity KK-compactified (and branes wrapped on) spindle orbifolds:

On orbifolds by 2-groups in view of sigma-models inspired from string theory:

category: Lie theory

Last revised on July 18, 2024 at 13:12:04. See the history of this page for a list of all contributions to it.