What exactly is an Orbifold?

Question

What exactly is an orbifold? I've come across orbifolds on several occasions and I know they are important to string theory, but what is an orbifold? I've seen some very technical mathematical definitions, but I was wondering if there was a more basic/intuitive definition. Also, what is the physical interpretation of an orbifold?

Answer 1

From the mathematical perspective: Orbifolds are locally quotients of differentiable manifolds by finite groups. They are smooth except for very simple, finite and tractable class of singularities.

The amazement is that despite its simplicity, orbifolds share strong similarities with much more general spaces such as stacks. It is also relatively easy (and suprisingly beautiful) to give explicit constructions of very important algebraic data such as coherent sheaves and use them to verify and impressive class of very deep phenomena such as equivalences of derived categories of coherent sheaves or the McKay correspondence.

Now, why are orbifolds so important for string theory? The answer is that orbifolds provide a wide class of examples of singular spacetimes at which strings can propagate in a demonstrably consistent way (see the classic Strings On Orbifolds).

A basic expectation is that a truly quantum theory of gravity should be able to deal with situations were the curvature of spacetime is very high (even planckian) as was the curvature of the very early universe.

You can read about propagation of strings even in elementary string theory textbooks such as the one of Zwiebach. You can also learn how branes provide physical mechanisms for singularity resolution or how string theory successfully deal with topology change in spacetime and how all the latter can be used to exactly compute black hole degeneracies or to provide phenomenologicaly realistic scenarios were famous no-go theorems are circumvented.

The understanding of the physics of black hole and cosmological singularities is one of the most greatest goals in theoretical physics. Even is possible that full quantum gravity can be understood purely in terms of high-curvature spacetime fluctuations (Wheeler's spacetime foam) and strings propagating on orbifolds are a beautiful example of how string theory is guiding us towards the achieve of those dreams.

Edit: I forgot to tell that ADE like singularities can be explicity defined as "Branes". For example, an $A_{N}$ singularity in type IIB superstring theory can be seen to be equivalent to n sparated M5-branes after a lift to M-theory . The dictionary can be found in Branes and toric geometry.

Answer 2

But only if the D-brane mechanism is viable, otherwise the local curvature can distort back into itself via supergravity, creating a conundrum that we still cannot rectify with current theory.

Answer 3

The simplest orbifolds are the global ones. These are the quotients of manifolds by the action of a finite group. And the simplest amongst these are the quotient of the real line by a finite group. In fact, the only symmetry group possible here is the reflection symmetry $\mathbb{Z}_2$. And it has ordinary manifold points in the interior and a point with an attached $\mathbb{Z}_2$ symmetry at its endpoint.

The next simplest global orbifold is the quotient of a plane by the action of a finite group. There is an action by each of $\mathbb{Z}_k$. The simplest action here is again the reflection symmetry $\mathbb{Z}_2$. This yields a half-plane whose points in the interior are smooth manifold points but on the edge have an attached symmetry of $\mathbb{Z}_2$. We also have the rotation action by $\mathbb{Z}_k$. The quotient of the plane by this is a wedge, this can be rolled up into a cone and all its interior points are ordinary manifold points but the origin is a cone point with attached symmetry group $\mathbb{Z}_k$. For example the quotient of the plane by $\mathbb{Z}_3$, is one third of the plane given by a wedge cut out by the ray from the origin by 0 degrees and the ray from the origin by 120 degrees. We can roll this wedge up and glue the latter ray to the former ray giving us a cone.

A general orbifold is one which is locally an orbifold of the above form.

Now in general relativity, although the metric is dynamical the topology isn't. By this, I mean it is impossible to change the topological type of the manifold by adding holes or handles. However, in quantum gravity, it is reasonable to expect that topology change occurs in the small. For orbifold singularities, string theory remains well defined at the singularity as it does on a smooth domain. Such singularities do not change the topology type.

However, there are conifold singularities which do yield topology change - and which note are not orbifold singularities. Here, string theory does become ill-defined at the singularity. But, it was realised in the mid-90s this was due to ignoring the effects of Dbranes. At a geometric singularity, a closed string can contract to zero which implies that new massless particles are produced. And these are what are required to cancel the infinities found at the singularity.