The String Coffee Table

I’ll write down my notes taken in the course of the lecture. Most of this material will probably not be new to experts.

Models for Orbifolds

The original definition of an orbifold goes like this (see this for more technical details)

An orbifold is a space $X$ in which every point has a neighborhood of the form $U/G$, where $U \subset \mathbb{R}^n$ and $G$ is a finite group acting on $U$. Here $n \in \mathbb{N}$ is fixed, but $G$ may change from point to point.

Orbifolds in this description were introduced originally by Satake. While this proved to be a very useful concept, it has the following problems:

1) The description in terms of local charts does not provide any global picture.

2) It is not manifest which notion of morphism between such structures would be natural.

Here are some examples of orbifolds:

i) the global orbifold [my terminology]. Let

globally, where $M$ is a manifold and $\Gamma$ a finite group acting by diffeomorphisms on $M$.

In the physics literature, this special case tends to be regarded as the definition of an orbifold.

ii) a first generalization of this example is obtained by considering the action of a Lie group $\Gamma$ acting properly on some manifold $M$, such that the subgroups

which fix a given point of $M$ (the stabilizer groups) are finite. In this case, $X = M/\Gamma$ is an orbifold for which the group $G$ appearing in the above definition in general changes from point to point.

iii) If $(M,F)$ is a foliated manifold with compact leaves and finite holonomy (see sections 1 and 2 of the above mentioned book for details), then the space of leaves $M/F$ has the structure of an orbifold. This is a consequence of a fact known as Reeb stability, which is explained in section 2.3 of the above mentioned book.

In a foliated space, one naturally has a notion of holonomy or, more generally, of transport as follows.

[ -digression-: Since this example will reappear further below, I’ll sketch some more details.

Given any two points $x$ and $y$ which are sitting on the same leave $L$ of the foliation, choose subspaces $T_x$ and $T_y$ which contain $x$ and $y$, respectively, and which are transversal to the leaf $L$. Now choose any path $\gamma$ from $x$ to $y$ which runs entirely inside of $L$. One can show that the homotopy class of such a path defines the germ (in $x$) of a diffeomorphism from $T_x$ to $T_y$. If the path is a loop, this defines a homomorphism from the fundamental group of $L$ to the group of germs (in $x$) of diffeomorphisms of $T_x$. - end of digression - ].

But, in order to study orbifolds as global objects together with maps between them (and possibly with extra stuff and structure, like for instance fiber bundles, over them) the most convenient language is that of Lie groupoids.

A groupoid is a category in which all morphisms are invertible. [As I have said, I am strictly transcribing that lecture… :-)]. A category is [Ok, I’ll make an exception here… ]

A Lie groupoid is a groupoid in the category of smooth spaces, where the space of objects and the space of morphisms are both smooth and source and target maps as well as the composition map are smooth maps.

Example: Consider again a foliated space $(M,F)$. The holonomy groupoid of a foliation $(M,F)$ is the groupoid whose space of objects is $M$ and whose morphisms are classes of paths $\gamma$ in leaves $L$ of the foliation, which have the same ‘holonomy’ (really: ‘transport’) in the sense of the above digression. Some work is involved in proving that this groupoid is in fact smooth.

The main message is now that

Orbifolds should be viewed as certain types of Lie groupoids.

Some definitions:

1) Given a Lie groupoid $G$, call $|G|$ the set of classes of objects of $G$ which are connected by any arrow. $|G|$ is the set of isomorphism classes of $G$.

2) Given any object $x$ in the groupoid $G$, Call the group $G_x = \{x \overset{g}{\to} x\}$ the isotropy group of $x$. [Elsewhere this would be called possibly the vertex group at $x$, but ‘isotropy’ is the term that suggests itself once we think of the groupoid as an orbifold.]

3) The groupoid $G$ is called étale if its source and target maps are local diffeomorphisms.

4) The groupoid $G$ is called proper if the map

is a proper map (i.e. closed map with compact fibers).

5) The groupoid $G$ is called a foliation groupoid if all isotropy groups $G_x$ are discrete.

Now, in terms of Lie groupoids and using the above terminology, one can give the following alternative definition of an orbifold.

Definition. An orbifold structure on a topological space $X$ is given by a proper étale groupoid $G$ together with a homeomorphism

The nice thing is that, once we know what some entity is in terms of abstract nonsense, we immediately know what the right notion of map between two such entites is: a morphism of groupoids is simply a functor. But since we do not just want morphisms between groupoids, but morphisms between smooth groupoids, there is an extra (though familar and well understood) subtlety.

Usually, we would regard a functor $G_1 \to G_2$ as ‘invertible’ if there is another functor $G_2 \to G_1$ such that their compositons are naturally isomorphic to the identity. However [if I understood correctly] the problem is that the weak inverse functor, while it may exist at the level of sets, will not in general be smooth. So we use the fact that, at the level of sets, an equivalence is the same as an essentially surjective fully faithful functor and impose a smoothness condition on this:

Definition. A functor between Lie groupoids is called an equivalence if it is a fully faithful and essentially surjective submersion.

Such a functor $G_1 \overset{\phi}{\to} G_2$ induces a homeomorphism

on the spaces of isomorphism classes of these groupoids (which, recall, ‘are’ the orbifolds represented by these groupoids).

We regard

as defining the same orbifold structure on $X$.

But this notion of equivalence needs to be weakened a little in order to correctly capture the notion of ‘equivalent orbifolds’.

Definition. Two Lie groupoids $G_1$ and $G_2$ are Morita equivalent if they are source and target of a span of ordinary equivalences. This means that there is a third Lie groupoid $H$ together with two equivalences from $H$ to $G_1$ and $G_2$:

Note: many properties are invariant under Morita equivalence, e.g. that $G$ is proper.

One can establish a dictionary between orbifolds as conceived originally and the corresponding groupoid. For instance

Fact. A groupoid $G$ is a foliation groupoid iff it is Morita equivalent to an étale groupoid.

Many invariants on orbifolds can be studied through propeties of groupoids which are invariant under Morita equivalence.

Example. Question: What would be a covering space of an orbifold?

Answer: A covering space of the orbifold represented by the groupoid $G$ is a covering space $S \to \mathrm{Obj}(G)$ equipped with a $G$-action $s \to gs$ covering every morphism $x \overset{g}{\to}y$ in $G$.

The category of such covering spaces, $\mathbf{Cov}(G)$, is equivalent to the category $\pi_1(G,x)-\mathbf{Sets}$ for a unique group $\pi_1(G,x)$. This group is known as the (Thurston) orbifold fundamental group.

(This fact is supposed to be good news if you already knew what Thurston’s definition was like using non-groupoid language.)

Example. Question: What is a vector bundle over an orbifold? Answer: This is an ordinary vector bundle $E$ over $\mathrm{Obj}(G)$ together with an isomorphism $E_x \overset{g(\cdot)}{\to} E_y$ of fibers covering every morphism $x \overset{g}{\to} y$ in $G$.

[So in other words, a vector bundle over an orbifold is a functor from the representing groupoid into vector spaces: $E : G \to \mathbf{Vect}$.]

The category $\mathbf{Vect}(G)$ of such vector bundles only depends on $G$ up to Morita equivalence. Decategorifying and forming the Grothendieck group, we arrive at $K(X)$, the orbifold $K$-theory of the orbifold $X$.

Finally, this $K$-theory can be related to the inertia groupoid $\Lambda(G)$ of $X$.

This groupoid has objects being automorphisms in the groupoid $G$ representing $X$ (i.e. elements of isotropy groups) and morphisms being morphisms in $G$ which relate to such isotropy group elements by conjugation.

One can apparently show that the orbifold K-theory is given by the cohomology of this inertia groupoid:

[But I realize that I don’t understand this yet. More tomorrow.]