The String Coffee Table

Before getting into any of the details I should emphasize the following.

In most of the existing physics literature, an ‘orbifold’ is regarded as the quotient $M/G$ of a manifold $M$ by the action of a finite group $G$. To mathematicians, an orbifold has always been more generally a quotient stack $[M/G]$ with $G$ not necessarily finite.

In as far as I understand, one point of papers like hep-th/0502044 is to point out that it is useful for ideas in physics to switch to the more sophisticated language of stacks.

In particular, one can ask the following type of question: There is more than one (2D) sigma-model for a given target space, obviously. Since a stack can encode more information than an ordinary space, does using stacks as targets maybe make the map from sigma models to targets ‘more injective’?

The answer is, yes, it does. One way for a stack to contain more information than a space is as follows.

Suppose we have a manifold $M$ and a group $G$ with $G$ acting trivially on $M$ - i.e. not at all. Then $M/G$, as a space, is clearly nothing but $M$ itself. However, the stack $[M/G]$, defined as in the previous entry, is not equivalent to the stack $M = [M/1]$. Therefore, we might expect that sigma models with target $[M]$ and target $[M/G]$ describe different sigma models on $M$ itself.

More generally, suppose that $G$ is an extension of a group $\bar G$ such that only $\bar G$ acts nontrivially on $M$. How is the sigma model on $[M/G]$ related to that on $[M/\bar G]$?

One example discussed by Eric Sharpe and Tony Pantev is that where $G= D_4$ is the dihedral group, which is a central extension of $\mathbb{Z}_2 \times \mathbb{Z}_2$ by $\mathbb{Z}_2$:

Suppose $D_4$ acts on $M$ such that the kernel of $p$ acts trivially.

In order to understand the physics described by that, look at the 1-loop (genus 1) partition function of any 2D sigma model into the stack. This is a sum over all ‘twisted sectors’, meaning over all ways to lift the torus from the quotient to the covering space $M$. Any such lift looks like a rectangle in $M$ whose left edge is mapped to the right edge by the action of an element $g\in D_4$, while the top edge is mapped to the bottom edge by some other element $h \in D_4$, such that $gh = hg$.

Given any such a pair of group elements $(g,h)$, call the partition function of this sector $Z_{(g,h)}(D_4)$. The full 1-loop partion functor is then

Since the center of $p$ acts trivially on $M$, one might naïvely expect that this equals the partition function made up from

with $\bar g = p(g)$ and $\bar h = p(h)$. However, not every pair $\bar g, \bar h$ lifts to a commuting pair $g,h$ in $D_4$. This means that $Z(D_4)$ contains only some of the twisted sectors of $Z(\mathbb{Z}_2\times\mathbb{Z}_2)$,

Hence the corresponding sigma models are different, even though $M/D_4$ is the same as $M/(\mathbb{Z}_2\times\mathbb{Z}_2)$. The stacks $[M/D_4]$, $[M/(\mathbb{Z}_2\times\mathbb{Z}_2)]$ know the difference.

There are more examples of this sort. Consider the situation with target manifold $M$ and group $\mathbf{Z}_k$ acting completely trivial on $M$. Then it turns out that the 1-loop partition function for target being the stack $[M/\mathbb{Z}_k]$

is $k$-times that with target being just $M$. Hence the sigma model for $[M/\mathbb{Z}_k]$ can be interpreted as that on $k$-copies of $M$.

This is a general pattern. It also appears in the last example which I shall mention here, that where the target is a gerbe.

A gerbe is a stack with extra properties. A banded $G$-gerbe is roughly a quotient stack where a group $G$ acts trivially. Since a gerbe is a stack, we may look at sigma models with that gerbe as target.

Everybody used to thinking of gerbes as generalizations of bundles and as describing Kalb-Ramond fields should watch out at this point. A sigma model on a gerbe in this sense has nothing to do with Kalb-Ramond fields. It just so happens that we can interpret a gerbe (alternatively to the higher gauge theory interpretation) as a generalized space – and we can have a sigma model on such a generalized space.

It turns out that a CFT on an abelian banded gerbe in this sense is the same as a CFT of a disjoint union of different target spaces, together with some extra data. But now I am too tired to check the precise details.

After all these examples of different stacks that can distinguish between different sigma models on the ‘same’ ordinary target space, I should finish by spelling out some more details of the complementary effect that I mentioned at the end of the last entry. There I mentioned that there may be different, but equivalent stacks which, when regarded as generalized target spaces, give rise to apparently different sigma models.

Stacks live in a 2-category. Hence for them to be equivalent is a subtle statement. It says that there are morphisms of stacks going back and forth between them which are mutually inverse up to a 2-isomorphism of stacks.

For this reason, it is maybe not too surprising that sigma models coming from different but equivalent stacks may be different, while at the same time being closely related.

One example is the ordinary orbifold $[\mathbb{C}/\mathbb{Z}_2]$, with the $\mathbb{Z}_2$ acting by sign reversal. This stack of a quotient by a finite group turns out to be equivalent to a stack $[X/\mathbb{C}^\times]$ of a quotient by the continuous group $\mathbb{C}^\times$ of non-vanishing complex numbers. Here $X$ is itself the manifold

In this expression the $\mathbb{Z}_2$ is supposed to act by rotations by $\pi$ on $\mathbb{C}^\times$. Hence the action here is free and $X$ is indeed a manifold.

As I said, the two stacks are equivalent as stacks

But the sigma models on them look quite different. The one on $[\mathbb{C}^2/\mathbb{Z}_2]$ is the ordinary free CFT on that orbifold background. The one on $[X/\mathbb{C}^\times]$ is instead a gauged linear sigma model, i.e. something with action looking like

where $D$ is a covariant derivative with respect to a $U(1)$-connection. The group $U(1)$ appears here as the compact part of $\mathbb{C}^\times$.

The conjecture is that following the RG flow of this gauged nonlinear sigma model will lead to the conformal fixed point given by the CFT on $[\mathbb{C}^2/\mathbb{Z}_2]$.

There are apparently many good pieces of evidence for the general conjecture that sigma models coming from different but equivalent stacks are related by RG flow. No proof, though.