The String Coffee Table

Let me start by saying some general things about stacks.

One curious thing about stacks is that they are motivated by and appear in two apparently rather unrelated contexts.

When you come from one point of view, a stack is a generalization of a sheaf. Since sheaves are a way of talking about gauge theory, this way of looking at stacks yields a generalization of gauge theory – namely ‘higher gauge theory’. Special sorts of stacks, called gerbes, are the right thing to describe the coupling of a string to a ‘2-form field’.

From another point of view, however, a stack is a generalization of the concept of a space, say of a topological space, or of a manifold. The approach here is similar in spirit to that in noncommutative geometry: 1) Pick some way to talk about spaces. 2) Realize that ordinary spaces are just a special case of this way of thinking. 3) Address the most general object obtainable by that sort of thinking as a ‘generalized space’.

In the case of stacks, step 1) is realized by the fact that every space (as in fact essentially every mathematical object) is completely characterized by the set of maps into the space. The precise statement of this fact is called the Yoneda Lemma.

Consider a topological space $M$. Given any other topological space $X$, we have the set $\mathrm{Map}(X,M)$ of all continuous maps from $X$ to $M$.

But there is more. Consider yet another topological space $Y$, together with a map $Y \overset{f}{\to} X$ from $Y$ to $X$. There is the set of maps $\mathrm{Map}(Y,M)$ as before. But in addition there is now a gadget that sends every element $X \overset{h}{\to} M \in \mathrm{Map}(X,M)$ to an element $Y \overset{h'}{\to} M \in \mathrm{Map}(X,M)$, simply by composing $h$ with $f$:

This mapping from maps from $X$ to maps from $Y$ is called $\mathrm{Map}(f,\mathrm{Id}_M)$. Hence we actually have a contravariant functor

which eats maps between topological spaces and spits out maps between the sets of maps between these spaces and the fixed space $M$.

Of course this is just what is called the $\mathrm{Hom}$-functor in general and all I have done is to spell out an elementary fact in category theory.

Anyway, the point is that knowing that gadget $\mathrm{Map}(-,M)$ is equivalent to knowing $M$ itself. This gives rise to step 2): $\mathrm{Map}(-,M)$ can also be thought of as a gadget that associates to every space $X$ the category whose objects are elements in $\mathrm{Map}(X,M)$ and which has a morphism between $X \overset{f}{\to} M$ and $Y \overset{g}{\to} M$ precisely if there is a map $X \overset{h}{\to} X$ such that

Usually one wants to allow only maps $h$ here which are invertible. In that case, every morphism in that category of maps from $X$ to $M$ is invertible. Hence $\mathrm{Map}(-,M)$ really associates a groupoid to every $X$.

This means that we have reformulated the information contained in the space $M$ itself in terms of a rule that to any other space $X$ associates the groupoid of maps from $X$ to $M$.

This is a step of the kind number 2) in the above, because it immediately generalizes: There are certainly other rules which associate groupoids to spaces. Rules that do not arise in the above fashion. Let’s think of all these rules as generalized spaces!

In fact, not quite all of them. The rule $\mathrm{Map}(-,M)$ has some nice properties, called gluing properties, that we want to retain. These are very similar to the gluing properties of a sheaf, just weakened a little.

But apart from that, we are now willing to address any rule which associates groupoids to spaces such that these gluing properties hold as a generalized space. And we call such a rule a stack.

(‘Why do we call it a ‘stack’?? I guess at one point mathematicians will run out of terms suggestive of ‘collections of stuff’. We have ‘sheaves’ and ‘bundles’, both of which probably originate in a time when mathematicians found themselves living in a more agricaltural environment. Similarly ‘gerbe’, which is french for ‘bouquet’ (and curiously similar to the german Garbe, which, however, means ‘sheaf’!). (But see David Robert’s comment on that point.) Then ‘stack’, which is always a nice way to waste some time in a talk which at the same time mentions ‘stacks of D-branes’. ‘Heap’ is also already a mathematical term, inherited from computer science (which has its own notion of stack). I wonder when somebody defines a mathematical concept called a ‘bunch’ and proves that every stack can be bunched. ;-)

Anyway, a stack, as I said, is a rule that associates categories (and groupoids in particular, if you use the more restrictive definition) to spaces such that some useful structure is preserved. One issue with stacks is that you can set them up on general sites. The category of open sets of some manifold constitutes a site, and that’s the context in which stacks are described for instance in Moerdijk’s text math.AT/0212266. Alternatively one can use the site of topological spaces or that of manifolds, which is the one used by Heinloth in his notes.

(Update: Thanks to David Roberts for making me correct the originally wrong statement in the above paragraph.)

If you feel like a physicist, and especially if you feel like a string theorist, E. Sharpe, in the introduction to his paper, proposes for you a way to get a better intuitive grasp of what the definition of a stack (when regarded as a way to talk about generalized spaces) really means.

In string theory one is familiar with the idea that target space is something secondary, while the worldsheet CFT is the primary entity. Concentrating on geometrical CFTs for a moment, namely on sigma-models, this means that one thinks of target space in terms of maps from the worldsheet into it. But that’s precisely the stack attitude. Think of a space in terms of maps into it.

This has some curious ramifications. Turns out that mathematicians have come up with the notion of a quotient stack. It’s definition at first sight tends to leave the standard physicist puzzled. But actually the standard physicist knows precisely this definition in different guise already under the name of a ‘twisted sector worldsheet map’.

Here is the abstract definition (Heinloth, example 1.5): Let $G$ be a Lie group acting on some space $M$. It doesn’t have to act nicely on it, neither freely nor transitively. This means that in general the naïve quotient $M/G$ won’t be a space of the type we started with. (Not a manifold, if $M$ is a manifold.) Instead it will be a (general) orbifold. This is something slightly more general than an ordinary space, and it is captured by the concept of stacks.

Namely one can define a stack capturing the generalized space “$M/G$” as follows. Let $[M/G]$ be the stack which is given by the rule that to each space $X$ associates a groupoid whose objects are $G$-bundles $P$ over $X$

equipped with $G$-equivariant maps $f^G$ from the total space of $P$ to $M$ (meaning that $f(g\cdot p) = g\cdot f(p)$ for $p \in P$). The morphisms between these objects in the groupoid are defined to be bundle isomorphisms between these bundles.

First of all one sees that in the case that $G= I$ is the trivial group, this rule is indeed the one of the space $M$ itself, regarded as a stack, which I talked about above.

In the case of nontrivial $G$ this rule simply reformulates the prescription for computing string partition functions for orbifold backgrounds. Lift the worldsheet ($X$) to the covering space by picking a $G$ bundle over it, identify it with a local section in that $G$-bundle and then map this into the target space in a way that respects the group action on everything. Summing over all possibilities of doing this is known as summing over all ‘twisted sectors’ of the string.

Hence, Eric Sharpe says, one can think of the sum over twisted sectors of a string $X$ as a sum over (isomorphism classes of) objects in the groupoid $[M/G](X)$ given by applying the stack $[M/G]$ to $X$.

Fine, but so far this is just a play with words. Is there any new insight available? Yes, plenty. In closing this part, I’ll just mention one of the main issues:

Given that we are prepared to allow stacks as string backgrounds, the question arises in how far stack technology and string technology mutually reproduce each other.

Given that we are prepared to allow stacks as string backgrounds, the question arises in how far stack technology and string technology mutually reproduce each other. In particular, it is well known that several different combinations of $M$ and $G$ yield stacks $[M/G]$ which are equivalent as stacks. But for some of them $G$ may be finite, while for others $G$ may be continuous. A sigma-model of $[M/G]$ for $G$ finite is just an ordinary free CFT on an orbifold. A sigma model on $[M/G]$ for non-finite $G$ however is some much more involved interacting gauged linear sigma model. Both sigma models don’t seem to be equivalent at all. But math tells us that the backgrounds they come from are equivalent in some sense. So what’s going on?

This issue has not been completely resolved, but Eric Sharpe says he has plenty of evidence for the conjecture that the sigma model on $[M/G]$ for $G$-non finite will flow by means of renormalization group flow to the RG fixed point given by the CFT on $[M/G]$ for $G$ finite.

In this respect the relation between equivalent but non-identical stacks would be somehow reminiscent of the relation betweenn isomorphic but non-identitcal objects in derived categories of coherent sheaves. The equivalence is realized physicall only after the ‘instable’ configuration has decayed to the stable physical background.

More later, if time permits.