The String Coffee Table

In the mathematically oriented literature it has long ago become conventional to address representations of cobordism ($\to$) categories as quantum field theories.

Topological quantum field theories are well understood from this point of view. Homotopy quantum field theories are defined to be slightly richer than topological ones, in that more “background” structure is assumed to be present.

I’ll concentrate on the case where the cobordisms are 2-dimensional, in which case we are talking about 2-dimensional quantum field theory.

Here a cobordims is just any 2-dimensional “space” $\Sigma$ (ordinarily (a diffeomorphism class of) a 2-dimensional manifold, but we will be interested in equipping this with extra structure), potentially with boundary $\partial\Sigma$.

($\partial \Sigma$ bounds $\Sigma$, hence $\Sigma$ cobounds $\partial\Sigma$. That’s the cologic here.)

We want to introduce a distinction between what are called incoming and outgoing boundaries, and interpret the cobordism as the worldsheet describing the process of the strings being the incoming boundaries propagating to the strings being the outgoing boundaries. Hence cobordisms may be composed in an obvious way by gluing their boundaries. Therefore we have some category $\mathbf{Cob}_2$ of 2-dimensional cobordisms. (Up to details, of course.)

A representation of that category, hence a 2-$d$ quantum field theory, is a functor sending it into vector spaces.

Hence this is something that assigns to each type of boundary (each type of string) a vector space, namely the space of states of that type of string.

In addition, the representation assigns to each cobordism a linear operator from the incoming to the outgoing spaces of string states. This is the propagator.

If you feel like a physicist you should think of this propagator as being given by some path integral. Otherwise, you shouldn’t.

In the simplest case our cobordisms are just that: (diffeomorphism classes of) manifolds without further structure. In this situation the representing functor can assign propagators only depending on the topology of the underlying worldsheet. Hence we get a topological string theory this way ($\to$, $\to$).

In this case the pair-of-pants cobordism yields a unique 3-point function of the three states at the three boundaries. This gives a product on these states, and, when read the other way around, also a coproduct. All in all one obtains a Frobenius algebra on the space of string states, and in fact this completely characterizes the representation functor, hence the QFT. ($\to$)

In order to get 2-$d$ field theories with more structure, we’d need to add more structure to our cobordisms.

We can justly expect that for every bit of structure added to the cobordisms, we will find an induced piece of extra structure on the Frobenius algebra, which we found in the bare, topological case.

For instance, we might equip the cobordisms with a complex structure (equivalently, a conformal structure). The result would be a conformal 2-$d$ field theory. (With vanishing central charge, unless we slightly modify the definition of our representation functor.)

But that extra structure already makes life pretty hard - at least as compared to the purely topological case. In some situations one can pull a trick and disentangle the complex analytic information from the purely topological one and solve two tractable seperate problems, the second of which looks again a lot like the topological string, but now yielding Frobenius algebras internal to the category of solutions to the first problem, in a way. That’s the discussion of rational conformal field theory as given by Fuchs, Runkel & Schweigert ($\to$).

A somewhat more modest proposal for extra structure on our topological morphisms, one that is not quite motivated by physics, but rather by the fact that it leads to interesting structures, was given by Turaev.

Turaev proposed to consider putting extra structure on cobordisms consisting of a choice of base point on each boundary, as well as a homotopy class of maps from each cobordims into some pointed topological space $B$, such that base points are mapped to base points.

In some sense this looks like the structure that would be interpreted as describing topological string worldsheets embedded into a target space $B$. However, this might not be the most fruitful way to look at this situation. This is one of the main points to be discussed below.

In any case, since the extra structure here involves homotopy classes of maps, representations of categories of cobordisms equipped with that extra structure were called HQFTs, homotopy quantum field theories. A review of these definitions can be found in section 1 of the second of the two papers cited above.

In fact, in this sort of game people usually restricted attention to rather special “target spaces” $B$. Obviously the homotopy type of $B$ is decisive. Hence people studied Eilenberg-MacLane spaces, for instance $B= K(G,1)$, the topological space which has first homotopy group isomorphic to $G$ and all other groups vanishing.

When I first learned about this idea in a talk by Jens Fjelstad ($\to$) I had two main impressions:

a) It was clear that this particular choice of extra structure on the cobordisms leads to tractable, yet interesting extra structure on the resulting representations.

b) It was not quite as clear to me what precisely the meaning of the extra homotopy structure was. How did it fit into the grand scheme of things, in particular into the grand scheme of things as seen from the point of view of a physicist? Why should we be interested in studying this particular extra structure?

The starting point for the two papers by Turaev and Porter mentioned above is essentially the desire to get a better handle on this point b).

As I have mentioned here every now and then (most recently, and featuring prominently there, in the context of String-2-bundles ($\to$)), there is a general connection between the world of topological spaces, continuous maps and homotopies between these and the world of categories, functors, and natural transformations between these.

To every category, we can associate its nerve, which is a simplicial space with an $n$-simplex for each collection of $n$ composable morphisms in the category. By identitfying each of these $n$-simplicies with the standard $n$-simplex in $\mathbb{R}^n$ we obtain a topological space, called the geometric realization of the nerve.

This relation between categories and topological spaces is not really bijective, in any sense, but often it is useful to realize that some topological space one comes across is really best thought of as the realization of the nerve of some category.

The most important example for the present purpose is also the most familiar one: the classifying space $BG$ for any group $G$ is really nothing but the realization of the nerve of $G$, regarded as a groupoid with a single object.

Closely related is the example that essentially is the starting point for Porter&Turaev’s considerations.

Let the “background space” $B$ of our 2-$d$ homotopy quantum field theory be $K(G,1)$, with a base point identitfied. Then a homotopy class of maps from HQFT cobordisms that respect the base point can be rather easily seen to have the following equivalent simplical formulation (sort of the pre-image under the operation of taking realizations of nerves):

Choose a triangulation on every cobordism. Assign to each edge an element in the group $G$, such that paths of edges starting and ending at the same vertex and being cobordant (inside the cobordism!) are assigned group elements which compose to the same result.

Equivalently, choose a triangulation of your cobordism and consider functors from its graph category into the group $G$, regarded as a category, such that the above condition holds.

The single object of $G$ ($G$ is a groupoid with a single object) plays the role of the basepoint of the background space $B$. Functoriality demands that every vertex gets send to this basepoint. Every edge hence becomes a loop starting and ending at the basepoint, hence a loop in $K(G,1)$, hence an element in $G$.

Porter&Turaev observe that this way of looking at things suggests an obvious generalization: Since we are in 2-dimensions, it is most natural to also associate data to the faces of the triangulation.

Hence, instead of just assigning group elements to edges, let’s also assign something like group elements to faces. Thinking functorially this means that we want a 2-functor from the 2-graph 2-category of the triangulation into some 2-group.

Notice that this is pretty much the same situation considered once, for instance, by Girelli&Pfeiffer ($\to$).

By passing to realizations of nerves one finds that now we are implicitly dealing with homotopy classes of base-point preserving maps from cobordisms into “target spaces” $B$ which have possibly nontrivial second homotopy groups. For some special cases examples of such a scenario had been studied before in the literature. Porter&Turaev therefore propose to generalize the definition of homotopy quantum field theory using this idea of the extra structure really being 2-functors from triangulations of cobordisms into 2-groups.

The resulting quantum field theory they call a formal homotopy quantum field theory.

So a formal HQGT is, roughly, a representation of the enriched cobordisms category whose cobordisms come equipped with triangulations together with 2-functors from the corresponding 2-graph 2-catgeories into some (strict) 2-group.

Once this definition has been written down, the game begins. We want to know how such representations look like. Not too surprisingly, the result is a generalization of the concept of a Frobenius algebra.

As special cases we reobtain first of all Turaev’s original result, which says that for 2-groups with only identity morphisms the corresponding formal HQFTs are equivalent to something called a “crossed $\pi$ algebra”, essentially a group algebra with an extra twist.

For other cases we obtain what are called $G$-Frobenius algebras, which are Frobenius algebras equipped with a certain action of the group $G$ by endomorphisms.

Most generally, and that’s the main result of the first of the two papers above, one finds that such formal HQFTs are equivalent to something called crossed $C$-algebras, where $C$ is the 2-group encoding the “target” of the formal HQFT.

Of course a crossed $C$ algebra is precisely defined to make this statement true, and hence its definition encodes all the structure implied by the definition of formal HQFTs.

Instead of saying anything else about this classification result, I would now like to finally turn attention to some more interpretational issues.

What does it all mean?

As soon as you draw any triangulation and begin coloring its edges with elements of a group $G$ in such a way that paths of edges with the same endpoints get assigned the same group element (after taking the product in the group for all elements), you’ll immediately be reminded of assigning transition functions to a local trivializations in some $G$-bundle.

In fact, choosing any good cover of a space $X$, it induces a groupoid, called the Čech groupoid of the good cover, and a choice of transition functions is just a functor from that groupoid to $G$ (more details can be found for instance in these notes).

Moreover, every choice of triangulation induces a choice of good cover and vice versa.

It is hence suggested (in section 6 of the above two papers) that we might want to think of a formal $C$-HQFT as something defined on cobordisms which carry (local trivializaitons of) $C$-2-bundles (nonabelian crossed module gerbes, if you like).

But is this exactly what is going on?

No doubt, one can decide to consider enriching cobordisms with the structure of $C$-2-bundles over them. With everything in sight simplical, it should be straightforward to define how to glue these enriched cobordisms, and we can study representations of the resulting enriched cobordism category.

But I don’t think that this would be related to the original definition of HQFT.

Instead, I’d think the following is true: a choice of triangulation of any space $X$ together with a functor from the graph category of that triangulation to a group $G$, such that parallel paths are sent to the same group element, specifies nothing but flat $G$-bundle with connection on $X$.

There are no transition functions at all. There are, after all, no edges going between the same point regarded as sitting in two different patches, hence no natural candidate to a assign a transition to in the first place. Instead, there are “real” edges, that actually correspond to paths in $X$, and assigning group elements to them coresponds to assigning parallel transport to them. The conditon that paths with the same endpoints have the same group elements associated to them (if they are cobordant) says that the connection inducing this parallel transport is flat. This again says that it is consistent to address group elements as the result of parallel transport.

So I think it would be consistent to say that an HQFT for target a $K(G,1)$ is a representation of a category of cobordisms which are equipped with flat $G$-bundles with connection.

Interestingly, a closely related notion is familiar in the case of purely topological 2-$d$ theories. A large class of these comes (using the Fukuma-Hosono-Kawai formulation) from a “state sum” (to be thought of as a path integral), where each state corresponds to an assignment of algebra elements to edges of a triangulation, such that three algebra elements associated to the three sides of any triangle of the triangulation are related by the product in the algebra.

In particular, one can choose this algebra to be the group algebra of $G$. Computing this sort of state sum then amounts to nothing but assigning all possible flat $G$-connections the the cobordism under consideration and summing over them. (For a nice review of this see John Baez’s lecture notes here: I, II. This is the 2D version of the famous Dijkgraaf-Witten TFT ($\to$)).

So there is a certain sense in which an HQFT is like a TQFT equipped with “background structure”. That background structure might be a flat $G$-bundle on cobordisms (instead of, say, a conformal structure on cobordisms, or a metric, or something else ). We might consider integrating out that background structure by summing over all its possible realizations, in a way. It looks like if we “integrate out” the background structure of an HQFT with target a $K(G,1)$ we obtain the 2-dimensional Dijkgraaf-Witten TQFT (i.e. Fukuma-Hosono-Kawai with algebra of states being the group algebra of $G$).

I would regard the general case of Porter&Turaev’s formal HQFT from the same point of view. I’d think that one way to describe what a 2$d$ formal $C$-HQFT (for $C$ a strict 2-group) is, would be to address it as a representation of the category of 2$d$-cobordisms which carry flat strict 2-bundles with connection.

This general statement is somewhat more interesting than the special case where $C$ has only identity morphisms (which leads to flat 1-bundles). The reason is that every 2-bundle in 2-dimensions is flat. (At least as long as we agree on reasonable definition of a 2-bundle.)

Hence it seems we can make the above characterization of 2$d$ formal $C$-HQFTs sound slightly more interesting:

An 2$d$ formal $C$-HQFT is a representation of the category of 2$d$-cobordisms which are equipped with (possibly local trivializations of) principal $C$-2-bundles with 2-connection.

This, then, is again close to what Porter&Turaev are pondering in section 6 of the second of the above two papers.

Looking from this point of view on formal HQFT, one might be tempted to enter into some obvious speculations.

For instance, when we write down the path integral for a conformal string propagating in a Kalb-Ramond background, the “topological” part of the action is nothing but a 2-bundle with connection present on spacetime, pulled back to the worldsheet (where it trivializes) and having its surface connection evaluated on the worldsheet.

This way of computing surface holonomy for gerbes by pulling the entire gerbe back to the surface, trivializing it there and then computing the “trivial” surface transport is often used as the definition of surface holonomy in the first place (see for instance Chatterjee’s thesis). Line transport in a bundle can be defined in the same way.

This illustrates maybe that it is quite natural to consider 2$d$ cobordisms equipped with (necessarily flat) 2-bundles with connection.