The String Coffee Table

Based on an insight by Dirk Kreimer, Alain Connes, Dirk Kreimer and others have developed a powerful algebraic language to describe the process of renormalization in quantum field theory.

As is nicely explained in the expository paper

K. Ebrahimi-Fard
Hopf algebra approach to Feynman diagaram calculations
hep-th/0510202

the crucial idea is to equip the set of all (Feynman)-graphs (or rather the vector space freely generated over it) with the structure of an algebra in a natural way.

I’ll say what this natural algebra structure is in a second. But in order to prepare the stage I would like to use an idea discussed in

K. Costello
The $A_\infty$ operad and the moduli space of curves
math.AG/0402015

in the context of stringy Feynman diagrams, also known as Riemann surfaces.

A Feynman diagram is essentially a finite graph. So let’s think about finite graphs. It has a finite set of vertices with a finite set of edges (unoriented, in the present context) stretching between these vertices. We also want to consider Feynman graphs with external lines and hence we allow also edges which are attached just to a single vertex, called “external edges”.

Now, there are two natural ways in which one could imaging “composing” two such finite graphs.

1) Given two finite graphs, match their external edges and glue them to obtain a new finite graph.

2) Given two finite graphs, “insert” one into the other.

It’s that second sort of composition which is interesting for the study of graphs that are to be thought of as Feynamn graphs.

Namely, quantum field theory suggests that we should think of every vertex of a graph as a hidden graph itself. We might “magnify” the vertex and realize that its “inside” consists of multitude of further vertices with further edges between these.

Hence, whenever there is a graph whose set of external edges matches the edges coincident on some vertex of a second graph, we may imagine that this vertex really “is” the first graph, shrunk to a point. As a reversion of this shrinking process, we may remove that vertex from the second graph and replace it by the entire first graph.

It’s like zooming in on a city with Google Earth. Here we zoom in on graphs and see further graphs appearing where formerly only vertices could be seen. Google Graph.

In more sophisticated language, this means that we have a category whose objects are sets of vertices with half-edges attached to them, and whose morphisms are finite graphs. The source of such a finite graph is the set of its vertices with the attached (half-)edges. The target is the set of connected components of the graph with the attached external edges.

Whenever the source, thus defined, of one graph matches the target, in this sense, of another, the latter may be inserted into the former in the way described above.

But there is a crucial subtletly. The set of connected components with attached external edges of one graph may in general be matched to the set of vertices with atteched half-edges of another graph in more than one way. In order to get a well defined category we need to do something about this.

In Kevin Costello’s paper this is not really discussed, it seems to me. But I think what we want to do is simply to equip all graphs with an ordering of their sets of vertices, internal and external edges.

I really want to talk about what this has to do with Connes/Kreimer. But maybe a couple of words on what Costello does with this are in order.

First of all, Costello notes that a functor from the category of graphs as defined above to a symmetric monoidal category $C$ is the same as an operad in $C$. (Actually, if we restricted to graphs that are rooted trees then we’d get an ordinary operad. For general graphs we get a “modular operad”.)

He then notes that there is an obvious functor from our zoom-in category of graphs to the category whose

- object space is the moduli space of possibly singular Riemann surfaces with boundary and marked points on their boundary

- whose morphisms are maps between Riemann surfaces.

Heuristically, given any graph this functor identifies every vertex and its $n$ attached edges with an incoming string with $n$-marked points on it, then glues all these incoming strings at their marked points and thus produces an outgoing string with a marked point for each external vertex of the graph.

(Since we glue at single points we have to deal with singular Riemann surfaces sitting on the boundary of moduli space.)

Kevin Costello’s main point is that the operad thus obtained is closely related to a simpler operad, the topological cyclic operad $A_\infty$. This is the operad one obtains from the same prescription as above, but restricting to graphs which are trees and mapping these only to those Riemann surfaces which have the topology of a (singular) disk with marked boundary points. More precisely, it is shown that the full operad described above is the “modular envelope” of this well known $A_\infty$-operad.

The main message here is that, in some precise sense, the information about the moduli space of all Riemann surfaces is already encoded in the space of singular surfaces that look like lots of disks glued at single boundary points.

Another perspective on this point is given in the more recent paper

K. Costello
A dual point of view on the ribbon graph decomposition of moduli space
math.GT/0601130.

There it is shown that the moduli space of all these singular, locally disk-shaped Riemann surface is weakly homotopy equivalent to the full moduli space of all (possibly singular) Riemann surfaces.

All right, now on to Connes/Kreimers. I mentioned that they consider a certain algebra on the space of (Feynman) graphs. Now, we already know now of a natural category structure on the space of graphs - the zoom-in-on-graphs category.

But given any category, we already have an algebra: its category algebra. This is the algebra generated by the morphisms of the category with the product being their composition if defined and zero otherwise.

We could apply that prescription directly to zoom-in-on-graphs. Thies would yield an algebra which is almost, but not quite that considered by Connes and Kreimer.

Instead, recall the above mentioned subtlety that, in order to get an honest category, we needed to specify an ordering on external and internal edges of our graphs. That’s a little less elegant than one might hope for. So let’s do away with it!

Let’s restrict attention to a slightly coarse grained sub-algebra of our category algebra, that obtained by “averaging” over the different choices of ordering of (ingoing-say) diagrams. Hence, let the generators of the algebra be sums of morphisms that correspond to the same graph, but with all choices of orderings. Furthermore, include in that sum all disjoint unions of our graph with all possible “identity graphs” (this are those without any internal edges).

A little reflection shows that the algebra on these “coarse-grained” generators is precisely the algebra defined in equation (6) of Kreimer’s hep-th/0510202!

This equation says that, given a graph $\Gamma_1$ and a graph $\Gamma_2$, their product $\Gamma_1 * \Gamma_2$ is the linear combination

where the sum is over all graphs $\Gamma$ and the coefficient $n(\Gamma_1,\Gamma_2;\Gamma)$ counts the number of ways in which $\Gamma_2$ can be inserted into $\Gamma_1$ such that the result is $\Gamma$.

Thus, up to a slight subtlety, the starting point for Connes-Kreimer is the category algebra of the zoom-in-on-graphs category. Or so I think. You should check that.

Of course, Ebrahimi-Fard/Kreimer go further than that. The above algebra $(\Gamma,*)$ is very special. While itself not even associative, antisymmetrizing the “$*$“-product yields a nice Lie algebra: the product

satisfies the Jacobi identity.

This is pretty remarkable given that $*$ can be interpreted as interting graphs into each other.

Once this Lie algebra is there, one simply turns the crank to obtain the advertised Hopf algebra structure. This is simply the (graded) dual of its universal enveloping algebra.

Using this Hopf algebra, it turns out that there is a simple formula which computes for each graph $\Gamma$ the corresponding counterterm in some renormalization scheme. (Equation (21) in Ebrahimi-Fard/Kreimer.)

Update, 24 Feb. 2006: After reading more in Tom Leinster’s book Higher Operads, Higher Categories I noticed that the category on graphs which I talk about above is probably best thought of as a version of the multicategory that Tom Leinster defines in example 4.2.12 of this book. Thinking of it as a multicategory also removes that ordering issue which I talk about. I believe that to every multicategory we can associate a multicategory algebra in straightforward generalization of the notion of category algebra (and to be distinguished from the concept “algebra for a multicategory” as in “algebra for an operad”). I think that the algebra in equation (6) of Kreimer’s hep-th/0510202 is nothing but the multicategory algebra of the more or less obvious multicategory of Feynman graphs.