A couple of posts ago, I mentioned some work by Rivasseau that touched on combinatorial species and QFT. Since then there have been a few mentions: at Arcadian Functor, where Kea further pointed out a post at U Duality which in turn pointed to the arXiv where there is a new paper by Rivasseau, Gurau and Magnen called “Tree Quantum Field Theory”. It claims to present a formalism for quantum field theory which is non-perturbative and based on species of marked trees.

(And here, incidentally, is a recent arXiv posting by Joachim Kock about decorated trees and polynomial functors, quite possibly related as we’ll see, but from a different point of view. I’d have to look at this more closely to see whether it’s related to species, but “analytic functors” certainly are. Analytic functors have the structure of power series as described below. As for “polynomial functors”, Dan tells me that the term also appears in homotopy theory in relation to the Goodwillie calculus, which talks about functors in categories of spaces. From a little examination of Kock’s paper, it seems like what he’s talking about deals with the case where the spaces you’re talking about happen to be discrete sets. I only mention this because this is a blog, so I don’t have to feel obliged to understand just how, or even whether, this is relevant – it just looks interesting, and I found it while searching around about the rest of this stuff.)

Since I have been thinking about species a bit recently anyway in the wake of a conversation I had with Jamie Vicary in Nottingham, I thought I’d try to lay out out one way of looking at the link between species and QFT, which is by way of groupoidification (described by John Baez in this draft paper) – more on that in part 2… For now, I’ll recap some basics about species.

Species

First off, what is the original definition of combinatorial species? This comes from Andre Joyal, I think originally in two main places: “Une theorie combinatoire des series formelles” (1981) and “Foncteurs analytiques et especes de structures” (1986). That word “especes” is the source of the term “combinatorial species”, but note that the more accurate translation of “especes de structures” is “type of structure” (or “sort of..”, or “species of…”). This is why some people, including John Baez (from whom I got the habit) call them “structure types” – . In any case, what are they?

The simple version is to say that they are functors $T$from $\mathbf{FinSet_0}$ (the category of finite sets and bijections, rather than any set maps) to $\mathbf{Sets}$, the category of sets. Every finite set $S$ gets assigned a set $T(S)$, interpreted as the set of $T$-structures on $S$. Every bijection of sets $f : S \rightarrow S'$ induces a bijection of the $T$-structured sets $T(f) : T(S) \rightarrow T(S')$ that comes by replacing each element in the underlying set of a given structured set.

A specific example should help: suppose $T$ is the type “combinatorial graphs”, so that $T(S)$ is the set of all graphs whose set of vertices is $S$. That is, each element of $T(S)$ amounts to a choice of edges, each of which is just a pair of vertices, so $T(S) \cong \mathcal{P}(S \times S)$. (I’m assuming a specific definition of “graph” here, and there are variants, but hopefully this is clear enough.) Then a bijection $f : S \rightarrow S'$ takes a graph and gives a new one with underlying set $S'$, where each vertex $s$ is replaced with $f(s)$.

Part of the point of this was to give a more sophisticated account of something combinatorialists had been doing for some time, which is using power series to represent “types of structure”, using a variable, say $z$, to “mark” elements of the underlying set (fancier versions use multiple variables, marking different things one might count). The “generating function” for $T$ has a coefficient for $z^n$ which is $\frac{\#T(S)}{n!}$ (the $n!$ refers to the number of self-bijections of an $n$-element set). So for instance, with the $T$ in our example, $\#T(S) = 2^{(\#S)^2}$, so the generating function for the structure type “graphs” is

$t(z) = \sum_{n=0}^{\infty} \frac{2^{(\#S)^2}}{n!} z^n$

Then there are combinatorial operations that correspond to sums and products of power series, composition of power series, and so on. You can read plenty more about this in various places apart from Joyal’s papers, such as Wilf’s “generatingfunctionology“, books on combinatorial enumeration by Richard Stanley, by Ian Goulden and David Jackson (whom I learned this stuff from long ago, so I’ll plug the book!), or even – why not? – in this paper of mine, which describes a relation between this classical stuff and a very simple (the simplest) QFT, namely the Hilbert space and algebra of observables for a single harmonic oscillator. A look at species as entities on their own can be found in Bergeron, Labelle and Leroux’s “Combinatorial Species and Tree-Like Structures”.

One example of the sort of thing one can do relies on the fact that the power series representation for $e^z$ represents the type “a set” (there is one of these for each finite set, since the coefficient for $z^n$ is $\frac{1}{n!}$). So if the generating function for a type $F$ is $f(z)$, then the type $T$, “sets of structures of type F\$, has generating function $t(z) = e^{f(z)}$. Conversely, since you can invert this to get $f(z) = ln(t(z))$, one can use the coefficients of the power series for this $f(z)$ to count the number of connected graphs on a set S (since a graph amounts to the same thing as a set of connected graphs). In particular, you can count them without having to find them all.

Well, so much for the classical theory of enumeration. One point of species is that one can deal with these functors from sets to sets directly, using various operations that correspond to the operations on power series. (A functor that can be represented in this “power series” sort of way is, naturally, an “analytic” functor.) One important such operation would be the derivative.

The natural way to define the combinatorial “derivative” is not too hard to predict: at the level of generating functions, it has to take $z^n$ to $nz^{n-1}$. The structure type whose generating function is $z^n$ is (naturally isomorphic to) the type “ordered $n$-element set” since there are $n!$ ways to make an $n$-element set with underlying set $S$ (presuming $\#S=n$). The derivative of this type has $n$ ways to define it on an $(n-1)$-element set. What this does is: given a set $S$, it formally adjoins a new element, and puts the structure “ordered $n$-element set” on the result, $S \cup \{\star\}$. This can only be done if $S$ (the “underlying set” of the new structure) has $n-1$ elements, and since there are $n!$ ways to order $S \cup \{\star\}$, the coefficient in the power series is $\frac{n!}{(n-1)!} = n$. In general, the combinatorial derivative takes a type $T$, and returns the type which gives $T$-structures on $S \{\cup \star\}$.

What does this have to do with QFT? Well, having a derivative operator sets us off in the direction of operator algebras, which is a key part of the answer. I’ll address that some more after I’ve explained how this relates to groupoidification in Part 2, upcoming…