This Week's Finds in Mathematical Physics (Week 191)

John Baez

Now I'm in Sydney, Australia, trying to learn a bit of category theory from the experts here. It's quite a change. Hong Kong was louder, faster, more densely packed and more commercial than the USA. Australia seems quieter, slower, sparser and less commercialized. Odd to think that all three were British colonies! They seem like different worlds.

Anyway, on to business. People are starting to get more interested in the role played by octonions and exceptional groups in superstring theory and supergravity. There are a lot of pretty patterns here that may boil down to pure algebra... in which case I might be able to understand them and maybe even come up with something cool!

Here are some of the papers I'm struggling to read about this. First, a nice introduction to how supergravity works in different dimensions:

1) Antoine Van Proeyen, Structure of supergravity theories, available as hep-th/0301005.

"We give an elementary introduction to the structure of supergravity theories. This leads to a table with an overview of supergravity and supersymmetry theories in dimensions 4 to 11. The basic steps in constructing supergravity theories are considered: determination of the underlying algebra, the multiplets, the actions, and solutions. Finally, an overview is given of the geometries that result from the scalars of supergravity theories."

Second, a interesting study of how you get supergravities in different dimensions by "oxidizing" 4-dimensional theories. This is a pun on "reduction", the process whereby we go down from high dimensions to lower ones by curling up the extra dimensions. It turns out that oxidation is deeply related to Dynkin diagrams:

2) Arjan Keurentjes, The group theory of oxidation, available as hep-th/0210178.

"Dimensional reduction of (super-)gravity theories to 3 dimensions results in σ models on coset spaces G/H, such as the E8/SO(16) coset in the bosonic sector of 3 dimensional maximal supergravity. The reverse process, oxidation, is the reconstruction of a higher dimensional gravity theory from a coset σ model. Using the group G as starting point, the higher dimensional models follow essentially from decomposition into subgroups. All equations of motion and Bianchi identities can be directly reconstructed from the group lattice; Kaluza-Klein modifications and Chern-Simons terms are encoded in the group structure. Manipulations of extended Dynkin diagrams encode matter content and (string) dualities. The reflection symmetry of the "magic triangle" for En gravities, and approximate reflection symmetry of the older "magic triangle" of supergravities in 4 dimensions, are easily understood in this framework."

Next, a tour of places where the octonions show up in string theory:

3) Luis J. Boya, Octonions and M-theory, available as hep-th/0301037.

"We explain how structures related to octonions are ubiquitous in M-theory. All the exceptional Lie groups, and the projective Cayley line and plane, appear in M-theory. Exceptional G2-holonomy manifolds show up as compactifying spaces, and are related to the M2 Brane and 3-form. We review this evidence, which comes from the initial 11-dim structures. Relations between these objects are stressed, when extant and understood. We argue for the necessity of a better understanding of the role of the octonions themselves (in particular non-associativity) in M-theory."

And here's an article about where the exceptional groups show up, from a true expert on the subject:

4) Pierre Ramond, Exceptional groups and physics, available as hep-th/0301050.

"Quarks and leptons charges and interactions are derived from gauge theories associated with symmetries. Their space-time labels come from representations of the non-compact algebra of Special Relativity. Common to these descriptions are the Lie groups stemming from their invariances. Does Nature use Exceptional Groups, the most distinctive among them? We examine the case for and against their use. They do indeed appear in charge space, as the Standard Model fits naturally inside the exceptional group E6. Further, the advent of the E8 x E8 Heterotic Superstring theory adds credibility to this venue. On the other hand, their use as space-time labels has not been as evident as they link spinors and tensors under space rotations, which flies in the face of the spin-statistics connection. We discuss a way to circumvent this difficulty in trying to generalize eleven-dimensional supergravity."

I haven't read this, but indeed, it's really annoying how structures like triality mix integer and half-integer spin objects in a way that doesn't seem to make physical sense. Does he really have a way to get around it?

Oh well. Let me talk about something I do understand.

Last week I said a bunch about "structure types", also called "species". A structure type is any sort of structure you can put on finite sets, but the cool part is that structure types act like power series. This fact has various spinoffs. Last week I sketched how people use it to solve problems in combinatorics. In "week185" I explained how it lets us categorify the harmonic oscillator! And now I want to explain how it gives a nice way of understanding operads.

But first I need to say what operads are. The slick way to define them uses structure types - but this is a bit devious, so it might fool you into thinking that operads are hard to understand. They're actually not, so I'll start out with an elementary introduction to operads, then give you some references for further study... and then pull out all the stops and explain how they're related to structure types.

So: what's an operad? An operad O consists of a set On of abstract `n-ary operations' for each natural number n, together with rules for composing these operations. We can think of an n-ary operation as a little black box with n wires coming in and one wire coming out:

```                  \    |    /
\   |   /
\  |  /
-----
|     |
-----
|
|
```
We're allowed to compose these operations like this:
```         \    /     \  |  /       |
\  /       \ | /        |
-----      -----      -----
|     |    |     |    |     |
-----      -----      -----
\        |        /
\       |       /
\      |      /
\     |     /
\    |    /
\   |   /
\  |  /
-----
|     |
-----
|
|
```
feeding the outputs of n operations g1,..,gn into the inputs of an n-ary operation f, obtaining a new operation which we call f o (g1,...,gn). We demand that there be a unary operation serving as the identity for composition, and we impose an "associative law" that makes a composite of composites like this well-defined:
```              \    /   |  \  |  /   \     /
\  /    |   \ | /     \   /
---   ---   ---       ---
|   | |   | |   |     |   |
---   ---   ---       ---
\   |   /          /
\  |  /          /
\ | /          /
-----      -----      -----
|     |    |     |    |     |
-----      -----      -----
\        |        /
\       |       /
\      |      /
\     |     /
\    |    /
\   |   /
\  |  /
-----
|     |
-----
|
|
```
(This picture has a 0-ary operation in it, just to emphasize that this is allowed.) We can permute the inputs of an n-ary operation and get a new operation:
```                      \ /   /
/   /
/ \ /
/   /
/   / \
\  |  /
-----
|     |
-----
|
|
```
We demand that this give an action of each permutation group Sn on each set On. Finally, we demand that these actions be compatible with composition, in a way that's supposed to be obvious from the pictures. For example:
```       \  |  /   |   \   /               \\\ /   / /
\ | /    |    \ /                 \\/   / /
---    ---   ---                  /\\ / /
| a |  | b | | c |                / \\/ /
---    ---   ---                /   / /
\     /   /                  /   / /\\
\   /   /                  /   | | \\\
\ /   /                  /    | |  \\\
/   /                  ---   ---   ---
/ \ /           =      | b | | c | | a |
/   /                    ---   ---   ---
/   / \                      \   |   /
\  |  /                       \  |  /
-----                         -----
|  d  |                       |  d  |
-----                         -----
|                             |
|                             |
```
That's all there is to it!

With this answered, your next question is probably: "why should I care about operads?" This gets a little more technical. For a detailed answer, the best place to look is this book:

5) Martin Markl, Steve Schnider and Jim Stasheff, Operads in Algebra, Topology and Physics, AMS, Providence, Rhode Island, 2002.

But if you just want a taste, try Stasheff's infamous "operadchik" paper - get it? - which for some reason isn't on the arXiv:

Another good introduction is this paper by Sasha Voronov:

7) Alexander Voronov, Notes on universal algebra, available as math.QA/0111009.

Tom Leinster is writing a book on the applications of operads to higher-dimensional algebra, but you'll have to wait a while for that.

Anyway, there are many reasons why you should care about operads. Historically, the first come from topology. In homotopy theory, the main way to probe a space X is by looking at maps from the k-sphere to X. We define the "kth loop space" of X, Ωk(X), to be the space of all such maps sending the north pole to a chosen point in X, called the "basepoint". The set of connected components of Ωk(X) is called the "kth homotopy group" of X; this is a group for k > 0 and an abelian group for k > 1.

Most homotopy theorists would gladly sell their souls for the ability to compute the homotopy groups of an arbitrary space. However, there is extra information lurking in the space Ωk X that gets lost when we consider only its connected components. Starting in the late 1950s, a large number of excellent topologists including Adams and MacLane, Stasheff, Boardman and Vogt, and May struggled to understand all the structure possessed by an k-fold loop space.

For example, Ω1(X) is something like a topological group, thanks to our ability to "compose" loops. (For details, see "week119".) However, the usual group laws such as associativity hold only up to homotopy. To make matters even trickier, these homotopies satisfy certain laws of their own, but only up to homotopy - and so on ad infinitum. Similarly, Ωk(X) is something like an abelian topological group for k > 1, but again only up to homotopies that themselves satisfy certain laws up to homotopy, and so on - and in a manner that gets ever more complicated for higher k!

After more than decade of hard work, it became clear that operads are the easiest way to organize all these higher homotopies. Just as a group can act on a set, so can an operad O, each abstract operation in On being realized as actual n-ary operation on the set in a manner preserving composition, the identity, and the permutation group actions. A set equipped with an action of the operad O is usually called an "algebra over O", though personally I'd prefer to call it an action of O on the set. It turns out that the structure of a k-fold loop space is completely captured by saying that it is an algebra over a certain operad!

Even better, if we choose this operad O to be "cofibrant" - whatever that means - any space equipped with a homotopy equivalence to a k-fold loop space will also become an algebra over O. This is the simplest example of how operads are used to describe "homotopy invariant algebraic structures", in which all laws hold up to an infinite sequence of higher homotopies.

For an operad to do this job, it must really have a topological space of operations On for each n, since the fact that various laws hold up to homotopy is expressed by the existence of certain continuous paths in these spaces. Similarly, composition and the permutation group actions should be continuous maps. Finally, we should only consider algebras that are topological spaces on which the operad acts continuously.

In short, topology really requires operads and their algebras in the category of topological spaces rather than sets. The ability to transplant the theory of operads to various different contexts is an important aspect of their power. So, it's good that Markl, Schnider and Stasheff treat operads in an arbitrary symmetric monoidal category. They also prove the worth of this level of generality by discussing many examples in detail. For example, they describe how operads in the category of chain complexes have been used to study deformation quantization - and also string theory, where the operations of gluing together Riemann surfaces are important. Indeed, these physics applications have led to a kind of renaissance in the theory of operads!

Okay. The last paragraph was packed with buzzwords, so now all the scaredy-cats are gone. Let me explain the relation between operads and structure types.

I said that a structure type is "any sort of structure you can put on finite sets", but let me make that more precise. A structure type is really a functor

F: FinSet0 → Set

where FinSet0 is the groupoid of finite sets and bijections, and Set is the category of sets and functions. FinSet0 is equivalent to the category that has one object, "the n-element set", for each n, with the morphisms from this object to itself forming the permutation group Sn. So, we can also think of a structure type as consisting of a set F(n) for each n, together with an action of Sn on this set F(n). This latter viewpoint is good for calculation, while the original viewpoint is better for conceptual work.

We also have morphisms between structure types, which are just natural transformations between functors of the above sort. So, the category of structure types is the functor category

hom(FinSet0, Set)

To understand why this category acts like the ring of formal power series in one variable, it's crucial to understand the analogy between ordinary set-based algebra and categorified algebra. The quickest way to get a feel for this may be a big chart, which starts like this:

```sets                            categories

monoids                         monoidal categories

commutative monoids             symmetric monoidal categories

commutative rigs                symmetric 2-rigs

the free commutative rig on     the free symmetric 2-rig on
no generators: N                no generators: Set

the free commutative rig on     the free symmetric 2-rig on
one generator: N[x]             on generator: Set[[x]] = hom(FinSet0, Set)
```
I'll assume you understand the first three lines of the chart, e.g. that just as a monoid is a set equipped with an associative multiplication and identity element, a "monoidal category" is a category equipped with the same sort of structure, but where all the laws hold only up to isomorphism, and these isomorphisms in turn satisfy some coherence laws. Similarly, a symmetric monoidal category is like a commutative monoid.

We can then throw in an extra operation, "addition". Recall that a "rig" is a set with two monoid structures + and x, where + is commutative and x distributes over +. Most algebraists prefer rings, where you can also subtract, but the natural numbers N are just a rig, and working over N instead the integers is important in combinatorics. The reason, ultimately, is that N is the free commutative rig on no generators!

No generators? Yes - since you get the numbers 0 and 1 for free in the definition of a rig, without needing to throw in any generators, and then the rig operations give you 1+1, 1+1+1, and so on.

Now, a 2-rig should be a categorified analogue of a rig. The classic example is the category of sets, where "addition" is disjoint union and "multiplication" is Cartesian product. It would be nice if this were the free 2-rig on no generators, to emphasize the analogy between natural numbers and sets.

There are various different ways to accomplish this, but one nice way is to define a "2-rig" as a monoidal category with colimits, where the monoidal structure preserves colimits in each argument. The colimits act like addition and the monoidal structure acts like multiplication. Given this, it's easy to check that the free 2-rig on no generators is the category Set.

(If we prefer an analogy between natural numbers and finite sets, we should say "finite colimits" instead of colimits in the definition of 2-rig: then FinSet will be the free 2-rig on no generators.)

Now, what's the free commutative rig on one generator?

It's N[x], the algebra of polynomials in one variable, with natural number coefficients.

If we complete this a bit, we get N[[x]], the algebra of formal power series with natural number coefficients. But let's categorify it, instead...

What's the free symmetric 2-rig on one generator?

It's the category of STRUCTURE TYPES!

I'll leave the proof of this as a puzzle for the budding category theorists out there. This is supposed to explain very precisely the sense in which structure types are a categorified version of formal power series.

(You might argue that structure types are the categorified version of polynomials, not formal power series, since the free commutative rig on one generator is an algebra of polynomials. But unlike in a rig, we have no trouble doing "infinite sums" in a 2-rig, since we've got arbitrary colimits. So, the difference between polynomials and formal power is not so big. Indeed, there's nothing "formal" about infinite sums in the categorified situation, since divergent sums aren't a problem: a sum will always converges to some set, though possibly an infinite set. This is one of the great reasons to categorify. Of course, the price you pay is that nobody is sure how to handle negative numbers in categorified mathematics.)

Now, formal power series can be multiplied in two ways: ordinary multiplication:

(FG)(x) = F(x) G(x)

which is commutative, and composition:

(FoG)(x) = F(G(x))

which is not. I talked about both of these and their combinatorial meaning for generating functions last time. Ordinary multiplication makes power series into a commutative rig; composition is noncommutative, and it doesn't give us a rig, since it only distributes over addition on one side:

(F+G) o H = FoH + GoH

Even worse, the composite F o G can diverge!

Similarly, structure types can be multiplied in two ways: ordinary multiplication and composition. I described how both of these work last time. Ordinary multiplication makes power series into a symmetric 2-rig; composition is not symmetric, and it doesn't give us a 2-rig, since it only distributes over colimits on one side. However, we don't have to worry about; composition really does put a well-defined monoidal structure on the category of structure types.

The "ordinary" multiplication is what makes structure types into the free symmetric 2-rig on one generator, but composition is also cool. It's related to operads. And here's how.

Recall from "week89" that we can define a "monoid object" in any monoidal category. This leads to another puzzle:

What's a monoid object in the category of structure types with composition as the monoidal structure?

Now, this took me quite a while to deeply understand, but when I did it was great. So, if you have enough category theory under your belt to have any chance at seeing why what I said is true, please work on it for a while and try to understand it. Just follow through all the definitions, until you see that indeed, what I'm claiming is true. It will strengthen your brain... you will literally grow new neurons.

Addendum: after an informally summarized list of axioms for the definition of an operad, I wrote above:

```That's all there is to it!
```

Alas, this isn't quite true. Peter May has subsequently pointed out to me that the book by Stasheff et al omits a crucial clause in the definition of operad, namely that operations like this are well-defined:

```  \    /   |  |    \ /
\  /    |  |     /
/     |  |    / \
/ \    |  |   /   \
\  |  /   |   \   /
\ | /    |    \ /
---    ---   ---
| a |  | b | | c |
---    ---   ---
\    |    /
\   |   /
\  |  /
-----
|  d  |
-----
|
|
```
Here we can either compose the operations a,b,c with d and then apply a permutation to the arguments of the result, or apply permutations to the arguments of a,b, and c and then compose the resulting operation with d - we get the same answer either way!

I hope in some future edition they'll be able to correct this mistake.