Also available at http://math.ucr.edu/home/baez/week191.html
January 11, 2003
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
hepth/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" 4dimensional 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
hepth/0210178.
"Dimensional reduction of (super)gravity theories to 3 dimensions
results in sigma 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 sigma 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;
KaluzaKlein modifications and ChernSimons 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 E_n 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 Mtheory, available as hepth/0301037.
"We explain how structures related to octonions are ubiquitous in
Mtheory. All the exceptional Lie groups, and the projective Cayley
line and plane, appear in Mtheory. Exceptional G2holonomy manifolds
show up as compactifying spaces, and are related to the M2 Brane and
3form. We review this evidence, which comes from the initial 11dim
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 nonassociativity)
in Mtheory."
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
hepth/0301050.
"Quarks and leptons charges and interactions are derived from gauge
theories associated with symmetries. Their spacetime labels come
from representations of the noncompact 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 spacetime labels has not been as evident
as they link spinors and tensors under space rotations, which flies
in the face of the spinstatistics connection. We discuss a way to
circumvent this difficulty in trying to generalize elevendimensional
supergravity."
I haven't read this, but indeed, it's really annoying how structures
like triality mix integer and halfinteger 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 O_n of abstract
`nary operations' for each natural number n, together with rules for
composing these operations. We can think of an nary 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 g_1,..,g_n into the inputs of
an nary operation f, obtaining a new operation which we call
f o (g_1,...,g_n). 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 welldefined:
\ /  \  / \ /
\ /  \  / \ /
   
       
   
\  / /
\  / /
\  / /
  
     
  
\  /
\  /
\  /
\  /
\  /
\  /
\  /

 



(This picture has a 0ary operation in it, just to emphasize
that this is allowed.) We can permute the inputs of an nary
operation and get a new operation:
\ / /
/ /
/ \ /
/ /
/ / \
\  /

 



We demand that this give an action of each permutation group S_n
on each set O_n. 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:
6) James Stasheff, Hartford/Luminy talks on operads, available
at http://www.math.unc.edu/Faculty/jds/operadchik.ps.
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
higherdimensional 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 ksphere
to X. We define the "kth loop space" of X, Omega^k(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 Omega^k(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 Omega^k 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 kfold loop space.
For example, Omega^1(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, Omega^k(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 f
in O_n being realized as actual nary 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
kfold 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 kfold
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 O_n 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 scaredycats 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: FinSet_0 > Set
where FinSet_0 is the groupoid of finite sets and bijections,
and Set is the category of sets and functions. FinSet_0 is
equivalent to the category that has one object, "the nelement
set", for each n, with the morphisms from this object to itself
forming the permutation group S_n. So, we can also think of a
structure type as consisting of a set F_n for each n, together
with an action of S_n 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(FinSet_0, 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 setbased 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 2rigs
the free commutative rig on the free symmetric 2rig on
no generators: N no generators: Set
the free commutative rig on the free symmetric 2rig on
one generator: N[x] on generator: Set[[x]] = hom(FinSet_0, 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 2rig 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 2rig 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 "2rig" 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 2rig 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
2rig: then FinSet will be the free 2rig 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 2rig 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 2rig, 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
2rig; composition is not symmetric, and it doesn't give us a 2rig,
since it only distributes over colimits on one side. However, we
don't have to worry about divergence; composition really does put
a welldefined monoidal structure on the category of structure types.
The "ordinary" multiplication is what makes structure types into
the free symmetric 2rig 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?
And the answer is: AN OPERAD!
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 welldefined:
\ /   \ /
\ /   /
/   / \
/ \   / \
\  /  \ /
\  /  \ /
  
 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.

Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumpingoff point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html