# August 10, 2006 {#week237}

This Week I'd like to talk about math books in Shanghai, and Urs
Schreiber's blog entry on the gauge $3$-group of M-theory. But first:
$$\includegraphics[max width=0.65\linewidth]{../images/klein_quartic_egan.png}$$

1) Greg Egan, "Klein's quartic equation", `http://gregegan.customer.netspace.net.au/SCIENCE/KleinQuartic/KleinQuarticEq.html`

I discussed Klein's quartic curve in ["Week 214"](#week214) and
["Week 215"](#week215). The idea is to take the nontrivial complex
solutions of
$$u^3 v + v^3 w + w^3 u = 0$$
and "projectivize" them --- in other words, count two as the same if one
is just a multiple of the other:
$$(u',v',w') = c(u,v,w)$$
The result is a $3$-holed Riemann surface with the maximum number of
symmetries! Here by a "symmetry" I mean a conformal transformation
mapping the surface to itself. Back in 1893 Hurwitz proved something
quite bizarre: an $n$-holed Riemann surface can't have more than $84(n-1)$
symmetries if $n > 1$. So, a $3$-holed Riemann surface can't have more
than 168 symmetries --- and Klein's quartic curve has exactly that many!

These 168 symmetries were constructed by Klein way back in 1879, but
Egan gives an elementary proof that uses only algebra and a bit of
calculus... and a lot of cleverness. And, his page has a wonderful
spinning picture of the *real* solutions of
$$u^3 v + v^3 w + w^3 u = 0.$$
This is what you see above.

As you can see, it consists of lots of lines through the origin,
including the $u$, $v$, and $w$ axes. When we "projectivize", we get one
point for each of these lines, so we get a curve which is the real
version of Klein's quartic curve. This curve has an obvious $3$-fold
symmetry, from cyclically permuting the coordinate axes. The rest of the
168 symmetries are only easy to visualize when we go to the complex
version --- as Egan explains.

It's great that Egan can draw this thing in Perth and I can easily see
it my apartment here in Shanghai --- I feel like I'm living in a
futuristic world, and I'm only 45. What it'll be like when I'm 64?

Another pleasant thing about life in Shanghai, at least for a well-off
visitor from America, is how cheap everything is. It's clear why the US
is running an enormous trade deficit: there's a vast economic
differential. Stopping the flow of goods one way and dollars the other
would be like damming the Niagara Falls.

For example, last night I saw this excellent hardcover book on sale for
78 yuan, or about \$10:

2) Yu. I. Manin and A. A. Panchishkin, _Introduction to Modern Number Theory_, second edition, Science Press, 2005.

It's a great overview of number theory, from the basics through class
field theory to $L$-functions, modular forms and the Langlands program!
It's wisely divided into three sections: "problems and tricks",
"ideas and theories", and "analogies and visions". Back when I used
to hate number theory, I thought it was all problems and tricks. Now
I'm beginning to learn some of the ideas and theories, and I hope
eventually to grasp the analogies and visions discussed here --- for
example, the analogy between Arakelov geometry and noncommutative
geometry.

\$10 is a nice price for a math book. If you buy this one from Springer
Verlag, you'll pay ten times that. Illegal knockoffs of Western books
are common in China, but I think the one I saw is legal, since Springer
has signed an agreement with Science Press, which is run by the Chinese
Academy of Sciences. In exchange for letting Science Press publish
Springer books in China at affordable prices, Springer gets to publish
translations of Chinese journals in the West at unaffordable prices.

By the way --- after checking out the bookstore, I went out to the street
vendors and bought an excellent dinner of rice, sausage and vegetables
for 35 yuan --- about 40 cents US. It was cooked by a husband and wife in
a wok on a cart.

Just after I bought it, someone yelled the Chinese equivalent of
"cops!", and all the street vendors suddenly dashed away with their
carts, leaving only the woman, who kindly handed me my dinner in a
styrofoam pack before walking off. They clearly had this down to a fine
art: it all happened faster than my brain could process. I guess the
cops don't allow street vendors there.

I only wish I'd noticed: did the street vendors turn off their gas
stoves before running, or run while still cooking?

Anyway, on to some serious math and physics.

You've probably heard of some mysterious thing called "M-theory" that
lives in 11 dimensions. Back in ["Week 158"](#week158) and
["Week 159"](#week159) I took a stab at understanding this. Now
I'll try again, with a lot of help from Urs Schreiber:

3) Urs Schreiber, "Castellani on free differential algebras in supergravity: gauge $3$-group of M-theory", `http://golem.ph.utexas.edu/string/archives/000840.html`

Calling M-theory a "theory" is a bit misleading, because nobody knows
what this theory is! There's just a lot of clues pointing to its
existence. It seems to be the quantum version of a well-defined
classical field theory called "$11$-dimensional supergravity". And, it
seems to involve $2$-branes and $5$-branes: $2$- and $5$-dimensional membranes
that trace out $3$- and $6$-dimensional surfaces in spacetime, just like
strings trace out $2$-dimensional surfaces.

Back in ["Week 158"](#week158) I wrote down a Lagrangian for 11d
supergravity. This is a truly monstrous thing involving three fields:

A) a frame field $e$ --- the "graviton",

B) a field $\psi$ taking values in the real spin-$3/2$ representation of the
11d Lorentz group --- the "gravitino",

C) a $3$-form $A$.

When it was discovered back in 1978, people were interested in 11d
supergravity mainly because it was the highest-dimensional theory they
could concoct that includes general relativity and supersymmetry --- a
symmetry that interchanges bosons and fermions, in this case gravitons
and gravitinos --- without including any particles of spin $> 2$. So, the
fact that it looked like a mess wasn't such a big deal. But now that
some people are taking it very seriously, it's worth trying to
understand the math behind it more deeply, to see what makes it tick.

For example: what's so great about 11 dimensions? And: what's the
reason for that $3$-form?

I'm not a huge fan of string theory, but I like puzzles of this sort ---
finding patterns that make certain things work only in certain
dimensions, and stuff like that. So, I got intrigued when I learned that
super-Yang-Mills theory and superstring theory are nice in dimension 10
because of special properties of the octonions --- see
["Week 104"](#week104). Maybe a little extra stretch could bring us
to dimension 11?

I got even more intrigued when I ran across two competing explanations
for that $3$-form in 11d supergravity. One was that it's a connection on
a twice categorified version of a $\mathrm{U}(1)$ bundle. The other was that it's
the Chern-Simons form for an $\mathrm{E}_8$ gauge theory.

Let me say a bit about what these means. I talked about categorified
$\mathrm{U}(1)$ bundles in ["Week 210"](#week210), so I'll be sort of brief
about those....

A connection on a $\mathrm{U}(1)$-bundle looks locally like a $1$-form, so we can
integrate it along a path and compute how the phase of charged particle
changes when we move it along that path:
$$
  \begin{gathered}
    x\xrightarrow{f}y
  \\[1em]\mbox{a path $f$ from the point $x$ to the point $y$;}
  \\\mbox{we write this as $f\colon x\to y$.}
  \end{gathered}
$$
Believe it or not, this is the basis of all modern ideas on
electromagnetism!

If we categorify this whole idea once, we get a kind of connection that
looks locally like a $2$-form. Folks call this a "connection on a $\mathrm{U}(1)$
gerbe", but don't let the use of French here intimidate you: they just
do that so they can charge more for the wine. It's just a gadget that
you can integrate over a surface, to compute how the phase of a charged
*string* moves when we slide it along that surface:
$$
  \begin{gathered}
    \begin{tikzpicture}
      \node (x) at (0,0) {$x$};
      \node (y) at (2,0) {$y$};
      \draw[->] (x) .. node[label={[label distance=-1mm]above:{\scriptsize$f$}}]{} controls (0.7,0.7) and (1.3,0.7) .. (y);
      \draw[->] (x) .. node[label={[label distance=-1mm]below:{\scriptsize$g$}}]{}controls (0.7,-0.7) and (1.3,-0.7) .. (y);
      \draw[double,double equal sign distance,-implies] (1,0.5) to node[label={[label distance=-1mm]right:{\scriptsize$F$}}]{} (1,-0.5);
    \end{tikzpicture}
  \\[1em]\mbox{a path $f$ from the point $x$ to the point $y$;}
  \\\mbox{we write this as $f\colon x\to y$.}
  \end{gathered}
$$
And, if we categorify once more, we get a "connection on a $\mathrm{U}(1)$
$2$-gerbe". This is something that looks locally like a $3$-form, which
describes what happens when we move *$2$-branes* around!

If you're wondering why I'm talking about "categorifying", it's
because this:
$$
  x\xrightarrow{f}y
$$
is also a picture of a morphism in a category, while this:
$$
  \begin{tikzpicture}
    \node (x) at (0,0) {$x$};
    \node (y) at (2,0) {$y$};
    \draw[->] (x) .. node[label={[label distance=-1mm]above:{\scriptsize$f$}}]{} controls (0.7,0.7) and (1.3,0.7) .. (y);
    \draw[->] (x) .. node[label={[label distance=-1mm]below:{\scriptsize$g$}}]{}controls (0.7,-0.7) and (1.3,-0.7) .. (y);
    \draw[double,double equal sign distance,-implies] (1,0.5) to node[label={[label distance=-1mm]right:{\scriptsize$F$}}]{} (1,-0.5);
  \end{tikzpicture}
$$
is a picture of a $2$-morphism in a $2$-category and so on. We're talking
about processes between processes between processes... so we're
climbing up the ladder of $n$-categories.

Anyway: since 11d supergravity has a $3$-form in it, and M-theory
apparently has $2$-branes in it, maybe we need to categorify the concept
of a $\mathrm{U}(1)$ bundle twice to understand what's going on here!

I came up with this crazy idea on my own back in
["Week 158"](#week158), but it's an obvious guess after you learn
that the $2$-form field called $B$ in 10d superstring theory really *is* a
connection on a $\mathrm{U}(1)$ gerbe:

4) Alan L. Carey, Stuart Johnson and Michael K. Murray, "Holonomy on D-branes", available as [`arXiv:hep-th/0204199`](http://arxiv.org/abs/hep-th/0204199).

Unfortunately there are some problems with naively pushing this idea up
a dimension. For example, a crucial factor of $1/6$ in the Lagrangian for
11d supergravity is not explained by thinking of $A$ this way.

Another possible explanation was that this $3$-form is the Chern-Simons
form of an $\mathrm{E}_8$ bundle over spacetime:

5) Emanuel Diaconescu, Gregory Moore and Edward Witten, "$\mathrm{E}_8$ gauge theory, and a derivation of K-theory from M-theory", _Adv. Theor. Math. Phys._ **6** (2003) 1031--1134. Also available as [`arXiv:hep-th/0005090`](http://arxiv.org/abs/hep-th/0005090).

6) Emanuel Diaconescu, Daniel S. Freed and Gregory Moore, "The M-theory $3$-form and $\mathrm{E}_8$ gauge theory", available as [`arXiv:hep-th/0312069`](http://arxiv.org/abs/hep-th/0312069).

This idea explains that factor of $1/6$. And, it might move towards an
explanation of how the octonions get into the act, because the group
$\mathrm{E}_8$ is deeply related to the octonions. But as the authors of the above
paper say, "the $\mathrm{E}_8$ gauge field plays a purely topological role and
appears, in some sense, to be a 'fake'." In particular, you don't
see any $\mathrm{E}_8$ connection staring you in the face in the Lagrangian for
11d supergravity that I wrote down in ["Week 158"](#week158).

Later, it started becoming clear that both ideas --- the twice
categorified $\mathrm{U}(1)$ connection and the $\mathrm{E}_8$ gauge theory --- fit together in
some way:

7) Paolo Aschieri and Branislav Jurco, Gerbes, "M5-brane anomalies and $\mathrm{E}_8$ gauge theory", _JHEP_ **0410** (2004), 068. Also available as [`arXiv:hep-th/0409200`](http://arxiv.org/abs/hep-th/0409200).

It all became a lot clearer to me when Urs Schreiber read these papers
and translated them into a language I like:

8) Leonardo Castellani, "Lie derivatives along antisymmetric tensors, and the M-theory superalgebra", available as [`arXiv:hep-th/0508213`](http://arxiv.org/abs/hep-th/0508213).

9) Pietro Fré and Pietro Antonio Grassi, "Pure spinors, free differential algebras, and the supermembrane", available as [`arXiv:hep-th/0606171`](http://arxiv.org/abs/hep-th/0606171).

The idea is to think of 11d supergravity as a twice categorifed gauge
theory --- not just the $3$-form field in 11d supergravity, but all the
fields, in a unified way!

For this, we need to do something much more clever than taking 11d
spacetime and slapping a $\mathrm{U}(1)$ $2$-gerbe on top of it. We need to combine
the graviton, the gravitino and the $2$-form field into a connection on a
*nonabelian* $2$-gerbe.

Here things get a bit technical, but Urs has covered the technical
points quite nicely in his blog, so right now I'll just try to give you
some hand-wavy intuitions.

Very roughly speaking, an connection on a bundle takes any path in
spacetime
$$x\xrightarrow{f}y$$
and gives you an element of some *group*, which says how a particle
would transform if you moved it along this path. This group could be
$\mathrm{U}(1)$ --- the group of phases --- or it could be something more fun, like a
*nonabelian* group.

If we categorify this concept, we get the concept of a connection on a
"$2$-bundle" (which is more or less the same as a gerbe). Such a
connection takes any path and gives you an *object* in some *$2$-group*,
but it also takes any surface like this:
$$
  \begin{tikzpicture}
    \node (x) at (0,0) {$x$};
    \node (y) at (2,0) {$y$};
    \draw[->] (x) .. node[label={[label distance=-1mm]above:{\scriptsize$f$}}]{} controls (0.7,0.7) and (1.3,0.7) .. (y);
    \draw[->] (x) .. node[label={[label distance=-1mm]below:{\scriptsize$g$}}]{}controls (0.7,-0.7) and (1.3,-0.7) .. (y);
    \draw[double,double equal sign distance,-implies] (1,0.5) to node[label={[label distance=-1mm]right:{\scriptsize$F$}}]{} (1,-0.5);
  \end{tikzpicture}
$$
and gives you a *morphism* in this $2$-group. You see, $2$-group is a kind
of category that acts like a group, and a category has "objects" and
"morphisms". The morphisms go between objects. For more on $2$-groups,
try:

10) "Higher-dimensional algebra V: $2$-Groups", with Aaron D. Lauda, _Theory and Applications of Categories_ **12** (2004), available at `http://www.tac.mta.ca/tac/volumes/12/14/12-14abs.html`. Also available as [`math.QA/0307200`](http://arxiv.org/abs/math.QA/0307200).

If we categorify once more, we get connections on a "$3$-bundle", which
is more or less the same thing as a "$2$-gerbe" --- unfortunately the
numbering systems are off by one. This gives us objects, morphisms and
$2$-morphisms in a *$3$-group*, which describe what happens when we move
particles, strings and $2$-branes.

And so on:

|    |    |
| :- | :- |
| group | point particles |
| $2$-group | point particles and strings |
| $3$-group | point particles, strings and $2$-branes |
| $4$-group | point particles, strings, $2$-branes and $3$-branes |

etc.

So, if 11d supergravity is a twice categorified gauge theory, we need to
know its symmetry *$3$-group*.

But actually, since we're doing geometry, this $3$-group should be a
"Lie $3$-group". In other words, very roughly speaking, a $3$-group that
has a *manifold* of objects, a manifold of morphisms, and a manifold of
2-morphisms, where all the operations are smooth.

But actually, since we're doing supersymmetric geometry, we need a
"Lie 3-supergroup"! In other words, very roughly speaking, a $3$-group
that has a *supermanifold* of objects, a supermanifold of morphisms, and
a supermanifold of $2$-morphisms, where all the operations are smooth.
(Maybe I should say "supersmooth", just to be consistent.)

If you don't know what a supermanifold is, now is probably not the time
to learn. I mean, not right this second. The point is just this:
supersymmetry infests everything once you let it in the door, just like
$n$-categories, and just like manifolds --- and now we're doing all three.

In fact, nobody has even written down a rigorous definition of a Lie
$3$-supergroup yet! But, Lie algebras are in some ways simpler than Lie
groups, and they're a good start, so we can be glad that people *do*
know what a Lie $3$-superalgebra is!

And Urs describes, in his blog, the relevant Lie $3$-superalgebra for 11d
supergravity!

I would like to say more about this, but it's getting a bit tough
trying to talk about this stuff in a fun, easily accessible style, and I
have the feeling I'm no longer succeeding. In fact, I don't think I
can give a "fun, easily accessible" description of this specific Lie
$3$-superalgebra --- at least not yet. So, now I'll completely give up
trying to be comprehensible, and simply state some facts.

As shown here:

11) "Higher-dimensional algebra VI: Lie $2$-Algebras", with Alissa Crans, _Theory and Applications of Categories_ **12** (2004), available at `http://www.tac.mta.ca/tac/volumes/12/15/12-15abs.html`   Also available as [`math.QA/0307200`](http://arxiv.org/abs/math.QA/0307200).

the category of Lie $n$-algebras is equivalent to the category of $L_\infty$
algebras which as chain complexes have only $n$ nonvanishing terms, the
$0$th to the $(n-1)$st. $L_\infty$ algebras are just algebras of Stasheff's $L_\infty$
operad in the category of chain complexes of vector spaces --- see
["Week 191"](#week191) and especially these:

12) Martin Markl, Steve Schnider and Jim Stasheff, _Operads in Algebra, Topology and Physics_, AMS, Providence, Rhode Island, 2002.

    James Stasheff, Hartford/Luminy talks on operads, available at `http://www.math.unc.edu/Faculty/jds/operadchik.ps`

But, we can replace vector spaces by $\mathbb{Z}/2$-graded vector spaces and
everything still works. Physicists call $\mathbb{Z}/2$-graded vector spaces "super
vector spaces". So, a "Lie $n$-superalgebra" is an algebra of the $L_\infty$
operad in the category of chain complexes of super vector spaces.

Given this, to specify a Lie $3$-superalgebra we first need to specify the
$0$-chains, then the $1$-chains, then the $2$-chains.

For the particular one Urs mentions, we have

- $\{0\text{-chains}\}$ = 11d Poincaré Lie superalgebra
- $\{1\text{-chains}\}$ = $\{0\}$
- $\{2\text{-chains}\}$ = $\mathbb{R}$

Here $\mathbb{R}$ is the real numbers, and this $1$-dimensional thing is what
ultimately gives the $3$-form field $A$ in 11d supergravity. As a vector
space, the 11d Poincaré Lie superalgebra is the direct sum of an even
part, which is the usual Poincaré Lie algebra $\mathfrak{iso}(11,1)$, and an odd
part, which is the $32$-dimensional real spinor rep of $\mathfrak{so}(11,1)$. These
give the graviton (or more precisely the Levi-Civita connection) and the
gravitino in 11d supergravity.

Next we need to make this stuff into a chain complex. That's easy: the
differential *has* to be zero.

Next, we need to specify the $L_\infty$ structure on this chain complex.
First, we need a binary bracket operation, like in an ordinary Lie
superalgebra. The bracket of $0$-chains is the usual bracket in the 11d
Poincaré Lie superalgebra. All the other binary brackets are zero.

Then, we need a ternary bracket operation, which expresses how the
Jacobi identity holds only up to chain homotopy. This is zero.

Then, we need a quaternary bracket operation (since that chain homotopy
satisfies its own identity only up to chain homotopy). This is nonzero:
when we take the quaternary bracket of four $0$-chains we get a $2$-chain,
and there's a nontrivial way to define this! This is the interesting
bit, since ultimately it relates the graviton/gravitino to the $3$-form
field.

How do we get that quaternary bracket? Well, here's where things get
funky: D'Auria and Fré dreamt up a formula that gives a number from 2
spinors and 2 vectors:
$$(\psi, \varphi, v, w) \mapsto \overline{\psi} \Gamma^{ab} \varphi v_a w_b.$$
And, magically, in 11
dimensions this gives a $4$-cocycle on the Poincaré Lie superalgebra! The
proof of this uses some Fierz identity in 11 dimensions:

12) R. D'Auria and Pietro Fré, "Geometric supergravity in $D = 11$ and its hidden supergroup", _Nucl. Phys._ **B201** (1982), 101--140. Also available at `http://www.math.uni-hamburg.de/home/schreiber/sdarticle.pdf`

And, from HDA6 we know that the $4$-cocycle condition is just what's
needed to make the quaternary bracket satisfy the identity we need for a
Lie $3$-superalgebra. (Alissa and I just did the calculation for Lie
$n$-algebras, but the "super" stuff should work too with a few signs
thrown in.)

So, this is all very cool, but I need to understand Fierz identities in
different dimensions to see what if anything is special to 11d here ---
or, alternatively, work out the cohomology of Poincaré Lie
superalgebras, to see when they can be deformed to Lie $n$-superalgebras.
Sounds like a lot of work --- maybe someone already did it. Actually
D'Auria and Fre make it look like a matter of understanding tensor
products of irreps of $\mathfrak{so}(n,1)$, which is not bad. A worthwhile project in
any event.

I also need to understand what all this has to do with $\mathrm{E}_8$. For that
the paper by Diaconescu, Freed and Moore should help.

Well, this is just the beginning, but Urs explains the rest.

------------------------------------------------------------------------

**Addenda**: I thank Noam Elkies for a correction. Aaron Bergman has
this to say about $\mathrm{E}_8$ and M-theory:

> John Baez wrote:
>
> > I also need to understand what all this has to do with $\mathrm{E}_8$.
>
> $\mathrm{E}_8$ might not have much of anything to do with this. As mentioned in
> Diaconescu, Freed and Moore, $\mathrm{E}_8$ appears to function solely as a
> convenient stand-in for $K(\mathbb{Z},3)$.
>
> On the other hand, the split form of the $\mathrm{E}$-series (up to $\mathrm{E}_{11}$ if
> you're feeling particularly speculative) is known to show up in
> describing the fields of 11D SUGRA, but the I don't think anyone
> knows of a connection between the two $\mathrm{E}_8$s. Just to add to the fun,
> $\mathrm{E}_8$ gauge fields also show up on the fixed points of M-theory on
> $S^1/\mathbb{Z}_2$ giving the $\mathrm{E}_8 \times \mathrm{E}_8$ heterotic string.
>
> Aaron

Urs Schreiber replied:

> Aaron Bergman wrote:
>
> > $\mathrm{E}_8$ gauge fields also show up on the fixed points of 
> > M-theory on $S^1/\mathbb{Z}_2$ giving the $\mathrm{E}_8 \times \mathrm{E}_8$ heterotic string.
>
> The topological part of the membrane action involves the integral of
> the sugra $3$-form over the worldvolume. By DFW, part of that $3$-form can
> be thought of as an $\mathrm{E}_8$ CS $3$-form.
>
> So part of the membrane action looks similar to an $\mathrm{E}_8$ CS-theory over
> the worldvolume.
>
> Now let the membrane have a boundary. A bulk $\mathrm{E}_8$ CS-theory is well
> known to induce an $\mathrm{E}_8$ WZW theory on the boundary.
>
> Could this be the connection between the DWF $\mathrm{E}_8$ and the
> Horava-Witten $\mathrm{E}_8$?
>
> I have asked this question before:
> `http://golem.ph.utexas.edu/string/archives/000791.html`. Jarah then
> agreed that this must be about right. But it is not completely clear
> to me yet.
>
> One problem is that in CS-theory we vary the connection, while in the
> topological membrane the $\mathrm{E}_8$ connection on the background is fixed
> and we vary the embedding by which we pull it back to the worldvolume.
> Under suitable assumptions that might be equivalent to varying an $\mathrm{E}_8$
> connection on the worldvolume itself?

Aaron Bergman replied:

> Urs Schreiber wrote:
>
> > Now let the membrane have a boundary. A bulk $\mathrm{E}_8$ CS-theory is
> > well known to induce an $\mathrm{E}_8$ WZW theory on the boundary.
> >
> > Could this be the connection between the DWF $\mathrm{E}_8$ and the
> > Horava-Witten $\mathrm{E}_8$?
>
> A similar observation was made by Horava way back in
> [`arXiv:hep-th/9712130`](http://arxiv.org/abs/hep-th/9712130) (in the
> final section).
>
> Aaron