Also available at http://math.ucr.edu/home/baez/week237.html
August 10, 2006
This Week's Finds in Mathematical Physics (Week 237)
John Baez
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:
1) Greg Egan, Klein's quartic equation,
http://gregegan.customer.netspace.net.au/SCIENCE/KleinQuartic/KleinQuarticEq.html
I discussed Klein's quartic curve in "week214" and "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.
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
6-fold symmetry, from permuting the coordinate axes. The rest of
the 168 symmetries are only evident 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 "week158" and "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 "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 "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 U(1) bundle. The other was that it's the Chern-Simons
form for an E8 gauge theory.
Let me say a bit about what these means. I talked about
categorified U(1) bundles in "week210", so I'll be sort of
brief about those....
A connection on a 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:
f
x------------>-----y a path f from the point x to the point y:
we write this as f: x -> y
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 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:
f
----------->-----
/ || \
x ||F y a path-of-paths F from the path f to the path g:
\ \/ / we write this as F: f => g
----------->-----
g
And, if we categorify once more, we get a "connection on a
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:
f
x------------>-----y
is also a picture of a morphism in a category, while this:
f
----------->-----
/ || \
x ||F y
\ \/ /
----------->-----
g
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 U(1) bundle twice to understand what's going
on here!
I came up with this crazy idea on my own back in "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 U(1) gerbe:
4) Alan L. Carey, Stuart Johnson and Michael K. Murray,
Holonomy on D-branes, available as 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 E8 bundle over spacetime:
5) Emanuel Diaconescu, Gregory Moore and Edward Witten,
E8 gauge theory, and a derivation of K-theory from M-theory,
Adv. Theor. Math. Phys. 6 (2003) 1031-1134. Also available as
hep-th/0005090.
6) Emanuel Diaconescu, Daniel S. Freed and Gregory Moore,
The M-theory 3-form and E8 gauge theory, available as
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 E8 is deeply related to the octonions.
But as the authors of the above paper say, "the E8 gauge
field plays a purely topological role and appears, in some
sense, to be a 'fake'." In particular, you don't see any
E8 connection staring you in the face in the Lagrangian for
11d supergravity that I wrote down in "week158".
Later, it started becoming clear that both ideas - the twice
categorified U(1) connection and the E8 gauge theory - fit
together in some way:
7) Paolo Aschieri and Branislav Jurco, Gerbes, M5-brane
anomalies and E_8 gauge theory, JHEP 0410 (2004), 068.
Also available as 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 hep-th/0508213.
9) Pietro Fre and Pietro Antonio Grassi, Pure spinors,
free differential algebras, and the supermembrane, available
as 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 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
f
x------------>-----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 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:
f
----------->-----
/ || \
x ||F y
\ \/ /
----------->-----
g
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.
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.
the category of Lie n-algebras is equivalent to the category
of L-infinity algebras which as chain complexes have only n
nonvanishing terms, the 0th to the (n-1)st. L-infinity algebras
are just algebras of Stasheff's L-infinity operad in the category
of chain complexes of vector spaces - see "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 Z/2-graded vector spaces and
everything still works. Physicists call Z/2-graded vector spaces
"super vector spaces". So, a "Lie n-superalgebra" is an algebra of
the L-infinity 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-chains} = 11d Poincare Lie superalgebra
{1-chains} = {0}
{2-chains} = R
Here 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 Poincare Lie superalgebra is the
direct sum of an even part, which is the usual Poincare Lie
algebra iso(11,1), and an odd part, which is the 32-dimensional
real spinor rep of 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-infinity 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 Poincare Lie superalgebra. All the other
binary brackets are zero.
Then, we need a ternary bracket operation (since 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 Fre dreamt up a formula that
gives a number from 2 spinors and 2 vectors:
(psi, phi, v, w) |-> psibar Gamma^{ab} phi v_a v_b
And, magically, in 11 dimensions this gives a 4-cocycle on
the Poincare Lie superalgebra! The proof of this uses some
Fierz identity in 11 dimensions:
13) R. D'Auria and Pietro Fre, 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 Poincare 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 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 E8.
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 E8 and M-theory:
John Baez wrote:
> I also need to understand what all this has to do with E8
E8 might not have much of anything to do with this. As mentioned in
Diaconescu, Freed and Moore, E8 appears to function solely as a
convenient stand-in for K(Z,3).
On the other hand, the split form of the E-series (up to E11 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 E8s. Just to add to the fun, E8 gauge
fields also show up on the fixed points of M-theory on S^1/Z_2 giving
the E8 x E8 heterotic string.
Aaron
Urs Schreiber replied:
Aaron Bergman wrote:
> E8 gauge fields also show
> up on the fixed points of M-theory on S^1/Z_2 giving the E8 x E8
> 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 E8 CS 3-form.
So part of the membrane action looks similar to an E8 CS-theory
over the worldvolume.
Now let the membrane have a boundary. A bulk E8 CS-theory is
well known to induce an E8 WZW theory on the boundary.
Could this be the connection between the DWF E8 and the
Horava-Witten E8?
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 E8 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 E8
connection on the worldvolume itself?
Aaron Bergman replied:
Urs Schreiber wrote:
> Now let the membrane have a boundary. A bulk E8 CS-theory is
> well known to induce an E8 WZW theory on the boundary.
>
> Could this be the connection between the DWF E8 and the
> Horava-Witten E8?
A similar observation was made by Horava way back in hep-th/9712130 (in
the final section).
Aaron
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