Natural Science Forum / Physics / Research / March 2005

Also available at http://math.ucr.edu/home/baez/week212.html

March 26, 2005
This Week's Finds in Mathematical Physics - Week 212
John Baez    

As you may know, theoretical particle physics is highly enamored of
"supersymmetry" these days.  This is not because there's a shred of
experimental evidence for it - there's not - but just because it's such
a cool idea from a mathematical point of view.  Mathematicians should
have gotten this idea and run with it first, but physicists did - and
maybe it's turned them into mathematicians.

The unarguable central core of this idea is that everything is made of
bosons and fermions.  In the Standard Model, most bosons are "force
carriers", like photons, which carry the electromagnetic force.  Fermions
are more like what we'd normally call "matter": leptons and quarks, for
example.  The one big exception is the Higgs boson, which gives elementary
particles their mass and... umm... hasn't been seen yet!

But, at a more fundamental level, the really important thing is that
bosons commute:

xy = yx

while fermions anticommute:

xy = -yx

Also, in case you're wondering, bosons commute with fermions.

But already, most mathematicians reading this will be confused and unhappy.
What does it mean for two particles to commute, much less anticommute?  
Does an apple commute with a grape?   Here in the suburbs of Los Angeles
almost everyone commutes, but that's not what we're talking about.

The whole idea of particles commuting or anticommuting only occurred to
people after they invented quantum theory, where the state of any
system is described by a unit vector in some Hilbert space.  In quantum
theory, if you have a system in some state x, and you check to see if
it's in the state y, your experiment gives you the answer "yes" with
probability

the square of the absolute value of the inner product of x and y.

There!  Now you know quantum theory.

Given this setup, when you have a system consisting of two particles, the
first in some state x and the second in some state y, it's natural to
write the state of the whole system as a kind of product xy.  But
then you have to figure out what rules you want this product to satisfy!

If you require it to be commutative:

xy = yx

you're saying that there's no difference between the FIRST particle
being in state x and the SECOND particle being in state y, and the
other way around.  In other words, the particles don't have little
name tags on them saying who they are.

This seems reasonable, and particles satisfying this rule are called
"bosons".  But, there's another popular option, called "fermions":

xy = -yx

Here again, the particles don't have name tags, since if we put
the whole system in the state xy and check to see if it's in the
state yx, we get the same answer as when we check to see if it's
in the state xy!  See:

           = |<xy,xy>|^2

thanks to the absolute value.  This means that the states xy and
yx are indistinguishable.  

Reading what I just said, you'd be forgiven for wondering what's the
big difference between fermions and bosons!  After all, that absolute
value in the formula probabilities just ignores minus signs.

One difference is the "Pauli exclusion principle".  Take a pair of
fermions and check to see if they're both in the state x.  The
probability is always zero, since

xx = -xx  

so xx = 0.   So, fermions are antisocial: that's why the electrons
in an atom form "shells" with different electrons in different states,
instead of all hanging out at the lowest energy state.

Bosons, by contrast, are gregarious: when a store clerk uses a laser
scanner to ring up your purchases, that beam of red light is a bunch of
photons all in the same state!   A laser is a quintessentially quantum-
theoretic gadget - we live in a marvelous world, where such things are
taken for granted.

After getting used to these ideas for a while - Bose and Einstein worked
out the idea of bosons in 1924, Pauli came up with his exclusion principle
in 1925, and Dirac systematized the whole business in 1926 - physicists
eventually started looking for symmetries that relate bosons and fermions.
Supersymmetries!  They're not seen in nature, but physicists were looking
to see if they're mathematically possible.  They turn out not only to
be possible, but fascinating.  

Formulating supersymmetries in a slick way requires that we take
everything we knew about linear algebra and generalize it by letting
all our vector spaces have both an "even" or "bosonic" part and an
"odd" or "fermionic" part.  Mathematically this just amounts to writing
our vector space as a direct sum

V = V_0 + V_1

where V_0 is the "even part" and V_1 is the "odd part".  Such a thing
is called a "Z/2-graded vector space", or "super vector space".  

So far this is pathetically simple.  But then - and this is the really
crucial part! - whenever we multiply things, we have to follow this rule:

      even    odd
    ----------------
even|  even    odd
odd |  odd     even

It's a little confusing, since this isn't what happens when you
multiply even and odd numbers - it's what happens when you ADD them.  
But, one quickly adapts.  

Also, when we generalize equations involving multiplication, we must
remember to stick in an extra minus sign whenever we switch two odd
vectors.  

So, for example, the usual concept of an algebra gets replaced
by that of a "superalgebra".  This is a super vector space A
equipped with an associative product and unit such that when we
multiply even and/or odd vectors, the rules in the above table hold.
We say a superalgebra is "supercommutative" if

xy = yx

when at least one of x,y lives in the even part A_0, while

xy = -yx

when both x and y live in A_1.  

Similarly we can define super Lie algebras, super Lie groups,
supermanifolds, and so on....

People have done a lot of work on this stuff: it would take me days to
explain it all - even longer if I actually knew something about it.  

But right now, I just want to zoom in the direction of super division
algebras.  These are not the most important aspect of "superalgebra" -
but they're pretty cool, and Todd Trimble has been explaining them to
me lately.  Everything interesting I'm about to say is due to him.

As you know, I'm inordinately fond of the normed division algebras:
the real numbers, complex numbers, quaternions and octonions.  They're
so beautiful, it's a little sad at times that there are only four!
Could superalgebra allow for more?

YES!  And, they turn out to be related to Bott periodicity.

Nobody seems to have pondered *nonassociative* super division algebras
yet, but Deligne has a nice article about the associative ones, which
I mentioned in "week211".  I'll give more references later.

So, what's the idea?

I've already told you what a superalgebra is.  We say it's a "super
division algebra" if every nonzero element that's purely even or
purely odd is invertible.  

That's pretty easy.  What are they like?  

Well, I don't completely understand all the options yet, so I'll
just list the "central" super division algebras over the real numbers,
namely those where the elements that supercommute with everything
form a copy of the real numbers.  There turn out to be 8, and their
beautiful patterns are best displayed in a circular layout:

                                   R

                                   0
          R[e] mod e^2 - 1    7          1   R[e] mod e^2 + 1


C[e] mod e^2 - 1, ei + ie   6               2   C[e] mod e^2 + 1, ei + ie


          H[e] mod e^2 + 1   5           3   H[e] mod e^2 - 1
                                   4

                                   H


What does this notation mean?  Well, as usual R, C, and H stand for
the reals, complex numbers, and quaternions.  In all but two cases,
we start with one of those algebras and throw in an odd element "e"
satisfying the relations listed: e is either a square root of +1 or
of -1, and in the complex cases it anticommutes with i.  

So, for example, super division algebra number 1:

R[e] mod e^2 + 1

is just the real numbers with an odd element thrown in that satisfies
e^2 + 1 = 0.  In other words, it's just the complex numbers made into
a superalgebra in such a way that i is *odd*.

The real reason I've arranged these guys in a circle numbered from
0 to 7 is to remind you of the Clifford algebra clock in "week210",
where I discussed the super Brauer group of the real numbers, and
said it was Z/8.  

Indeed, the central super division algebras are a complete set of
representatives for this super Brauer group!  In particular, the
Clifford algebra C_n is super Morita equivalent to the nth algebra
on this circle:

C_0 = R           ~ R
C_1 = C           ~ R[e] mod e^2 + 1
C_2 = H           ~ C[e] mod e^2 + 1, ei + ie
C_3 = H + H       ~ H[e] mod e^2 - 1
C_4 = H(2)        ~ H
C_5 = C(4)        ~ H[e] mod e^2 + 1
C_6 = R(8)        ~ C[e] mod e^2 - 1, ei + ie
C_7 = R(8) + R(8) ~ R[e] mod e^2 - 1

where ~ means "super Morita equivalent".

I think this is cool.  I'm not quite sure what to do with it yet,
though.  How much of what people ordinarily do with division algebras
can be done with super division algebras?  For example, can we define
projective spaces over super division algebras?  (See "week106" and
"week145" for why that would be interesting.)

To read more about this, try:

1) Pierre Deligne, Notes on spinors, in Quantum Fields and Strings:
A Course For Mathematicians, volume 1, American Mathematical Society,
Providence, 1999.  Also available at
http://www.math.ias.edu/QFT/fall/spinors.ps

A lot of the ideas go back to here:

2) C. T. C. Wall, Graded Brauer groups, J. Reine Angew. Math. 213
(1963/1964), 187-199.

and here's another good reference:

3) Peter Donovan and Max Karoubi, Graded Brauer groups and K-theory
with local coefficients, Publications Math. IHES 38 (1970), 5-25.
Also available at http://www.math.jussieu.fr/~karoubi/Donavan.K.pdf

I should admit that I have a yearning to classify *nonassociative*
super division algebras.  Has anyone ever tried this?  It's already
plain to see that we have two 16-dimensional nonassociative super
division algebras:

O[e] mod e^2 + 1

and

O[e] mod e^2 - 1

where e is an odd element that commutes with all the octonions.
(I should have mentioned this before, when talking about H[e]:
even though the quaternions are noncommutative, we assume that
e commutes with all of them.)  Maybe one of these algebras
deserves to be called the SUPEROCTONIONS.  I bet these or
something awfully similar are lurking around in string theory.  

Hmm... next I wanted to write something about the topology of Bott
periodicity and how *that* fits into what I've been discussing,
but I'm running out of energy.  Let me say it briefly, in a way
that only experts will understand, just in case I never get
around to a decent explanation.  

Two super algebras are super Morita equivalent precisely when
they have equivalent categories of super representations.  So,
the super Brauer group really consists of 8 different *categories*:
the categories SuperRep(C_n), where Bott periodicity says

SuperRep(C_{n+8}) ~ SuperRep(C_n)

Moreover these are symmetric monoidal categories, since direct
summing lets us "add" objects in these categories in a nice way.

A long time ago, Graeme Segal figured out how to take a symmetric
monoidal category and get an infinite loop space from it.
I explained this construction in "week199", but for a much more
detailed and intense treatment with lots of references to earlier
work, try:

4) R. W. Thomason, Symmetric categories model all connective
spectra, Theory and Applications of Categories 1 (1995), 78-118.
Available at http://www.tac.mta.ca/tac/volumes/1995/n5/1-05abs.html
 
If we do this to SuperRep(C_n), I think we get something like

Omega^n(kO)

that is, the n-fold loop space of something called kO, the "connective
K-theory spectrum", which I explained in "week105".  The fact that this
repeats with period 8:

Omega^{n+8}(kO) ~ Omega^n(kO)

is the topological version of Bott periodicity - see "week105" for more.
So, we get the topological version of Bott periodicity from the algebraic
version by turning symmetric monoidal categories into infinite loop
spaces!  

But, the interesting puzzle here is: what process can we do to SuperRep(C_n)
to get SuperRep(C_{n+1}), which is the algebraic version of looping?
And I think the answer is: "taking super representations of C_1 in it".
You see,

C_1 tensor C_n = C_{n+1}

where I'm using the super tensor product of superalgebras, and this
means that the category of representations of C_1 *in* SuperRep(C_n)
is SuperRep(C_{n+1}).  

And, if I were trying to really explain this instead of merely scribbling
notes about it, I would try to say why this is because C_1 is the complex
numbers, and the unit circle in the complex numbers is related to LOOPS.

But, sigh, that will have to wait.

One more thing before I quit for today...

I just saw a cool paper by Dror Bar-Natan, Thang Le and Dylan Thurston
about the "Duflo isomorphism".  This is a cousin of the Poincare-Birkhoff-
Witt theorem, which in its best form says that the universal enveloping
algebra UL of a Lie algebra L is isomorphic *as a coalgebra* to the
symmetric algebra SL.  You'll often see worse versions of the PBW theorem
in textbooks, and ugly proofs, but James Dolan showed me the nice version
and proof a while back.

The kernel of the idea is this: if L is the Lie algebra of a group G,
UL consists of left-invariant differential operators on G, and there's
a map UL -> SL sending any differential operator to its "symbol".  
This is an isomorphism of vector spaces and even of coalgebras, but
not of algebras.  

Anyway, there's something vaguely similar relating the invariant
subalgebras of UL and SL.  By "invariant" here, I mean that since
L acts as derivations of UL and SL, we can look at the subalgebra of
either one consisting of guys who are killed by these derivations;
such guys are called "invariant".   Physicists call invariant elements
of UL "Casimirs", after the first physicist to think about this stuff.
They form commute with everything else in UL.  Invariant elements of
SL are like classical Casimirs: there's a Poisson bracket on SL, and
these are the guys whose Poisson bracket with everyone vanishes.

The Duflo map is an *algebra isomorphism* from SL to UL.  So, it's
like a very nice way to quantize Casimirs, one that gets along with
multiplication.  It's called the "Duflo map" because it was invented by
Harish-Chandra for semisimple Lie algebras and for Kirillov in general.
Kirillov conjectured that it was always an isomorphism; what Duflo
did is prove it:

5) Michel Duflo, Operateurs differentiels bi-invariants sur un groupe
de Lie, Ann. Sci. Ecole Norm. Sup. 10 (1977), 265-288.

Apparently all known proofs are sort of hard!  According to Bar-Natan,
Le and Thurston:

  In the book of Dixmier, the proof is given only in the last chapter
  and it utilizes most of the results developed in the whole book,
  including many classification results (a situation Godement called
  "scandalous").  As discussed below, there have been several recent
  proofs that do not use classification results, but they all use tools
  from well outside the natural domain of the problem.

The proof by Bar-Natan, Le and Thurston uses the connection between
knot theory and Lie algebras - namely, the theory of Vassiliev
invariants.  I think there's still something slightly scandalous about
this, but it's awfully interesting.  Anyway, take a look:

6) Dror Bar-Natan, Thang T. Q. Le and Dylan P. Thurston, Two
applications of elementary knot theory to Lie algebras and Vassiliev
invariants, Geometry and Topology 7 (2003), 1-31.  Available at
http://www.maths.warwick.ac.uk/gt/GTVol7/paper1.abs.html and also
as math.QG/0204311.

For more, try Thurston's thesis:

7) Dylan P. Thurston, Wheeling: a diagrammatic analogue of the Duflo
isomorphism, math.QG/0006083.

and, just for fun, Deligne's handwritten letter to Bar-Natan:

8) Pierre Deligne, letter to Dror Bar-Natan about the Duflo map,
available at http://www.math.toronto.edu/~drorbn/Deligne/

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