The Magic of 24
Aaron Bergman
discussed the origin of the 26 in bosonic string theory,
eventually tying in the cancellation of the conformally anomaly
(the sum of the central charges of the gauge-fixing bc CFT and
the matter CFT) into monstrous moonshine. I'm not sure I got of
all that, but having worked my way up to chapter 6 in Polchinki's
book, I'm curious about the magic of the number 24. This seems to
me, at first glance to be a disparate issue. It seems like it
should tie into the 26, but I can't see off the top of my head
how.
Regardless, the famed Dedekind eta function appears in the
calculation of the partition function on the torus. More
specifically, we get (from Pol. eq 7.2.9)
Z(t) = (4\pi^2\alpha't_2)^{-1/2} |\eta(t)|^{-2}
Now, because eta is a modular form (for something like
\Gamma_0(4)) of weight (1/2,0), and t_2 is a non-holomorphic form
of weight (-1,-1), this has a total weight of zero thus being
invariant under the modular group as one needs in string theory.
Now, the eta function is quite an interesting object. The
expression that is arrived at in Polchinski's book is the
infinite product
q^{1/24} \Product_{n=1}^{\Infinity} (1 - q^n)
Now, what surprised me upon seeing this is how the weird but
necessary q^(1/24) appears. Now, I have to admit that the 24 here
seems rather magical even without the string theory figuring in.
Now, when one deals with nice simple modular forms, it turns out
the the space of forms is 1-dimensional, spanned by the
Eisenstein series, up to the 12th dimension where there is the
discriminant
(2\Pi)^{12}
\Delta = ----------- (E_4(z)^3 - E_6(z)^2)
12^3
(using the notations and normalizations of Zagier's book). This
formula is overflowing with 12s, which appears to me because 12
is the least common multiple of 4 and 6, the first two nontrivial
forms. That's dreadfully bland, though. (Twelve also appears, of
course, in the zeta-function regularization of \Sum n. Any
connection?)
Anyways, Zagier introduces the Dedekind eta function seemingly
out of nowhere and shows that its 24th power is the discriminant
using its transformation properties and the previous derivation
of the dimension of the space of weight 12 forms.
I find this horribly unelightening. Any suggestions? I don't
remember finding Gunning's book any more helpful.
Back to the physics. The partition function on the torus is
derived in Polchinski 7.2:
Consider the path intergral with no vertex operators,
<1>_{T^2(t)} \defeq Z(t). We can think of the torus with modulus
t [really a tau] as formed by taking a field theory on a circle,
evolving for Euclidean time 2\pi t_2, translating in \sigma^1 by
2\pi t_1, and then identifying the ends. In operator language
this gives a trace,
(7.2.5) Z(t) = Tr[ exp(2\pi i t_1 P - 2\pi t_2 H]
= (q\qbar)^{-d/24} Tr(q^{L_0} \qbar^{\Lbar_0})
The L's here are the Virasoro generators and Polchinski has used
the expressions P = L_0 + \Lbar_0 and H = L_0 + \Lbar_0 +
(1/24)(c+\cbar). This is where the magic 24 comes from, which
makes everything nice and modularly invariant.
So, now, we hit conformal field theory. My knowledge of that
subject is unfortunately fairly sparse, limited to chapter 2 of
Polchinski and a few _very_ brief forays into Di Francesco et
al. I think I can use the stuff in Polchinski to formulate my
question. The above relations for P and H ultimately related to
the non-tensorial transformation of the stress-energy tensor in a
conformal field theory, Polchinski's equation 2.4.26:
(@_z z')^2 T'(z') = T(z) - (c/12) {z',z}
where the funny bracket thing an ugly thing called the Schwarzian
derivative which is left markedly unmotivated in Polchinski's
book. Anyways, that 12 there comes from the TT OPE and the Ward
identity and ultimately comes from the fact that 12 = 2*3!. An
extra factor of two comes from {e^z, z} and, poof, 24.
Eek. What does this all mean? It feels like there should be
something deep going on here. In the process of writing this
post, I noticed that if I squint _really_ hard, the Schwarzian
derivative looks a bit like the reciprocal of the j-function. Or
maybe not. It's all very weird, so, after hopefully not insulting
anyone's intelligence with all this exposition (which I also hope
is all correct) :
What's so magic about 24?
Aaron
--
Aaron Bergman
<http://www.princeton.edu/~abergman/>
John Baez
Aaron Bergman <aber...@princeton.edu> wrote:
>Now, the eta function is quite an interesting object. The
>expression that is arrived at in Polchinski's book is the
>infinite product
>
>q^{1/24} \Product_{n=1}^{\Infinity} (1 - q^n)
>
>Now, what surprised me upon seeing this is how the weird but
>necessary q^(1/24) appears. Now, I have to admit that the 24 here
>seems rather magical even without the string theory figuring in.
>Now, when one deals with nice simple modular forms, it turns out
>the the space of forms is 1-dimensional, spanned by the
>Eisenstein series, up to the 12th dimension where there is the
>discriminant
>
> (2\Pi)^{12}
>\Delta = ----------- (E_4(z)^3 - E_6(z)^2)
> 12^3
>
>(using the notations and normalizations of Zagier's book). This
>formula is overflowing with 12s, which appears to me because 12
>is the least common multiple of 4 and 6, the first two nontrivial
>forms. That's dreadfully bland, though.
It's not really so bland. A modular form is almost a function
on the space of lattices in the complex plane - where we count
two lattices as "the same" if they differ by rotation and/or
dilation. But actually a modular form is a section of a certain
line bundle over the space of lattices. In lowbrow physics lingo,
this means that a modular form suffers from a certain "phase
ambiguity": as we march around a loop in the space of lattices,
the modular form needn't come back to where it started. Instead,
it may be multiplied by a phase. A humbler example of this
phenomenon is the function z^{1/n} on the complex plane, which
isn't really a function, because it gets multiplied by an nth
root of 1 when we march all the way around the origin. The origin is
a nasty place when dealing with this function, and similarly,
when dealing with modular forms there are two nasty places
in the space of lattices, corresponding to the lattices with
special symmetry. These are the square lattice, which has
4-fold symmetry, and a lattice with lots of equilateral triangles
in it:
* * * *
* * *
* * * *
which has 6-fold symmetry. The modular form you're calling
E_4 picks up a phase when we march around the lattice with
4-fold symmetry - in fact, it picks up a phase equal to a
4th root of unity. And similarly, the modular form E_6
picks up a phase equal to a 6th root of unity when we march
around the lattice with 6-fold symmetry.
If you look at the classification of modular forms you'll
see that what really matters are the phases one gets when
one marches around these two specially symmetrical lattices,
and this is how one proves that E_4 and E_6 freely generate the
ring of all modular forms. The ring of modular forms is
graded in such a way that E_4 has grade 4 and E_6 has grade
6 - though people use the term "weight" instead of "grade".
It follows that the space of modular forms of weight w is
at most one-dimensional until w = 12. Since 12 is 4+4+4 but
also 6+6, both E_4^3 and E_6^2 have weight 12! And this is
what makes the formula you wrote down so interesting.
>(Twelve also appears, of
>course, in the zeta-function regularization of \Sum n. Any
>connection?)
Yes indeed! If you look at the vacuum energy of all the
right-moving modes of the massless scalar field on a cylinder,
you get
(1 + 2 + 3 + ....)/2 = -1/24
if you do zeta function regularization, and this is why one
gets that 1/24 in the partition function of this system,
which is
Z(b) = exp(b/24) product_{k=1}^{infinity} 1/(1 - exp(-bk)),
none other than the Dedekind eta function (slightly disguised)!
>Anyways, Zagier introduces the Dedekind eta function seemingly
>out of nowhere and shows that its 24th power is the discriminant
>using its transformation properties and the previous derivation
>of the dimension of the space of weight 12 forms.
Right. From the physics point of view, one might naively
suspect that the Dedekind eta function was a well-defined
function on the space of lattices mod rotation and dilation -
since it's the partition function of what was *classically*
a conformally invariant system. But it's not quite: instead,
its 24th power is the discriminant, which is a modular form
of weight 12. So conformal invariance is getting slightly
screwed up after quantization - in physics jargon, we've
got an "anomaly" on our hands. You can see from the above
stuff that this is somehow related to the phenomenon of
vacuum energy! But the real challenge is to understand
exactly what's going on and to get an intuitive feel for
it.
>I find this horribly unelightening. Any suggestions? I don't
>remember finding Gunning's book any more helpful.
You might try rereading "week124"-"week127" now that you've
been battering your head on this stuff for a while - stuff
that previously seemed abstruse will probably seem easy now.
You can get these quickly here:
http://math.ucr.edu/home/baez/twfshort.html
And then try this book: it's full of really nice expository
discussions of modular forms which emphasize intuition and
leave out the yuckier details:
M. Waldschmidt et al, editors, From Number Theory to Physics:
lectures given at the meeting 'Number Theory and Physics' held
at the Centre de physique, Les Houches, France, March 7-16, 1989,
Springer-Verlag, New York, 1992.
>So, now, we hit conformal field theory.
Wham!
>I think I can use the stuff in Polchinski to formulate my
>question. The above relations for P and H ultimately related to
>the non-tensorial transformation of the stress-energy tensor in a
>conformal field theory, Polchinski's equation 2.4.26:
>
>(@_z z')^2 T'(z') = T(z) - (c/12) {z',z}
>
>where the funny bracket thing an ugly thing called the Schwarzian
>derivative which is left markedly unmotivated in Polchinski's
>book.
I've never understood the Schwarzian derivative as well as
I should, and it's probably crucial to this whole business,
but basically it's my impression that it is a very natural
conformally invariant way of differentiating functions on
the complex plane. At this point I should probably mutter
something about "cross-ratios", but I can't do it in a
sufficiently authoritative tone of voice to fool you into
thinking I know what I'm talking about. I believe there
are some very helpful words of wisdom about cross-ratios,
the Schwarzian derivative, and the number 12 in de Francesco
et al's book on conformal field theory.
>Eek. What does this all mean?
It means that string theory must be true, even if there's
no experimental evidence for it, because god would never have
made mathematics be so full of weird coincidences concerning
complex analysis and the number 24 if he hadn't been meaning
to *do* something with it all. :-)
>It feels like there should be something deep going on here.
There clearly is. The hard part is figuring out exactly
*what it is*. I spent a long time trying to boil it down
to the simplest possible form and "week124" - "week127".
But I'm sure if I knew more and thought harder I could do
a better job - I don't think I got to the bottom of it, if
bottom there be. Indeed, this stuff really makes one question
what it means to "understand" a piece of mathematics: the
deeper you dig, the more you seem to find. Why are there
24 even unimodular lattices in 24 dimensions? Why are there
26 sporadic finite simple groups?
>In the process of writing this
>post, I noticed that if I squint _really_ hard, the Schwarzian
>derivative looks a bit like the reciprocal of the j-function. Or
>maybe not.
Hmm. See, you're becoming paranoid, just like I did after
spending a few months thinking about this stuff! The worst
thing is, there may really be a connection!
Aaron J. Bergman
Two corrections:
1) It's chapter seven of Polchinski.
2) All reference to Zagier should be references to Koblitz. Beats me why
I confuse the two.
torqu...@my-deja.com
aber...@princeton.edu wrote:
> The above relations for P and H ultimately related to
> the non-tensorial transformation of the stress-energy
> tensor in a
> conformal field theory, Polchinski's equation
> 2.4.26:
>
> (@_z z')^2 T'(z') = T(z) - (c/12) {z',z}
>
> where the funny bracket thing an ugly thing called the
No no no! It's beautiful! Schwarzian derivatives have
*lots* of amazing properties. For example they share many
of the properties of the usual derivative even though they
are defined in a rather complex manner. Unfortunately
the physics papers tend to leave out the nice stuff and just
give a completely obscure looking definition.
> Schwarzian derivative which is left markedly unmotivated
> in Polchinski's book.
I believe I can help motivate the appearance of the
Schwarzian derivative. I had a hard time trying to make
sense of all these mysterious numbers years ago but I
eventually managed to figure something out. Unfortunately
I'm a pure mathematician, not a physicist, so my idea of
'understanding' might not agree with yours! My memory is
hazy on this stuff so I hope you check up on anything I
say in my references!
Suppose we have a holomorphc function f(z) and we try to
approximate it in the neighbourhood of some point p by a
fractional linear function (az+b)/(cz+d) varying a,b,c,d
so that the first three terms of the Taylor expansion
around p vanish. The coefficient of the next term is the
Schwarzian derivative (up to some constant). In other
words the Schwarzian derivative is the amount by which a
function deviates from being fractional linear. (This is
just like the usual derivative that measures how close
a function is to being linear.) Functions with zero
Schwarzian derivative are precisely the fractional linear
ones. Schwarzian derivatives, therefore, have a tendency to
pop up in calculations involving comparisons of holomorphic
functions with rational functions.
A nice expression for the Schwarzian derivative {f(v),v}
is this:
(*) 6 lim_{u -> v} (f'(u)f'(v)/(f(u)-f(v))^2 - 1/(u-v)^2}
(it looks prettier if you write it out properly without
pseudo-TeX notation)
Consider the meromorphic tensor (du \otimes dv)/(u-v)^2
on the complex manfold CxC with coordinates (u,v). Then
(*) describes how the singularity along the diagonal
behaves under a holomorphic change of coordinate.
(see Hawley and Schiffer, Half-order differentials on
Riemann surfaces, Acta Math, 115, 1966).
Let me skip all the details and say where this appears
in conformal field theory. We are often trying to obtain
the value of a partition function which is something that
varies with the conformal structure of the underlying
Riemann surface. In other words we expect it to be a
scalar function on the moduli space of Riemann surfaces.
Generally the partition function involves the reciprocal
of the (regularised) determinant of a differential
operator. Consider how this varies with the change of
underlying Riemann surface - this is described by a
differential form on moduli space. Now derivatives of
determinants involve traces and if you work out the details
it turns out you're trying to find the trace of the inverse
of a differential operator. The trace of a differential
operator is essentially an integral along its diagonal
in some representation - and the trace of the inverse of
a diferential operator is essentially an integral along
the diagonal of its Green's function. In practice we are
dealing with differential operators acting on spaces of
differential forms so it is a Green's tensor rather than
a Green's function.
In one of the simplest CFT's the Green's function that
appears takes the form: G(x,y) ~ dxdy/(x-y)^2+holomorphic
terms. Like most Green's functions it's singular along the
diagonal so we need to regularise it to find a trace. So
we do the usual physicist thing and subtract of the
singularity - in other words we work with
(*') lim_{x -> y} (G(x,y)-dxdy/(x-y)^2) instead of G(x,x)
But there's a catch - the choice of subtraction depends on
coordinate system and if we work with (f(x),f(y)) instead
of (x,y) we'll get a different trace. The cool thing is
we know how this behaves because of (*) above. In other
words the Schwarzian derivative is simply a measure of
how the Green's function singularity varies with change
of coordinate system and that's the key to everything :-)
Of course it shouldn't really change with coordinate so
this is actually an anomaly. Note how a fractional linear
coordinate change has no effect on the above expression so
although we lose invariance under the full conformal group
we keep SL(2,Z) invariance. (I learnt much of this stuff
from the PhD thesis of Jay Jorgensen - I can't remember the
title but he does have a book on regularised determinants).
We started out assuming we were working with the derivative
of a function on moduli space but we've had to subtract of
a regularisation term. When we integrate (*') back up
again to get an object on moduli space it turns out that we
no longer have the derivative of a function but have added
an extra twist and so the partition function now becomes
a section of a non-trivial line bundle. The consequences
of this have already been discussed in John Baez's TWFs.
BTW The expression (*') is essentially the stress-energy
tensor. A good summary of how Green's functions are related
to partition functions and the stress-energy tensor is,
if I remember correctly, in Ginsparg's notes on Applied
Conformal Field Theory which are available on the web
somewhere.
I did a lot of this stuff in my PhD thesis. Along the
way I found an elementary generalisation of (*) that
contains the 6j^2-6j+1 term that appears in the Mumford
isomorphism theorem (again see Baez's TWFs). Given
the connection with line bundles on moduli space I've
mentioned above this gives an intuitive explanation for
that theorem. So we find Schwarzian derivatives intimately
connected with some rather deep algebraic geometry.
Unfortunately I had no idea how to turn the physics side
into into a proper proof and now I'm too busy with real work...
--
Torque
http://travel.to/tanelorn
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Share what you know. Learn what you don't.
torqu...@my-deja.com
torqu...@my-deja.com wrote:
Oops! Some errata to my previous post:
Replace:
> so that the first three terms of the Taylor expansion
> around p vanish. The coefficient of the next term is the
by
> so that the first three terms of the Taylor expansion
> *of the difference*
> around p vanish. The coefficient of the next term
> *in the difference*
> (*) 6 lim_{u -> v} (f'(u)f'(v)/(f(u)-f(v))^2 - 1/(u-v)^2*}*
by
> (*) 6 lim_{u -> v} (f'(u)f'(v)/(f(u)-f(v))^2 - 1/(u-v)^2*)*
> we keep SL(2,*Z*) invariance. (I learnt much of this stuff
by
> we keep SL(2,*C*) invariance. (I learnt much of this stuff
Plus numerous others I'm sure.