August 8, 2003

This Week's Finds in Mathematical Physics (Week 197)

John Baez

I've been away from This Week's Finds for a long time, so I have a lot to talk about... so much that I scarcely know where to begin!

In June I went to a big general relativity conference at Penn State, and I have a lot to say about that, but at the end of July I went to two conferences in Lisbon, and I want to talk about those a bit now.

One was a workshop on "categorification and higher-order geometry". This was run by Roger Picken and Marco Mackaay, and it brought together a bunch of people interested in how n-categories are affecting our notions of geometry. If you're interested in this, you might enjoy looking at the talk titles here:

1) Workshop on categorification and higher-order geometry, http://www.math.ist.utl.pt/~rpicken/CHOG2003

The other was the "Young Researcher's Symposium", a section of the International Congress of Mathematical Physics. This symposium allows old geezers to pass on their accumulated wisdom to young researchers before they go senile and forget it all. The youngsters also give talks, but I was invited as one of the old geezers. It's a bit scary!

Anyway, at these conferences I learned some cool stuff about elliptic cohomology from Stephan Stolz, and also some cool stuff about "Monstrous Moonshine" from Terry Gannon. It turns out they're more related than I realized - and the relation involves string theory! I always love it when two things I'm studying turn out to be related. So, I'd like to tell you about this stuff... before I forget it.

I gave a very sketchy introduction to elliptic cohomology in "week149" and "week150". One reason I'm interested in this subject is that it seems to be a categorified version of something topologists are already fond of: K-theory. In K-theory, you study a space by looking at all the vector bundles over this space. By trying to categorify the concept of "vector space", Kapranov and Voevodsky were led to the concept of "2-vector space", which is a category that acts sort of like a vector space. You can think of elliptic cohomology as a souped-up version of K-theory where you study a space by looking at all the "2-vector bundles" on it!

I'll warn you right away, this isn't how most people think about elliptic cohomology - this is a fairly new approach due to Nils Baas, Bjorn Dundas and John Rognes. Most people think of elliptic cohomology as being related to string theory. But the two viewpoints seem to be compatible.

Here's why: if you have a connection on a vector bundle, it gives a way to parallel transport a vector along a curve. People use this to study how the state of a point particle changes when you move it around in a "gauge field" - which is just physics talk for a connection.

So now let's imagine you categorified this whole story. If you had a connection on a 2-vector bundle, and you believe that categorification increases the dimensions of things by one - which it often does - you might hope that this connection would tell you how to do parallel transport over a 2d surface! And this in turn might tell you how strings change state when you move them around.

Well, nobody has worked out all the details yet, but something like this seems to be going on... and I want to know what it is!

I'd like to explain what Stephan Stolz told me about this. I have to warn you, though: this stuff applies to a new improved version of elliptic cohomology, which became popular after the one I was talking about in previous Weeks. Some of the old stuff I said no longer applies to this new version. To minimize confusion, people call this new version the theory of "topological modular forms".

So, what is this thing?

First of all, it's a generalized cohomology theory.

Hmm. To make sure you understand that sentence, I need to give the world's quickest course on generalized cohomology theories. For a more leisurely introduction see "week149".

Here goes:

A "spectrum" is an infinite list of spaces E(n) where n ranges over all integers, such that each space in the list is the space of loops in the next space on the list. Given any space X, we can define the "generalized cohomology groups" of X to be

hn(X) = [X,E(n)]

If you know about the good old familar "ordinary" cohomology groups Hⁿ(X) of a space X, you'll be pleased to know that these are an example of a generalized cohomology theory. You'll also be happy to know that lots of the basic theorems about ordinary cohomology theory hold for these generalized ones. The main one that doesn't hold is the one that says:

Hn(point) = Z  if n = 0
            0  otherwise

Okay, now you're an expert on generalized cohomology theories.

As I said, the theory of "topological modular forms" is one of these things. So, to completely describe it, I just need to give you an infinite list of spaces tmf(n) forming a spectrum. Then for any space X we can define a list of abelian groups

tmfn(X) = [X,tmf(n)]

Unfortunately, at present it's a lot of work to define these spaces tmf(n). Mike Hopkins and Haynes Miller figured out how, and it was a great achievement:

2) Michael J. Hopkins, Topological modular forms, the Witten genus, and the theorem of the cube, in Proceedings of the International Congress of Mathematicians (Zurich, 1994), Birkhauser, Basel, 1995, pp. 554-565.

But, they used a lot of heavy-duty algebraic topology that simple-minded folks like me have almost no chance of understanding.

Fortunately, Stephan Stolz told me what people secretly think these spaces must be! Nobody has proved this yet or even made it into a precise conjecture, but it's so audacious - and it would explain so much - that I can't resist saying it:

tmf(n) is the space of supersymmetric conformal field theories of central charge -n.

Zounds!

Before we blow our minds contemplating the space of all superstring vacua, let me back off a bit and try to explain what any of this has to do with "modular forms". Modular forms are a famous old concept from complex analysis. These days people do complex analysis not just on the complex plane but on more general Riemann surfaces, and this turns out to be crucial for understanding modular forms. We also use these surfaces to describe the "worldsheets" traced out in spacetime by the motion of a strings. So, it should not come as a shock that modular forms should show up in a generalized cohomology theory involving strings! But I'd like to make this connection considerably more precise.

To do this, I'll reveal that the spectrum for topological modular form theory is a ring spectrum, and the abelian groups

tmfn(point) 

Well, at least after we tensor it with the complex numbers, it is... but before we worry about that, I should say what modular forms are.

I'll start with a quick but unenlightening definition. First, a "modular function of weight n" is an analytic function on the upper half of the complex plane, say

f: H → C                      

f((az+b)/(cz+d)) = (cz+d)n f(z)

(a b) 
(c d)

There are only nonzero modular forms when the weight is a natural number. It's easy to see that these form a graded ring: if you add two modular forms of weight n you get another one of weight n, and if you multiply two modular forms of weights n and n', you get one of weight n+n'.

This graded ring is the same as what you get by tensoring the graded ring tmfⁿ(point) by the complex numbers!

In case you're wondering what this "tensoring with the complex numbers" business is all about: it's mainly just a way of killing off elements of a group that become zero when you multiply them by some integer. If you're a topologist these so-called "torsion elements" are really interesting. They make topological modular forms a lot more subtle than traditional modular forms as defined above. Topologists really go into raptures over torsion! But if you're a lowly mathematical physicist such as myself, struggling to understand even a little of what's going on, you go ahead and kill the torsion by tensoring with C. And, I'm pretty sure the new "topological modular form" theory is the same as the old version of elliptic cohomology except for stuff involving torsion.

So, ignoring these subtleties, let's just say that tmf is a generalization of cohomology theory in which the integers get replaced by the modular forms when we calculate the cohomology of a point... where modular forms are some weird functions that show up in complex analysis!

But what does this have to do with the idea that tmf is related to the space of all string theories?

To understand this, we need a better understanding of modular forms: we need to see how they're related to "elliptic curves", and we need to see how these are related to conformal field theory. Then things will start to make sense.

To do this, let's start with the world's quickest course on elliptic curves. For a more leisurely introduction, see "week13", "week125", and "week126".

An "elliptic curve" is what you get when you take the complex plane and mod out by a lattice, like this:

                *       *      *      *

                             
                    *      0*      *                


                *       *      *      *

With this definition, it turns out that we can rotate or dilate our lattice without changing the elliptic curve we get from it. More precisely, we get an isomorphic elliptic curve. So, any elliptic curve is isomorphic to one coming from a lattice like this:

                                   z         z + 1 
                      *            *           *
 
                                0           1
                   *            *           *
               
 
                *           *           *

But, lots of different choices of z give the same elliptic curve! For example, we can replace z by z + 1 and still get the same lattice, hence the same elliptic curve. We can also replace z by -1/z. This turns the short squat right-leaning parallelogram in the above picture into a tall skinny left-leaning one - but after rotating and dilating this, we get back the parallelogram we started with, so we get the same elliptic curve.

In fact, though it's not obvious from this way of thinking about the problem, it's easy to show that all the different choices of z that give the same elliptic curve are related by these two transformations.

Now, the group of transformations of the upper half-plane generated by

z |-> z + 1

z |-> -1/z

z |-> (az+b)/(cz+d)

(a b)
(c d)

H/PSL(2,Z)

Okay, now you're an expert on elliptic curves.

A while back, I defined a "modular function of weight n" to be an analytic function on the upper half-plane

f: H → C                  

f((az+b)/(cz+d)) = (cz+d)n f(z)

So, if you're trying to explain modular functions to your friends, just tell them they're functions that depend on the shape of a doughnut - what could be simpler than that? Of course "shape" needs to be interpreted in a subtle way to make this true.

Similarly, a modular function is a "modular form" if it doesn't blow up when Im(z) → +∞, which means that it doesn't blow up when your doughnut gets really long and skinny, more like a circle than an honest doughnut. The circle is like the ultimate low-calorie doughnut. In the language of string theory, where the surface of your doughnut is the worldsheet of a string, the limit Im(z) → +∞ corresponds to the "particle limit", where the worldsheet of the string degenerates to the worldline of a particle.

Of course, when n is nonzero, modular forms of weight n aren't really invariant under PSL(2,Z): they're only invariant "up to a phase". I put this physics jargon in quotes because the fudge factor (cz+d)ⁿ isn't really a unit complex number. But the moral principle is the same - and in string theory, this fudge factor really does come from a quantum mechanical phase ambiguity, called the "conformal anomaly".

(To make this "up to a phase" idea precise, we can think of modular forms of weight n as sections of some line bundle on the moduli stack of elliptic curves... but I explained this already in "week125", and I don't want to say more about it now.)

Now that we understand modular forms a bit better, we can begin to vaguely see why

tmfn(point) tensor C

Here's how. If you know a little about the path-integral approach to quantum field theory, you'll know that one of the basic things you compute in any quantum field theory is a number called the "partition function". You'll also know that this number is often infinite, or defined only up to some ambiguities... that's why quantum field theory is tough.

So, given that a conformal field theory is something like a string theory, and given that the worldsheet of a string is a Riemann surface, you shouldn't be surprised that given any compact Riemann surface and any conformal field theory we can try to compute a number called the "partition function". Nor should you be surprised that this "number" is sometimes afflicted with ambiguities!

So, restricting attention to the case where our Riemann surface is an elliptic curve, you should not be surprised that the partition function of any conformal field theory is a MODULAR FORM!

If this modular form has weight 0, the partition function is an honest-to-goodness function on the moduli space of elliptic curves: for any elliptic curve the partition function is an actual number. But if the modular form has nonzero weight, the partition function is afflicted with "phase ambiguities" - where "phase" is in quotes for the same reason as before.

In particular, if the partition function is a modular form of weight n, we say our conformal field theory has "central charge -n". The central charge just tells us how the phase ambiguity works... though some jerk put in a minus sign to confuse us.

Now think what this implies! Remember that tmf(n) is space of conformal field theories with central charge -n. Since the partition function of any such thing is a modular form of weight n, we get a map

Z: tmf(n) → {modular forms of weight n}

tmfn(point) tensor C = {modular forms of weight n} 

tmfn(point) = [point,tmf(n)]

[point,tmf(n)] 

To see this, we'd just need to show four amazing things:

• The partition function doesn't change as we trace out a continuous path in the space of conformal field theories of central charge -n. Thus, the partition function defines a map

  Z: [point,tmf(n)] → {modular forms of weight n}
  

• The set of connected components of the space of conformal field theories of central charge -n forms an abelian group, and the above map is a group homomorphism.

• The kernel of the above homomorphism consists precisely of the torsion elements, so we get a 1-1 homomorphism

  Z: [point,tmf(n)] tensor C → {modular forms of weight n}
  

• Any modular form of weight n is a linear combination of partition functions of conformal field theories of central charge -n, so the homomorphism

  Z: [point,tmf(n)] tensor C → {modular forms of weight n}
  

  is also onto.

Sorry, I'm getting a little carried away... it's not good to put in so much detail when you're explaining stuff, but I just realized that we need these four amazing things to be true, and I couldn't resist writing them down. Learning by teaching is great for the teacher; sometimes less so for the student.

Anyway:

The first amazing thing must come from "index theory" and how the "index of a Fredholm operator" doesn't change when we deform it continuously. It must also use the fact that the partition function we're talking about can be written as such an index. This only happens because we're considering supersymmetric theories! Stephan Stolz emphasized to me that we really need to be using "N = 1/2 supersymmetric conformal field theories"; I haven't gotten around to understanding the N = 1/2 part.

The second amazing thing is not really amazing. In fact, it's easy to see the whole graded ring structure of modular forms coming from operations on conformal field theories. I'll explain that in a minute.

The third amazing thing is a total mystery to me. It's obvious that all torsion elements must lie in the kernel of a homomorphism from a group to a vector space, but it's utterly mysterious why the kernel consists precisely of the torsion elements.

The fourth amazing thing is presumably some sort of calculation: you just need to find enough conformal field theories to make sure their partition functions generate the ring of modular forms. In fact, the ring of modular forms is generated by one of weight 4 and one of weight 6: these are both "Eisenstein series", which are well-understood, so we just need someone to cook up conformal field theories having these as partition functions. Does anyone reading this know how to do it?

(Irrelevant digression: The previous paragraph implies that all nonzero modular forms have even weight. To correct for this, some people stick in a factor of 1/2 when defining the weight of a modular form. I mention this only so you're forewarned when you read the literature.)

Okay, let me round off this story by saying a little about how you add and multiply conformal field theories... and why.

A "conformal field theory" assigns a Hilbert space to any compact oriented 1-manifold, and a linear operator going between Hilbert spaces to any Riemann surface with boundary going between such 1-manifolds. There are a bunch of axioms it needs to satisfy, invented by Graeme Segal. I won't list these here, but the category theorists among you will quiver with delight upon learning that the most important of these axioms say a conformal field theory is a "symmetric monoidal functor".

Anyway, it's easy to take direct sums and tensor products of Hilbert spaces and also operators. This gives a way of defining the direct sum and tensor product of conformal field theories. When we take the direct sum of conformal field theories their partition functions add. When we take their tensor product the partition functions multiply. So, these operations on conformal field theories correspond precisely to the graded ring structure on modular forms!

To see why this graded ring structure is interesting in string theory, I should be more precise about the relation between string theory and conformal field theory. Perturbatively, string theory in a given background is described by a conformal field theory. We can use this to calculate an operator for any Riemann surface with boundary: we think of this operator as saying how the string changes state given the conformal structure on its worldsheet. When a conformal field theory plays this role we call it a "string vacuum".

But, not any old conformal field theory will serve as a string vacuum! It has to be one with central charge 0, in order to have a partition function without any ambiguities. If the central charge is nonzero we say there's a "conformal anomaly" and turn up our noses in disgust. However, people often build conformal field theories with central charge 0 out of ones with nonzero central charge. The simplest ways to build new conformal field theories from old are direct sums and tensor products. So, the graded ring structure on modular forms is sort of lurking around in string theory!

To learn more about elliptic cohomology and its relation to conformal field theory, you should read this paper that Stephan Stolz is in the process of writing with Peter Teichner:

3) Stephan Stolz and Peter Teichner, What is an elliptic object? Available at http://math.ucsd.edu/~teichner/papers.html

This paper is almost 80 pages long and they aren't even done yet! The main goal is to define a concept of "elliptic object" on a space X such that tmfⁿ(X) is built from formal differences of elliptic objects with central charge n over X, just as the K-theory of X is built from formal differences of vector bundles over X. In fact you can built K-theory using formal differences of vector bundles equipped with a connection, and an elliptic object is really a categorified version of a vector bundle equipped with connection. In particular, it lets you do "parallel transport" over 2d surfaces in your space X. The funny part is that these surfaces need to be Riemann surfaces. Indeed, an elliptic object is very much like a conformal field theory, but where the surfaces are mapped into X.

The concept of elliptic object goes back to Graeme Segal. His idea was roughly that an elliptic object should be a functor assigning a Hilbert space to any compact oriented 1-manifold mapped into X, and a linear operator to any Riemann surface with boundary mapped into X. Stolz and Teichner's big realization is that an elliptic object needs to be not just a functor, but a 2-functor! In other words, it needs to assign data not just to Riemann surfaces and 1-manifolds in X, but also to points in X! Thus it's a lot like a 2d extended topological quantum field theory, as explained in "week35 ". The big difference is that the surfaces are Riemann surfaces, and everything is happening "in X".

For how elliptic cohomology is related to 2-vector spaces, read this:

4) Nils A. Baas, Bjorn Ian Dundas and John Rognes, Two-vector bundles and forms of elliptic cohomology, available as math.AT/0306027.

I'll quote the abstract because it will be enlightening to a few of you:

In this paper we define 2-vector bundles as suitable bundles of 2-vector spaces over a base space, and compare the resulting 2-K-theory with the algebraic K-theory spectrum K(V) of the 2-category of 2-vector spaces, as well as the algebraic K-theory spectrum K(ku) of the connective topological K-theory spectrum ku. We explain how K(ku) detects v₂-periodic phenomena in stable homotopy theory, and as such is a form of elliptic cohomology.

The point is, there's an infinite tower of generalized cohomology theories, called the "chromatic filtration". This has ordinary cohomology tensored with the complex numbers on the 0th level, complex K-theory on the 1st level, elliptic cohomology on the 2nd level, and so on up to infinity, where something called "complex cobordism theory" sits grinning down at us. Theories on the nth level "detect v_n-periodic phenomena". Despite the best efforts of several homotopy theorists, I still don't understand what this means. But, Bott periodicity for complex K-theory is the paradigm of a "v₁-periodic phenomenon", so we're talking about some heavy-duty generalization of that!

Note that Baas, Dundas and Rognes don't talk about connections on their 2-vector bundles. The closest thing to this that people have used in elliptic cohomology is the notion of "elliptic object", invented by Graeme Segal and improved by Stolz and Teichner. An elliptic object on a manifold M is like a way of moving strings around in M, so you can think of it as a recipe for 2d parallel transport. The funny part is, you need a conformal structure on your surface before you can do parallel transport over it!

Stolz and Teichner do a great job of working out the following analogy:

complex K-theory                       elliptic cohomology
connections on complex vector bundles  elliptic objects
supersymmetric 1d field theories       supersymmetric conformal field theories

some cohomology theory that detects vn-periodic phenomena
connections on complex "n-vector bundles"
some supersymmetric field theories on n-dimensional spacetime

Next I should say a word about the "new" versus "old" versions of elliptic cohomology. At this point things are going to get... ahem... a bit technical. Then I'll talk about the connection to Monstrous Moonshine, and things will get really vague, and downright bizarre.

The old version of elliptic cohomology was a specially nice sort of generalized cohomology theory called a "complex oriented cobordism theory". I explained what these were in "week149", and in "week150" I explained how each of these things gives a "formal group law".

If you want an easily understood example of a formal group law, just take a group, pick coordinates near the identity of this group, and write out the group operation in terms of these coordinates as a power series. This works whenever your group is an analytic manifold and the group operations are analytic functions. The result is a "formal group law". The word "formal" comes from the fact that we'd actually be satisfies if the group operations were described by formal power series.

Anyway, now consider the torus. A torus is a group in an obvious way - just a product of two copies of the group U(1) - but there are different ways to make it into a complex manifold where the group operations are complex analytic functions. A way of doing this is nothing other than an "elliptic curve"!

In fact, each elliptic curve corresponds to a complex oriented cobordism theory, and we could call any one of these "an elliptic cohomology theory", if we wanted.

But it's better, actually, to glom all these different theories into one big "universal" theory. The most obvious way to attempt this is to take the moduli space of elliptic curves and cook up a formal group law over the algebra of functions on this space by stitching together all the formal group laws for each specific elliptic curve. This formal group law corresponds to a complex oriented cobordism theory called Ell. This is what I was calling the "old version" of elliptic cohomology.

The "new version", namely "tmf", is a bit sneakier. I think it's the "limit" - in the sense of category theory - of the elliptic cohomology theories for all specific elliptic curves. The reason this is different than Ell is that some elliptic curves have nontrivial symmetries! Unlike Ell, tmf is not a complex oriented cobordism theory. But the difference is very subtle, and only involves "2-torsion" and "3-torsion", that is, elements that vanish when you multiply them by some power of 2 times some power of 3.

The reason the numbers 2 and 3 show up is apparently because the elliptic curves with nontrivial symmetries come from the square lattice:

             *     *     *     *


             *     *     *     *


             *     *     *     *

            *       *      *      *


                *       *      *                


            *       *      *      *

In fact, they also show up in other devious ways, which I would love to understand better. For starters, they give a certain "period-12" pattern in the theory of modular forms, which becomes a "period-24" pattern if you define weights using the convention that I'm using here. Lots of people know about this - see any introduction to modular forms, like this one:

5) Neal Koblitz, Introduction to Elliptic Curves and Modular Forms, 2nd edition, Springer-Verlag, 1993.

I already vaguely explained this in "week125".

But, more deviously, these symmetries are also related to a certain "period-576" pattern in topological modular form theory! The number 576 is 24 x 24. According to my vague memories of what Stephan Stolz said, the first 24 is the usual one in bosonic string theory. In particular, if we ignored subtleties involving torsion, elliptic cohomology would have period 24, with the periodicity generated by a conformal field theory of central charge 24 having an enormous group called the Monster as its symmetries! This is where Monstrous Moonshine comes in, and especially the work of Borcherds.

(This can't be exactly right, because the most famous conformal field theory whose symmetries form the Monster is not supersymmetric, and its partition function is the j-function, which is modular function of weight 0, not a modular form of weight 24. So, my brain must have been a bit fried by the time we got to this really far-out stuff.)

Where does the extra 24 come from? I don't know, but Stephan Stolz said it has something to do with the fact that while PSL(2,Z) doesn't act freely on the upper half-plane - hence these elliptic curves with extra symmetries - the subgroup "Γ(3)" does. This subgroup consists of integer matrices

(a b)
(c d)

(1 0)
(0 1)

So, if we form

H/Γ(3)

SL(2,Z)/Γ(3) = SL(2,Z/3)

In fact, I think this is just another way of explaining the period-24 pattern in the theory of modular forms, but I like it.

I especially like it because SL(2,Z/3) is also known as the "binary tetrahedral group". To get your hands on this group, take the group of rotational symmetries of the tetrahedron, also known as A₄. This is a 12-element subgroup of SO(3). Using the fact that SO(3) has SU(2) as a double cover, take all the points in SU(2) that map to A₄. You get a 24-element subgroup of SU(2) which is the binary tetrahedral group.

In fact, if you think of SU(2) as the unit sphere in the quaternions, the binary tetrahedral group becomes the vertices of a 4-dimensional regular polytope called the 24-cell!

I'm very fond of this polytope, and have already extolled its charms in "week91" and "week155". So, what pleases me now is that I've found a trail directly from the 24-cell to the appearance of the number 24 in string theory... and even the fact that topological modular form theory has periodicity 24 x 24.

Of course I can barely follow this trail myself, and I probably got some stuff wrong - I hope the experts correct me! But the trail seems to be real, not just a will o' the wisp, so I can now try to widen it and make it less twisty.

There's more to say but I'll stop here. I have given other references to monstrous moonshine in "week66", but here's a very pretty website about it:

6) Helena A. Verrill, Monstrous moonshine and mirror symmetry, http://hverrill.net/pages~helena/seminar/seminar1.html

and here is a nice easy paper by Terry Gannon about it:

7) Terry Gannon, Postcards from the edge, or Snapshots of the theory of generalised Moonshine, available as math.QA/0109067.

I thank Allen Knutson and Peter Teichner for help with this issue.

Addenda: After posting this article, Aaron Bergman helped solve my puzzle about a supersymmetric conformal field theory with the Monster as symmetries, and Stephan Stolz explained why topological modular form theory has period 24².

Aaron Bergman writes:

John Baez wrote:

>(This can't be exactly right, because the most famous conformal
>field theory with the Monster as symmetries is not supersymmetric,
>and its partition function is the j-function, which is a modular
>function of weight 0, not a modular form of weight 24.  So, my
>brain must have been a bit fried by the time we got to this really
>far-out stuff.)

You might be interested in:

BEAUTY AND THE BEAST: SUPERCONFORMAL SYMMETRY IN A MONSTER MODULE. 
By Lance J. Dixon (Princeton U.), P. Ginsparg (Harvard U.),
Jeffrey A. Harvey (Princeton U.). HUTP-88-A013, PUPT-1088, Apr
1988. 30pp. 
Published in Commun. Math. Phys. 119 (1988), 221-241.

There's a scanned version on line. Note that they are working in
lightcone gauge so c=24.

Aaron
-- 
Aaron Bergman
http://www.princeton.edu/~abergman/

In reply to an email of mine, Stephan Stolz wrote:

 John Baez wrote:

 >Do you have a reference on the period-24^2 behavior of tmf?
 >That's one of the things I'm having trouble understanding,
 >even heuristically.  Actually I saw something about
 >it having period 192.  That's not 24^2.

 One reference is the course notes of a course Charles Rezk taught at 
 Northwestern University in 2001. You can find them on his home page
 http://www.math.uiuc.edu/~rezk/papers.html

 Let me make some remarks on periodicity: the ring M* of 
 integral modular forms is 24-periodic with the discriminant Δ
 being the periodicity element.  Explicitly:

 M* = Z[c4,c6,Δ]/(c43 - c62 - 123 Δ).

 There is a ring homomorphism tmf* → M*; the periodicity 
 of tmf* is then determined by the smallest power of Δ in 
 the image of this map.  After localizing at 2, this is Δ8 
 (see Thm. 19.3 in Rezk's paper) which makes a period of 8 x 24 = 
 192.  However, this is only after localizing at 2!  Localized 
 at the prime 3, the smallest power of Δ in the image is 
 Δ3 (see Thm. 17.2); hence after inverting all the primes not
 equal to than 2,3, the smallest power in the image is Δ24! 
 Since localized at any other primes the above map is an isomorphism, 
 this shows that integrally tmf* has period 242.

 Best regards,
 Stephan
 -- 

George described his routes to me with a steady, comprehensive tone, telling me about places of incredible hazard and reward. He had walked the entire length of the Grand Canyon when he was fifty-seven years old, in eighty days, all of it done in the puzzling confines of the inner reaches. I had seen some of his routes before, and had used a number of them, his meager catwalks and handholds. They were like spider's silk, lines across the landscape that were not visible until I touched them. - Craig Childs, Soul of Nowhere

© 2003 John Baez
baez@math.removethis.ucr.andthis.edu