The String Coffee Table

0) elliptic curves and formal groups

The Weierstrass form of an elliptic curve is an equation in two variables $x$ and $y$ of the form

There is something called the discriminant $\Delta$ of $E$ and if it is nonvanishing we have a smooth curve.

Thinking of the above equation as living over the real numbers, such smooth curves are certain smooth curves in $\mathbb{R}^2$. Straight lines in $\mathbb{R}^2$ which coincide with this curve in three point $P$, $Q$ and $R$ define an abelian group structure on points by setting

For many applications it is convenient to perform a coordinate transformation from $(x,y)$ to $(w,z)$ with

Then there is an $f$ such that the above equation for the ellitptic curves reads equivalently

By iteratively re-inserting $f$ into itself according to this equation, we find that

which is a power series in $z$ starting in degree 3, with coefficients being polynomials in the $a_i$ over the integers.

Using this, we can understand the above mentioned addition on the elliptic curve as given by a power series in two variables. Namely, if $(w_1,z_1)$ and $(w_2,z_2)$ are two points on the smooth elliptic curve $E$ (which means that the $z$ coordinate is determined by $w$), then the result of adding them has a $z$-coordinate which is given by a power series

This $F_E$ is a formal group law, which implies (as I mentioned last time) that it satisfies equations

1) $F(z_1,0) = z_1$

2) $F(z_1,z_2) = F(z_2,z_1)$

3) $F(F(z_1,z_2),z_3) = F(z_1,F(z_2,z_3))$

for all $z_i$.

Form this one can show that inverses of all elements exist.

The prototypical example of such a formal group law is obtained by taking a 1-dimensional real Lie group, looking at the tangent space $T_e$ at a given point $e$, using the exponential map to identify a neighbourhood of $e$ in the group with the tangent space and expanding for $x,y \in T_e$ the multiplication in the group as

In general, formal groups are local expansions of group laws. The power series $F_E$ associated to a smooth elliptic curve as described above is similarly the expansion of the additve group law defined by the elliptic curve.

Quillen explained that formal groups are related to complex cobordisms.

Let $MU_*$ be the complex bordsim ring, which is the ring whose elements are cobordism classes of (stably) complex manifolds with multiplication being cartesian product and additon being disjoint union. We write

This ring is universal for formal group laws in the sense that there is a formal group law

over $MU_*$ such that for every formal group law $F$ over any ring $R$ there is a unique ring homomorphism

such that

which means that if $F_{MU}(x,y) = \sum_{n,m} a_{nm}x^n y^m$ then

Recalling from above that every elliptic curve $E$ gives rise to a formal group law $F_E$ over the ring $\mathbb{Z}[a_1^,\cdots,a_6]$, we find that for every elliptic curve there is a unique ring homomorphism

Using this homomorphism we get an action of $MU_*$ on $\mathbb{Z}[a_1,\cdots,a_6]$. We want to use this to form a generalized cohomology theory ($\to$) by tensoring $\mathbb{Z}[a_1,\dots,a_6]$ with the universal cohomology $MU_*$ theory defined by complex cobordisms.

Instead of describing the ring spectrum which represents $MU_*$, we here just say how the $MU$ cohomology $MU_*(X)$ of any space $X$ looks like.

We set $MU_n(X)$ to be the ring of maps

from stably complex $n$-manifolds $M^n$ to $X$, where we identitfy two maps if their domain manifolds are cobounded by a stably complex $(n+1)$-manifold.

Here stably complex means that we can embed $M^n$ in some $\mathrm{R}^N$ for $N$ sufficiently large, such that the normal bundle of $M^n$ in $\mathbb{R}^N$ is a $\mathbb{C}$-vector bundle.

The entire ring $MU_*(X)$ is just the direct sum

and in particular the bare $MU_*$ from above is shorthand for the $MU$-cohomology of a point

It is important for the following construction that there is a natural graded action of $MU_*(\text{pt})$ on any $MU_*(X)$

simply given by taking a map

and

and forming the obvious map

In summary, we have an action of the ring $MU_*(\text{pt})$ both on the ring $\mathbb{Z}[a_1,\cdots,a_6]$ and the ring $MU_*(X)$, for all $X$.

Hence, for each elliptic curve $E$ and each space $X$, we can form the graded ring

This is the elliptic cohomology ring of $X$ with respect to the elliptic curve $E$.

As an example for this we recover ordinary integral cohomology and K-theory as degenerate cases of elliptic cohomology.

Namely, if our elliptic curve happens to be

with a bad singularity at $(0,0)$, the corresponding group law is simply (this is not supposed to be obvious)

As I reviewed last time, this is the group law which corresponds to ordinary integral cohomology.

The elliptic curve

has a singularity which is not quite as bad. It gives rise to the group law

As you can see from the table given last time, this is the group law which identifies complex K-theory.

1) classical elliptic cohomology

A special case of elliptic curves are the Jacobi curves, which are of the form

depending on two parameters $\delta$ and $\epsilon$.

The discriminant of these is

Using

we can alternatively write

which however works only in characteristic $\gt 3$, which is problematic in particular when applied to speher spectra, cause homotopy classes there have lots of 2- and 3-torsion.

Anyway, the formal group law corresponding to these curves is

where

This formula was originally found by Euler, even though he did not call it a formal group law.

We can rewrite $F(x,y)$ as

where

Now let $\tilde M_*$ be the ring of modular forms under the subgroup of $SL(2,\mathbb{Z})$ generated by $\tau \mapsto \tau + 2$ and $\tau \mapsto -\frac{1}{\tau}$.

There is a theorem due to Landweber, Ravenal and Stong which says that for $\delta$ and $\epsilon$ algebraically independent over $\mathbb{Q}$ we have

and for all of the rings in the diagram

there exists a generalized homology theory $h_*$ such that $h_*(\text{pt})$ is that given ring.

This is constructed by noticing that the formal group law defes an action of the oriented cobordism ring on the given ring $R_*$ from above, which allows us to form the homology ring of some space $X$ as

This is the homology which does the job.

The relation to genera is as follows.

A genus is a ring homomorphism

from oriented cobordisms to any other ring $\Lambda$ that is also a $\mathbb{Q}$-algebra.

Since

it suffices to specify $\phi$ on $[\mathbb{C}P^{2n}]$.

One calls the expression

the logarithm of the genus.

Ochanine defined an genus to be elliptic if this logarithm is of the form

for suitable ring elements $\delta$ and $\epsilon$, algebraically independent over $\mathbb{Q}$ and $\Delta \neq 0$.

There is a theorem by Landweber, Ochanine and Stong which says that if a genus $\phi$ is elliptic, then its image is

with

Furthermore, the image of spin cobordisms is

Again, looking at degenerate cases we find famliar examples.

1) In the case that $\epsilon = \delta$ we get

This corresponds to the signature genus of L-genus (which I also mentioned last time).

Here, too, the corresponding Jacobi-curve is singular.

2) Another example is $\epsilon = 0$ and $\delta = -\frac{1}{8}$. Here one gets

and this corresponds to the $\hat A$-genus.

As Atiyah and Singer found with their famous index theorem, for $M$ compact and spin and $\mathrm{dim} M = 2n$ the $\hat A$-genus

is the index of a Dirac operator on $M$, taking values in the integers.

We want some lifting of this statement to the loop space of $M$.

It turns out that this is possible if $M$ is string ($\to$), which means, according to a theorem by Laughlin, that it sfirst two Stieffel-Whitney classes and one half of the first Pontryagin class vanishes.

In this case there is something like a Dirac operator on loop space $LM$ and a theorem due to Witten and Zagier says that its index is a genus which is the $q$-series of a modular form - the Witten genus (partition function of the heterotic string).

What we are after is the homology theory which corresponds to this genus.

tmf - topological modular forms

According to Birgit Richer, in her experience it takes a group of experts a full week to discuss the construction of tmf. At that point 4 minutes time were left.

Apart from that, what is important about tmf is that, as Jacob Lurie describes on pp. 9-10 of his “survey” ($\to$), tmf is something like the universal elliptic cohomology.

We can get a glimpse of what this means by realizing that the way elliptic cohomology was defined above depended on a choice of coordinates (in the Weierstrass form) for an elliptic curve. In a vague sense tmf is the coordinate-free version of alliptic cohomology. Or something like that.

The point is that the Weierstrass form

is invariant under the coordinate transformations of the form

where $\lambda$ is a “unit” (invertible). Call the group of these transformations

Let

and form the cohomology theory

Then, according to a theorem by Hopkins, Miller, Goerss which has been given a more conceptual proof by Jacob Lurie, the $G$-invariant part $(E_A)^G$ of $E_A$ is a model for tmf.

(I am only 30 per cent convinced that this statement makes good sense as stated. Need to check that.)

There is a map from the tmf cohomology of a point to modular forms

whose kernel and cokernel are annihilated by 24.

Moreover, it is known that

Finally, the Witten genus is the composition map

from string cobordisms over $\mathrm{tmf}$ to power series in $q$ with coefficients in K-theory, restricted to the $0$-dimensional string manifold.

Oh dear, you can tell that this last part was transmitted and received in mere 4 minutes.