Elliptic Cohomology making a splash

[Kea]

All right. I've slowly come to the conclusion that the paper

A Survey of Elliptic Cohomology
J. Lurie
www.math.harvard.edu/~lurie

mentioned on various physics blogs already, along with anything else Lurie might have written, is going to be well worthwhile spending time on. Lurie is undoubtedly one of those very rare young wizards that find their natural home surfing the waves on the deserted beach of future mathematics. Funnily enough, he was mentioned to me only recently by a very respectable older mathematician whom Lurie had managed to impress. I'd never heard of him.

It would be nice to leave him to it, but it looks like he's quite keen on things like higher dimensional toposes. This is frustrating, because he's really a homotopy theorist and treats categories accordingly.

To put the reader in the picture: Urs Schreiber at http://golem.ph.utexas.edu/string/index.shtml has been talking quite a bit lately about Elliptic Cohomology. This is because the non-perturbative Stringy people, such as the authors of

Type II string theory and modularity
Igor Kriz, Hisham Sati
http://arxiv.org/abs/hep-th/0501060

have become interested in refining the K-theory partition function in type II string theories using elliptic cohomology (to quote the abstract). It turns out that to be 'anomaly-free' one needs a pretty fancy type of cohomology called TMF (topological modular forms). This is reviewed in

Algebraic topology and modular forms
Michael J. Hopkins
http://arxiv.org/abs/math/0212397

but the more mathematically inclined should look in Lurie's paper to see how it arises from the search for some kind of universal elliptic theory - and universal is meant in the categorical sense. This is all in part 1 of the paper. By part 2 we're thrown into Derived Algebraic Geometry, which is what Lurie's thesis is on. He wants a G-equivariant type of general cohomology theory. Recall that ordinary G-equivariant cohomology came crashing together with quantum field theories when Witten looked at 2D Yang-Mills (in particular) path integrals in such terms.

Back later
Kea

 

[Kea]

A Survey of Elliptic Cohomology J. Lurie
www.math.harvard.edu/~lurie

All of this fancy maths starts out by doing things with commutative number fields and commutative formal group laws etc. Associated (one dimensional) algebraic groups for an algebraically closed field can only be one of the following:

1. the additive group
2. the multiplicative group
3. an elliptic curve

Hence the desire for some sort of cohomology associated to elliptic curves, given a commutative ring $R$.

What happens when one tries to define an equivariant version? It turns out (page 11) that one needs to think of functions on an algebraic group $G$ not in terms of sheaves of commutative rings, but in terms of $E_{\infty}$-rings. Lurie says: this is a very important notion which we will need to use throughout the paper.

Unfortunately, he then says that a precise definition would waste too much time, but the basic idea is: start by thinking of a space as an object $A$ in the category of topological spaces with some special additional structure that knows about higher homotopies which weaken, for instance, relations between $A$ and $A \times A$ expressing basic properties of multiplication. Sound familiar? It's higher dimensional category theory! A morphism $A \rightarrow B$ is an equivalence if it gives isomorphisms $\pi_{n} A \rightarrow \pi_{n} B$. Given any such $E_{\infty}$-space one can define a cohomology theory, which has the property that $H^{n}$(pt) = 0 for $n$ > 0.

But that theory isn't quite right either, because it isn't periodic (like K-theory) so what one actually needs is something called an $E_{\infty}$-ring. So he's calmly generalising commutative algebra. Why not now do Algebraic Geometry the $E_{\infty}$ way? Hence derived schemes etc.

 

[Kea]

What are these $E_{\infty}$-rings doing? Can we understand the analogue of the category Rng of commutative rings? Rng has an initial object, namely the integers $\mathbb{Z}$. The initial object for $E_{\infty}$-Rng is the sphere spectrum $\Sigma$ of n-dimensional spheres.

Using the group algebra $\Sigma [\mathbb{C}\mathbb{P}^{\infty} ]$ in a way that remains a mystery to me for now, Lurie can define an $E_{\infty}$-ring that is equivalent to the usual complex K-theory! And he can get real K-theory as well. Quoting Lurie from page 20:

If we view $\mathbb{C}\mathbb{P}^{\infty}$ as the classifying space for complex line bundles, then the group algebra $\Sigma [ \mathbb{C}\mathbb{P}^{\infty} ]$ can be viewed as a universal cohomology theory in which it is possible to add line bundles. The above result can be viewed as saying that if we take this universal cohomology theory and invert the Bott element $\beta$ then we obtain a theory which classifies vector bundles. A very puzzling feature of the result is the apparent absence of any direct connection of the theory of vector bundles with the problem of orienting the multiplicative group.