Atiyah-Singer style index theorem for elliptic cohomology?

Question

In 1994, Mike Hopkins wrote a paper called Topological Modular Forms, the Witten Genus, and the Theorem of the Cube. As usual, the introduction was fantastic, explaining the power of various cobordism invariants and connections between homotopy theory and other fields. He states "it is believed that there is an "index" theorem relating analysis on loop space to elliptic cohomology. So far, a satisfying mathematical theory is lacking."

What is the status of this question? Has such a theory been developed?

Answer 1

The status of this question is OPEN.

This theory has NOT been developed yet.

That being said, the evidence is as compelling as ever, I don't know of any obstructions to making this work, and I'm convinced that there's an awesome theory out there, waiting to be discovered.

Roughly 8 years ago, I wrote an unsuccessful ERC proposal, where I outlined a program. This proposal can be found on my Utrecht website here and here (warning: that website will probably stop existing one year from now, so the links will become broken -- but the linked material will still be there to be found on whatever new website I end up having in the future).

There are small bits and pieces of what one might call progress, which I've made available on my website:

Here's one. In this draft, I take a compact simply connected Lie group $G$ of dimension $d$, and I consider the map $p:G\to \{pt\}$. I construct, geometrically, the $TMF$-pushforward $p_!(1)\in TMF^{-d}(\{pt\})=\pi_d(TMF)$ of the element $1\in TMF^0(G)$ along the map $G\to \{pt\}$.

Here's another one. In this draft, I show that there's a new type of 2-equivariance for $TMF$, where the group of equivariance gets replaced by a fusion category.

Answer 2

What little background I have in this area is probably outdated, but I can share a few thoughts. The "index theorem" to which Hopkins was most likely referring was in Witten's 1987 paper The Index of the Dirac Operator In Loop Space. Witten's idea was to apply Atiyah and Singer's equivariant index theorem to the natural action of $S^1$ on the loop space of a manifold. This doesn't strictly speaking make sense since the loop space is infinite dimensional, but he nevertheless did a formal computation which broadly speaking gave the results one might expect from similar computations elsewhere in mathematical physics. He found that the index of the "loop space Dirac operator" is what we now call the Witten genus, a genus in the sense of Hirzebruch.

This triggered a lot of mathematical work to try to properly make sense of Witten's computation, and to the best of my knowledge mathematicians have not yet succeeded. But there is a rich and suggestive collection of analogies (alluded to in David Roberts' comment), some of which have been made precise:

1. KO-theory spectrum $\leftrightarrow$ tmf spectrum
2. Spin structure $\leftrightarrow$ String structure
3. Spinor Dirac operator $\leftrightarrow$ Loop space Dirac operator (??)
4. Index of Dirac operator $\leftrightarrow$ Poincare duality pairing in tmf
5. $\hat{A}$-genus $\leftrightarrow$ Witten genus

A lot of this is worked out: the tmf spectrum has been constructed in various different ways, and it is known that string structures are the right notion of orientation in tmf just as spin structures are orientations in KO-homology. There are various constructions which deserve to be called a loop space Dirac operator (for instance here), and there are a lot of strong analogies between the $\hat{A}$-genus and the Witten genus - that's more or less the point of the Hopkins paper linked to in the original question, for instance.

Others can correct me if I'm wrong, but I think the big gap is in point 4 above: the analogy between index theory and Poincare duality for tmf. In traditional index theory this analogy is made possible by the fact that there are good analytic models of the KO-theory spectrum - involving spaces of Fredholm operators, say - but I'm not sure there are similar models for tmf. It takes quite a bit of work to prove that the spinor Dirac operator gives the KO-fundamental class, but with a good model for the spectrum it's "just" an analysis problem. I'm by no means a tmf expert, but my impression is that it is still quite hard to explicitly describe (co)homology classes in the TMF groups of a manifold.

One promising candidate for this is the Stolz-Teichner program which proposes to construct generalized (co)homology theories as spaces of supersymmetric topological field theories (up to concordance). They were able to construct ordinary cohomology and K-theory using 0|1 and 1|1 TFT's, respectively, and there is some good evidence that 2|1 TFT's could give TMF classes, but I don't think this has been proven rigorously.