In 1994, Mike Hopkins wrote a paper called [Topological Modular Forms, the Witten Genus, and the Theorem of the Cube](http://www.mathunion.org/ICM/ICM1994.1/Main/icm1994.1.0554.0565.ocr.pdf). 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?



Atiyah-Singer style index theorem for elliptic cohomology?
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](https://web.archive.org/web/20161027180429/http://www.staff.science.uu.nl/~henri105/Grants/CNTMF-B1.pdf) and [here](https://web.archive.org/web/20161027180429/http://www.staff.science.uu.nl/~henri105/Grants/CNTMF-B2.pdf) (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](https://web.archive.org/web/20170702204522/http://www.staff.science.uu.nl/~henri105/PDF/OW-report-STcocycles.pdf). 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](https://web.archive.org/web/20170702204744/http://www.staff.science.uu.nl/~henri105/PDF/EquivariantElliptic.pdf). 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.


