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

Question

I read this intriguing statement in John Baez' week 197 the other day, and I've been giving it some thought. The post in question is from 2003, so I was wondering if there has been any progress in formulating or even settling the conjecture in the title.

Here tmf(n) is the spectrum of topological modular forms, defining a sort of generalized elliptic cohomology theory. These have a very nice construction by Lurie involving a certain moduli stack, so I was hoping one could use this construction to give a description of conformal field theory. Even if the statement "tmf(n) is the space of supersymmetric conformal field theories of central charge -n" is just a moral statement I am interested in the intuition behind it.

This post has been migrated from (A51.SE)

Comments

Could you please elaborate a bit on what the conjecture is?

This post has been migrated from (A51.SE)

---

Unfortunately the only reference I have is from http://math.ucr.edu/home/baez/week197.html as I said. Maybe it does not even deserve to be called a conjecture, but I would like to understand the intuition behind the statement. The connection between modular forms and vertex operator algebras seems very deep, mostly witnessed by the solutions to specific problems, such as the "monstrous moonshine". The comparative generality of the statement in the title is what is so interesting.

This post has been migrated from (A51.SE)

---

Thanks. I was just looking for a brief explanation or definition of the terms in the title and whatever references you have, just as a starting point for whomever answers.

This post has been migrated from (A51.SE)

---

Thanks a lot, that's perfect. Hoping for some interesting and useful answers.

This post has been migrated from (A51.SE)

---

You're unlikely to get a description of conformal field theory from the tmf spectrum. It seems more likely that any interesting functors go the other way, and that elliptic cohomology exhibits some kind of shadow of CFT.

This post has been migrated from (A51.SE)

Answer 1

This is a conjecture of Stoltz and Teichner (see, for example, this paper or this survey). The best evidence is that they do define a notion of the space of 1D field theories and show that it is a classifying space for K-theory. One might suspect that elliptic cohomology (i.e., tmf) would come from one dimension up. If there was a better motivation for it than that (other than the obvious connection with the Witten genus, etc.), I've forgotten it. I last looked at this around 2006, so there might have been some progress in the interim.

This post has been migrated from (A51.SE)

Comments

Thanks. These look like just the kind of thing I was looking for. I'm still hoping someone can weigh in on any recent developments, though.

This post has been migrated from (A51.SE)

Answer 2

Besides the paper and survey pointed out by Aaron, which are the best things to read, there are also these talks:

http://online.itp.ucsb.edu/online/strings05/teichner/

http://online.itp.ucsb.edu/online/strings05/stolz/

This post has been migrated from (A51.SE)