# “tmf$(n)$ is the space of supersymmetric conformal field theories of central charge $-n$”

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.

-
Could you please elaborate a bit on what the conjecture is? –  user566 Dec 14 '11 at 5:32
Unfortunately the only reference I have is from 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. –  Ryan Thorngren Dec 14 '11 at 5:39
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. –  user566 Dec 14 '11 at 5:45
Thanks a lot, that's perfect. Hoping for some interesting and useful answers. –  user566 Dec 14 '11 at 6:02
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. –  Scott Carnahan Jan 6 '12 at 5:47

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.

-
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. –  Ryan Thorngren Dec 15 '11 at 3:41

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/

-

I happen upon this old thread now. Maybe it is still worth giving an update, and more of an answer.

The latest account (as of the time of this writing) of the conjectural statement in question here appears as Conjecture 1.17 in

(See section 1.8 there for explicit comparison to the earlier accounts which have been cited in the other replies.)

As Aaron hints at in another reply, this conjecture is motivated from two facts:

first, the partition function of the (heterotic) superstring -- the Witten genus being a ring homomorphism

$$\Omega^{String}_{\bullet} \longrightarrow MF_\bullet$$

from the cobordism ring of Green-Schwarz quantum anomaly-free spacetimes to the ring of modular forms has a homotopy-theoretic refinement to what is called the String-orientation of tmf, which is now a homomorphism of coherently homotopy-commutative rings

$$M String \longrightarrow tmf$$

to the tmf-spectrum.

Second, Stefan Stolz and Peter Teichner in their earlier work on (1|1)-dimensional field theories and K-theory have shown that in some sense the "space" of $(1|1)$-dimensional quantum superparticle theories is K-theory spectrum.

This is suggestive, because by chromatic homotopy theory there is a precise mathematical sense in which tmf is the 2-dimensional (stringy) version of K-theory theory.

Therefore the above Conjecture 1.17 says very roughly that the "space" of suitably 2-dimensional super-string theories should "be" $tmf$.

For the time being it seems fair to say that the conjecture really remains a conjecture. What the above article establishes rigorously is a rederivation of the Witten genus in such a way that now the conjecture may be phrased precisely.

(Over the last years, much of the attention in this "Stolz-Teichner program" has been on lower dimensional analogs of the general idea. Much work has appeared on the lower-dimensional case of $(0|1)$-dimensional field theories (physically: super-instantons, super D(-1)-branes). )

-