I’ve given some detailed technical arguments about this and you’re not addressing them or showing any signs of having looked into the details of what is going on in these two-loop calculations enough to even understand what I am talking about.

If you talk to people who have done these recent two-loop calculations (d’Hoker-Phong, another group is Zheng-Wu-Zhu) or read their papers they’ll explain to you the subtleties that happen at two loops and that they don’t know what happens at three loops. As for covariant approaches like that of Berkovits, it is simply undeniable that he has yet to calculate a 2-loop amplitude.

If you don’t believe me, contact any of these experts, or sit down and read some of their papers. Until you’ve done that, you have no business going around making claims that finiteness of multiloop amplitudes is well-established fact. It isn’t and such claims are extremely unfair to people who are working hard to actually understand what is going on, rather than making unsupported claims about the issue.

Many, many thanks for your explanations and for pointing out a nice (very readable!) paper!

The D=11 supergravity starts having divergences at 2-loops.

The conclusion seems quite strong (for perturbative supergravity, at least):
“…there is no local, unitary (ghost/tachyon-free) quantum field
theory whose action reduces to QGR or classical GR that is also free of infinities; the latter are
almost certainly there at every order, requiring an infinite number of input parameters to define
these theories. The conclusion includes all possible SUGRA models, i.e., from D=4 through D=11, as well. Although the presence of new counterterms at all loop orders (or at an infinite set of them) cannot reasonably be rigorously demonstrable, the fact that the ones we did see appeared at lowest permitted order (so that no “hidden” invariances prevented them), is quite convincing evidence.”

Again, there’s no evidence of a problem that will occur at higher loops in superstring multi-loop amplitudes, there just isn’t a solid argument showing that no problems can occur. I’d have to look up the precise facts about supergravity, I believe they depend on N, the number of supersymmetries. But it has been shown that the known symmetries of the theory are such that at high enough loops you get terms that have no reason to be zero and are presumably divergent. It is true that these problems are at high enough loops that no one has been able to actually compute the diagrams to be sure the infinities are really there.

A quick search turned up this reference
http://arxiv.org/abs/gr-qc/9911073

It is pretty sad to see (some) string theorists being unreliable even about technical aspects of the subject. Prior to this, I thought string theory had already been “proven finite” to all orders: thanks for pointing out D’Hoker-Phong’s remarakable paper.

A couple of questions:
“…even if the multiloop amplitudes are finite, their sum isn’t”

Does that mean string perturbation theory series is an asymptotic series?

“Maybe someone will find that, just like in supergravity, if you go to high enough order, you run into trouble. ”

Surprising result to me. Is it easy to understand why this happens? A reference perhaps?

But for the last 20 years this idea has been beaten to death by thousands of theorists writing tens of thousands of papers on the subject. The bottom line of all this work is that, even if the multiloop amplitudes are finite, their sum isn’t, so this idea doesn’t give a well-defined, finite theory of quantum gravity. There’s a lot of wishful thinking about hypothetical M-theory and non-perturbative versions of string theory, but wishful thinking is all it is right now. The main achievement of this whole program, the supposed calculation of the entropy of certain special black-hole like configurations using branes, is pretty underwhelming.

You can’t just keep repeating the initial motivation for a speculative idea, long after a huge amount of effort has gone into working on it, with extremely disappointing results. At some point you have to admit failure and move on, something that most string theorists seem incapable of.

1. The particle like limit of the string corresponds to various boundaries of moduli space. I was under the impression that these limits are simple and one is able to make general statements about them, though I am far from being an expert. The tricky parts have to do with the interior of moduli space. If there were some problems there they would not be called UV or IR diveregences, they would be inherently stringy. No reason to suspect they are there, but in any event I am more impressed with cancelling the divergences we already knew about, not with the elimination of all logical possibilities for problems.

2. Divergences are wonderful since they give rise to proper understanding of field theories via the renormalization group. This very understanding tells you GR cannot be treated as QFT, as it is not renormalizable. This is a concrete and straightforward problem, not an interpretational issue or an aesthetic displeasure (as in “background independence”), one simply cannot calculate finite and unambigous results.

What’s his argument that divergences in the interior of moduli space can’t occur in his formalism? Do you mean the footnote where he says that in conformal gauge, there are no obvious potential sources of divergences since the amplitudes are independent (up to surface terms) of the locations of picture-changing operators?

This isn’t a very solid argument. How do I know there isn’t a “non-obvious” potential source of divergences? And how do I know that the picture-changing operators are really well-defined and do exactly what they are supposed to? If Berkovits could explicitly work out what happens at two loops so one could be sure that at least in that case he really had well-defined amplitudes and there were no “non-obvious” problems, that would be a lot more convincing.

That’s about all I have to say about this, if you really are so interested in this topic, you should read D’Hoker and Phong’s papers carefully. I think if you do this you’ll see that this is an exceedingly tricky business and that it is far from obvious that Berkovits has a solid argument.