I read Lee Smolin's book "The trouble with physics" and the book says that the finiteness of string theory ( or string pertubative theory) is by no means a proven mathematical fact, despite that the string community widely believes it to be so.
However, some string theorists do pronounce in a very strong term that the string theory is indeed proven to be finite, such as this website:
The names associated with the available proofs of the finiteness include Martinec; Mandelstam; Berkovits; Atick, Moore, Sen; d'Hoker, Phong, and others. Some of these papers are more complete - or quite complete - or more constructive than others and there are various causal relations between the papers. Many of these results are secretly equivalent to each other because of the equivalences between the approaches that are demonstrated in other papers. Many of these papers were preceded by less successful papers or papers with flaws - flaws that were eventually fixed and settled.
Also, I assure Jacques that he has met people who consider Mandelstam's proof to be a proof, and besides your humble correspondent, this set includes Nathan Berkovits who confirms Mandelstam's proof on page 4 of his own proof in hep-th/0406055, reference 31, even though Nathan's proof is of course better. ;-)
At any rate, the question of perturbative finiteness has been settled for decades. Many people have tried to find some problems with the existing proofs but all of these attempts have failed so far. Nikita will certainly forgive me that I use him as an example that these episodes carry human names: Nikita Nekrasov had some pretty reasonably sounding doubts whether the pure spinor correlators in Berkovits' proof were well-defined until he published a sophisticated paper with Berkovits that answers in the affirmative.
So? Who is right on this? Are there rigorous proofs that show that string theory is always finite, as opposed to proofs that only show the second, or third term of the series is finite?
Edit: This website says that, in Remark 1:
The full perturbation series is the sum of all these (finite) contributions over the genera of Riemann surfaces (the “loop orders”). This sum diverges, even if all loop orders are finite.
So I guess this says-- in a very strong term--that String Theory is proven to be infinite... am I right?
According to here, it is a good thing that the string theory is infinite, because if the sum is finite, this would indicate negative coupling constants which are not physical.
But I still don't get it. The reason why we use an infinite series to represent a physical quantity is because we believe that after summing up the series, we will come to get a finite number. If not, we would say that the theory breaks down and the physical quantity is not computable from the theory. So in order to avoid negative coupling constant which is unphysical, then we allow the sum to be infinite? Then what does this tell us of string theory's predictive power? If a theory can't predict physical values, then it is quite as useless as any meta-reasoning.