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?

Edit 2:

According to here,

it is a good thing that the string theory is, because if the sum is finite, this would indicate negative

infinite

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.

## Best Answer

A commented list of literature with claimed results on (super-)string perturbative finiteness is here:

http://ncatlab.org/nlab/show/string+scattering+amplitude

Notice the technical caveats in remarks 1 and 2 at the beginning of this entry.

In summary the statement is: there are plenty of arguments that the (super-)string is UV-finite at each order and this argument is regarded as robust. There are much more recently only computations of the actual integrals over (super-)moduli space which also come out finite (hence IR finite) but which have been done in detail only at low loop order (since this is technically much more demanding). Arguments by Berkovits that the pure spinor formulation helps here seem to have not been further followed up much (?).

One issue apparent from the list of literature is that theoretical physics is suffering here a bit from its lack of mathematical certainty: it is not always clear whether a claimed result has really been established, or just made very plausible, and what exactly has been claimed. For instance often one sees people point to Madelstam's article (listed at the above link) as a proof of finiteness, while Mandelstam himself, according to his Wikipedia article, says he only showed the absence of one of several possible divergences.