The problem with this question, for mathematicians, and actually for anyone, is that the term "string theory" is not well-defined, making the question of falsifiability much more complicated.

The most well-defined interpretation of "string theory" would be the superstring in 10 flat space-time dimensions, which is defined by a series expansion. The details of the precise definition of the higher order terms here are very tricky, see the multiple hundreds of pages of Witten's recent papers. The standard conjecture is that this series expansion does not converge, but may be useful as an asymptotic expansion. This interpretation of "string theory" is not only falsifiable, but false: we don't live in a world of 10 flat space-time dimensions.

Less well-defined is the extension of the above to non-flat space-times. This is a long story, but the standard conjecture is that such an extension exists for manifolds with curvature satisfying a condition close to vanishing of the Ricci tensor. Looking for examples with 4 flat conventional space-time dimensions and six compact dimensions, one gets the famous 6d Calabi-Yau manifolds, which come in families of various dimensions ("moduli"). Early attempts to connect string theory to reality focused on finding an appropriate 6d compact manifold, assuming it had a size of order the Planck scale, and trying to conjecture what the effective theory at long distances would look like, trying to match this to the Standard Model. One major problem here is that you need some new mechanism to "stabilize moduli", fixing the values of the moduli parameters. The early hope was that some simple such mechanism could be found, giving a small number of solutions for each class of Calabi-Yaus (the number of classes may be finite, but this is unknown).

In this interpretation of string theory, if you make a bunch of assumptions, including that some parameters are small so that one has a useful asymptotic series, you could imagine getting falsifiable predictions: in the right parameter region, imagine you could calculate the low energy limit for each stabilized moduli solution in each Calabi-Yau class, this should give a list of possibilities, which you could compare with the real world, falsifying the theory if none matched.

In the mid-nineties, the standard conjecture among string theorists became that the above series expansions were asymptotic expansions to an unknown theory called "M-theory", a theory which remains undefined to this day, although there are a large number of conjectures about the properties of this unknown theory. One part of this conjecture is that the spectrum of the theory contains not just excitations of strings, but also "branes", which correspond in some sense to possible boundary conditions one can impose on the ends of strings. The M-theory conjecture dramatically increased the range of possibilities for observable low energy physics, as well as making it exceedingly unclear what exactly one meant when one said "string theory". The best definition now might be something like "an undefined theory conjectured to have the following list of properties", where the list might vary from person to person, with different strengths of confidence in different things in the list. At this point it seems to me that falsifiability in principle starts to become a huge issue, with it very unclear what if any constraints on possible low energy physics come from the M-theory conjecture (or for that matter, what the constraints on high energy behavior are). The sorts of arguments you now hear for M-theory falsifiability are sometimes things like "well it's a quantum theory, so if quantum theory is wrong it's falsified", but I find it difficult to take this kind of thing seriously.

A conventional argument for string theory falsifiability goes something like "we don't have any low energy predictions, but if you could build a big enough accelerator and probe Planck scale processes, you would see characteristic properties of the amplitudes that make up terms in the asymptotic series". The problem with this is that it assumes that you are in the limiting case of M-theory where it is a theory of superstrings. For a generic M-theory solution, we have no real idea what the possibilities are, making the theory non-falsifiable not just at low energies but at all energies.

The current state of affairs is that various mechanisms invoking branes were found a decade or so ago which could be used to conjecturally stabilize moduli. These mechanisms however lead to exponentially large numbers of solutions, raising a problem of falsifiability, independent of the one you already have from not knowing quite what the theory is. This is the so-called "landscape" problem. It is rather deadly for prospects of getting predictions from string theory, but is part of a larger problematic situation that I have tried to describe.

So, I'd claim that "string theory", as the term is used now, is not falsifiable in any conventional scientific usage of the term. For more details, see my book "Not Even Wrong", or the blog with the same name.

**Addendum:** In other answers to this question, the argument is being made that string theory is not falsifiable, but this is no different than other physical theories. Obviously there's something funny about such an argument, since it is claiming that a theory renowned for its failure to be testable by experiment is no different than our most successful theory in physics, which has been tested over and over again with dramatic success.

It is in general true that if you pick any "theory" in physics, if it disagrees with experiment you can find some more complicated version of it that evades this disagreement. In this sense, most physics theories are "unfalsifiable", but this just shows that "falsifiability" is a somewhat more subtle issue than one might naively think. I'll leave it as an exercise for the reader to think through these issues, and see for themselves why "falsifiability", while a subtle concept, is not an empty one. Here's a hint: theories can fail in two ways. One way is by making a wrong prediction that falsifies the whole thing. The other is by turning out to be an empty idea, always requiring that you put more complexity into your model to match the reality is is supposed to explain. String theory is an example of a theory that has failed in the second of these failure modes (which actually is the most common one).

The argument is also being made that string theory is "more predictive" than other theories. This argument is based on comparing two very different things: the conjectural properties that its proponents would like an unknown theory to have with the actual properties of a mathematically well-defined theory.

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