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Like many other mathematicians, I think string theory very attractive. This theory has wonderfully influenced many new topics in mathematics (I myself have worked on one of them), but it's not the issue here (see for example there for this point).

Unfortunately, it is well known that string theory is not appreciated by some of the physics community, because of its alleged non-falsifiability. Nevertheless, here and there, I hear some say that there are some possible indirect experiments...

So, to do justice to string theory, I ask those who can to explain here (for mathematicians) one reason why this theory is non-falsifiable or one experiment that would test it.

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closed as off topic by Felipe Voloch, Misha, Andres Caicedo, Andy Putman, Igor Belegradek Jun 11 '13 at 10:57

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of course string theory is falsiable, or it would not be physics; it predicts a new generation of particles (so called "supersymmetric" counterparts of the known particles) and it predicts extra spatial dimensions; the accelerator experiments at CERN support neither of these predictions, so this does not look hopeful, and this is why many physicists are critical of string theory; at the same time, as a mathematical toolbox, it is proving to be very useful in other branches of physics, for example providing a new approach to a theory of superconductivity –  Carlo Beenakker Jun 9 '13 at 10:26
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@JHI There is a difference between being falsifiable and being falsified. –  Woett Jun 9 '13 at 16:00
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Voted down because this isn't a math question. Nonetheless, string theory could be falsified if we could figure out how to accelerate elementary particles up to the Planck scale. Since that seems to be slightly out of the realm of experimental possibility, the answer is no at the moment. Someone could come up with something clever, though. –  Aaron Bergman Jun 10 '13 at 14:35
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Voted to closed since this is not a math research question. Specifically, I do not see how falsifability of the string theory affects mathematics. It it quite clear that the string theory has already had a huge impact on (some areas of) math regardless of whether it describes physical reality. –  Igor Belegradek Jun 11 '13 at 11:07
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6 Answers 6

up vote 21 down vote accepted

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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a) No interesting perturbation series is supposed to converge, for it would mean that it converges also for negative coupling. The important aspect of the string perturbation series is that it is degreewise finite, hence renormalized. b) CY-compactifications "one gets" only when demanding N=1 global susy after compactification. c) The "string landscape" is conserably more constrained than the "QFT landscape", as there are considerably more consistency constraints on a string background. d) Complaints about the size of the landscape confuse theories with models built inside theories. –  Urs Schreiber Jun 10 '13 at 8:42
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The nLab has a page devoted to this question: ncatlab.org/nlab/show/string+theory+FAQ –  Urs Schreiber Jun 10 '13 at 9:07
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First let us recall why string theory is attractive. As of now, we have two experimentally verified, but mutually incompatible theories describing fundamental physical phenomena. The standard model of particle physics is a quantum field theory describing all the elementary particle interactions except for gravitation. General relativity is a classical theory describing gravitation and classical electrodynamics, but none of the other fundamental interactions in the realm of the standard model. The fact that it is a classical theory also tells us that it is necessarily an approximation of reality.

String theory is a quantum theory which in certain limits can be described by quantum field theories similar to the standard model, and in other limits by classical gravitational theories akin to general relativity. It seems therefore that string theory offers a realm to unify our two apparently incompatible descriptions of fundamental physical phenomena.

Now there are two issues concerning the falsifiability of string theory.

The first one is common to any theory of quantum gravity. Very generic arguments indicate that "quantum gravity effects", i.e. effects that would mix quantum physics and gravitational physics, for which neither the standard model nor general relativity would be appropriate descriptions, occur at energies well beyond experimental reach. Because of this, there is no serious hope to test experimentally a theory of quantum gravity which is consistent with the standard model and general relativty. Of course these arguments are not water-tight, and we can imagine various scenarios where quantum gravity effects would kick in just at the limit of our current experimental reach, but such scenarios are necessarily unlikely. Let us emphasize that this has nothing to do with string theory, and is a problem common to any theory of quantum gravity that does not obviously contradicts what we already know about Nature. Also, we have computational methods allowing us to extract the physics from solutions of string theory. So should we have experimental data of physics at high enough energies, we would be able to rule out of confirm most of the solutions of string theory. Therefore this first issue is really about our limited experimental reach.

The second issue is specific to string theory. While we do not understand the theory completely yet, there are good indication that the space of solutions of the theory is very large (the so-called "landscape"). For instance, in the limit of low energy and zero gravitational coupling, where in principle we would expect to recover the standard model, one can find a huge variety of field theories, most of which have nothing to do with the standard model. There is therefore a feeling that "anything goes" and that there is no way of explaining from string theory the low energy physics we are used to. When you look at the details, it is in fact not true that anything goes, but there are still a large variety of field theories that can be obtained in this way, and it is not clear how the standard model would have been selected over all the other possibilities.

There is nothing we can do about the first issue. The best we can do about the second issue is to further our understanding of string theory and understand better its space of solution.

It should be also emphasized that confrontation with experiment, while of crucial importance, is not the only way we have to construct physically relevant theories. Einstein's theory of general relativity was confronted to experiment only after it had been fully formulated. Einstein devised it using consistency requirements, essentially requiring the compatibility of classical electrodynamic and gravitational physics. The development of string theory has been very much in this spirit, and string theory has passed an amazing number of very non-trivial consistency tests (anomaly cancellation, consistency of the web of dualities, etc...). The fact that consistency constraints seem to fix a unique form of the theory is an encouraging sign that it should have something to do with reality.

Addendum to answer the comments below: One has to be aware that a single theory, such as string theory or a quantum field theory, can have several states looking like the vaccuum, whose excitations above the vaccuum obey different physics. So from a low energy point of view, what we can easily falsify are the vacua of a theory. Only when one has exhausted all the vacua, and that none of them is seen to coincide with whatever experimental data is available can we conclude that the theory has been falsified.

Tools have been developed to extract the physics from the known vacua of string theory. The immense majority of them has already been falsified: for instance many of them describe spacetimes with a number of macroscopic dimensions different from 4, if they describe a spacetime at all. All these vacua are ruled out. A whole domain of research of string theory consists in trying to isolate the vacua of string theory whose physics is close or identical to the physics we experience, i.e. to isolate the vacua which cannot be falsified with our current knowledge of low energy physics and characterize them.

What we cannot produce yet is an exhaustive list of these vacua together with all their physical properties. But that's partly why there are still people working on string theory, and why it is an interesting subject to work on. I hope this shows clearly that to the extent of our theoretical and experimental abilities, string theory is falsifiable and no different from any other physical theory.

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$n > 0$ seems like a good start. –  Woett Jun 9 '13 at 18:00
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You didn't actually answer the question. Your argument seems to be that the theory is falsifiable, so you should provide what is asked for: a specific experimental prediction. Note that if your prediction is that "at high enough energy one will see the soft scattering amplitudes of perturbative string theory", you really haven't taken into account M-theory in general, so you are talking about falsifying only one corner of the theory. –  Peter Woit Jun 9 '13 at 23:41
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Why is a large space of solutions a problem? Do we think we are going to "explain" why reality is the way it is? As mathematicians don't we know that this is inherently hopeless? There are infinitely many mutually incompatible consistent theories. The job of physics should be to describe the world in a systematic way, not search for "ultimate explanations" which is infinitely regressive and hopeless. –  Monroe Eskew Jun 10 '13 at 0:20
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mbsq is right, a large number of solutions has never been a problem of any theory. Compare for instance general relativity, which certainly has a highly infinite dimensional moduli space of solutions (certainly not a finite set, and be it of size 10^500). A theory alone never makes any prediction, it is only a theory together with a choice of model in the theory that does (a choice of some of the paramters). E.g. general relativity does not predict that we live in an FRW universe. That is the model we have chosen because it fits observation. –  Urs Schreiber Jun 10 '13 at 12:06
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... vacua reproducing convincingly the standard model. These solutions of string theory cannot be ruled out with our current experimental knowledge, as the standard model is not yet ruled out. They predict a well-defined completion of the standard model that we will be able to test with experiments probing higher energies. –  Samuel Monnier Jun 10 '13 at 14:01
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The nLab has a page devoted to this question:

ncatlab.org/nlab/show/string+theory+FAQ

I'll be glad to further expand this as need be.

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The page you link to does not answer the question, with the term "falsifiability" not on the page. It appears to me that your answer to the question would be that string theory is unfalsifiable, but you believe all theories are unfalsifiable. This is an extreme argument that few physicists would agree with, as is your argument that "string theory is actually more testable than other existing theoretical frameworks". You are well aware of the counter-arguments, so if you want nLab to be taken as an authoritative source of information, you should not do this sort of thing. –  Peter Woit Jun 10 '13 at 13:26
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That there are more constraints on a consistent perturbative string theory background than on any old QFT is well known. That's why one started describing that "landscape" of solutions in the first place, since for general QFT there are two few constraints to say anything of interest. And if anyone points out a mistake on an nLab page, it will be corrected. –  Urs Schreiber Jun 10 '13 at 17:56
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I was encouraged to post this answer from Physics Stack Exchange to the nearly equivalent question "What experiment would disprove string theory?" here as well.

One can disprove string theory by many observations that will almost certain not occur, for example:

  1. By detecting Lorentz violation at high energies: string theory predicts that the Lorentz symmetry is exact at any energy scale; recent experiments by the Fermi satellite and others have showed that the Lorentz symmetry works even at the Planck scale with a precision much better than 100% and the accuracy may improve in the near future; for example, if an experiment ever claimed that a particle is moving faster than light, string theory predicts that an error will be found in that experiment

  2. By detecting a violation of the equivalence principle; it's been tested with the relative accuracy of $10^{-16}$ and it's unlikely that a violation will occur; string theory predicts that the law is exact

  3. By detecting a mathematical inconsistency in our world, for example that $2+2$ can be equal both to $4$ as well as $5$; such an observation would make the existing alternatives of string theory conceivable alternatives because all of them are mathematically inconsistent as theories of gravity; clearly, nothing of the sort will occur; also, one could find out a previously unknown mathematical inconsistency of string theory - even this seems extremely unlikely after the neverending successful tests

  4. By experimentally proving that the information is lost in the black holes, or anything else that contradicts general properties of quantum gravity as predicted by string theory, e.g. that the high center-of-mass-energy regime is dominated by black hole production and/or that the black holes have the right entropy; string theory implies that the information is preserved in any processes in the asymptotical Minkowski space, including the Hawking radiation, and confirms the Hawking-Bekenstein claims as the right semiclassical approximation; obviously, you also disprove string theory by proving that gravitons don't exist; if you could prove that gravity is an entropic force, it would therefore rule out string theory as well

  5. By experimentally proving that the world doesn't contain gravity, fermions, or isn't described by quantum field theories at low energies; or that the general postulates of quantum mechanics don't work; string theory predicts that these approximations work and the postulates of quantum mechanics are exactly valid while the alternatives of string theory predict that nothing like the Standard Model etc. is possible

  6. By experimentally showing that the real world contradicts some of the general features predicted by all string vacua which are not satisfied by the "Swampland" QFTs as explained by Cumrun Vafa; if we lived in the swampland, our world couldn't be described by anything inside the landscape of string theory; the generic predictions of string theory probably include the fact that gravity is the weakest force, moduli spaces have finite volume, and similar predictions that seem to be satisfied so far

  7. By mapping the whole landscape, calculating the accurate predictions of each vacuum for the particle physics (masses, couplings, mixings), and by showing that none of them is compatible with the experimentally measured parameters of particle physics within the known error margins; this route to disprove string theory is hard but possible in principle, too (although the full mathematical machinery to calculate the properties of any vacuum at any accuracy isn't quite available today, even in principle)

  8. By analyzing physics experimentally up to the Planck scale and showing that our world contains neither supersymmetry nor extra dimensions at any scale. If you check that there is no SUSY up to a certain higher scale, you will increase the probability that string theory is not relevant for our Universe but it won't be a full proof

  9. A convincing observation of varying fundamental constants such as the fine-structure constant would disprove string theory unless some other unlikely predictions of some string models that allow such a variability would be observed at the same time

The reason why it's hard if not impossible to disprove string theory in practice is that string theory - as a qualitative framework that must replace quantum field theory if one wants to include both successes of QFT as well as GR - has already been established. There's nothing wrong with it; the fact that a theory is hard to exclude in practice is just another way of saying that it is already shown to be "probably true" according to the observations that have shaped our expectations of future observations. Science requires that hypotheses have to be disprovable in principle, and the list above surely shows that string theory is. The "criticism" is usually directed against string theory but not quantum field theory; but this is a reflection of a deep misunderstanding of what string theory predicts; or a deep misunderstanding of the processes of the scientific method; or both.

In science, one can only exclude a theory that contradicts the observations. However, the landscape of string theory predicts the same set of possible observations at low energies as quantum field theories. At long distances, string theory and QFT as the frameworks are indistinguishable; they just have different methods to parameterize the detailed possibilities. In QFT, one chooses the particle content and determines the continuous values of the couplings and masses; in string theory, one only chooses some discrete information about the topology of the compact manifold and the discrete fluxes and branes. Although the number of discrete possibilities is large, all the continuous numbers follow from these discrete choices, at any accuracy.

So the validity of QFT and string theory is equivalent from the viewpoint of doable experiments at low energies. The difference is that QFT can't include consistent gravity, in a quantum framework, while string theory also automatically predicts a consistent quantum gravity. That's an advantage of string theory, not a disadvantage. There is no known disadvantage of string theory relatively to QFT. For this reason, it is at least as established as QFT. It can't realistically go away.

In particular, it's been showed in the AdS/CFT correspondence that string theory is automatically the full framework describing the dynamics of theories such as gauge theories; it's equivalent to their behavior in the limit when the number of colors is large, and in related limits. This proof can't be "unproved" again: string theory has attached itself to the gauge theories as the more complete description. The latter, older theory - gauge theory - has been experimentally established, so string theory can never be removed from physics anymore. It's a part of physics to stay with us much like QCD or anything else in physics. The question is only what is the right vacuum or background to describe the world around us. Of course, this remains a question with a lot of unknowns. But that doesn't mean that everything, including the need for string theory, remains unknown.

What could happen - although it is extremely, extremely unlikely - is that a consistent, non-stringy competitor to string theory that is also able to predict the same features of the Universe as string theory can emerges in the future. (I am carefully watching all new ideas.) If this competitor began to look even more consistent with the observed details of the Universe, it could supersede or even replace string theory. It seems almost obvious that there exists no "competing" theory because the landscape of possible unifying theories has been pretty much mapped, it is very diverse, and whenever all consistency conditions are carefully imposed, one finds out that he returns back to the full-fledged string/M-theory in one of its diverse descriptions.

Even in the absence of string theory, it could hypothetically happen that new experiments will discover new phenomena that are impossible - at least unnatural - according to string theory. Obviously, people would have to find a proper description of these phenomena. For example, if there were preons inside electrons, they would need some explanation. They seem incompatible with the string model building as we know it today.

But even if such a new surprising observation were made, a significant fraction of the theorists would obviously try to find an explanation within the framework of string theory, and that's obviously the right strategy. Others could try to find an explanation elsewhere. But neverending attempts to "get rid of string theory" are almost as unreasonable as attempts to "get rid of relativity" or "get rid of quantum mechanics" or "get rid of mathematics" within physics. You simply can't do it because those things have already been showed to work at some level. Physics hasn't yet reached the very final end point - the complete understanding of everything - but that doesn't mean that it's plausible that physics may easily return to the pre-string, pre-quantum, pre-relativistic, or pre-mathematical era again. It almost certainly won't.

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In point 1, you write, "Lorentz symmetry works ... with a precision much better than 100%". What did you mean with that? –  rem Jun 11 '13 at 9:45
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The original author of this answer, Lubos Motl, should be credited here. Also, it is chock full of hype and there is nothing particularly mathematical about its content or its intended audience. –  Igor Khavkine Jun 11 '13 at 11:34
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Point 3 makes no sense at all. –  G. Rodrigues Jun 11 '13 at 11:51
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+1, Igor. How much actual sense point 3 makes might even be beside the point, as it may be read merely as more nakedly polemical than some of the others. –  Todd Trimble Jun 11 '13 at 12:33
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Perhaps it should have been clearer that my comment went along with a downvote. Point 3 is a pure equivocation fallacy. The word "consistent" is thrown around quite liberally in theoretical physics and has at best an extremely tenuous connection to "consistency" in mathematical logic. –  Igor Khavkine Jun 11 '13 at 12:48
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From a very lowbrow point of view, the question rests on the new ingredients string theory mandates not already embedded on either the Standard Model or General Relativity, namely supersymmetric partners and extra dimensions. The critique then goes that since we currently don't know the energy scales at which these things turn relevant (i.e. the mass or couplings of the susy fields or the radius of the extra dimensions) some have been tempted to trust string theory and claim that these energies are larger than the currently acessible through experiments. The fact that some people thought these energies should be probed firts in Tevtron and then LHC and now propose are evn higher amounts to the sentiment that if nothing is found one could always suppose it is in even higehr energies. A more detailed answer should adress the landscape issue, viz. Samuel Monnier's answer.

On the issue of experimental confirmation, there are two alternatives: get lucky and see new particles in colliders/cosmic rays and check if they fit in some extension contained in string theory (not easy) or any precise cosmological prediction not explicable in current terms. For instance one could explain the value of the cosmological constant.

Just to make sure, the falsifiability question should not bother too much (especially mathematicians), there is a long literatutre discussing the fact that is a not very well defined condition, or that it does not adress the way science works (one could look up here for starters http://en.wikipedia.org/wiki/Falsifiability).

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comment (as i dont have enough rep): for explanations by/for mathematicians, what about further requesting a mathematical reason whether string theory is falsifiable in your original question? for example, i would find it very interesting to read answers to "what parts of string theory are (currently) not mathematically rigorous?". concrete examples would also be nice here

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Perturbative string theory has a rather solid formalization: first formalize 2d CFT, then formalize the computation of correlators in 2d CFT by integration over moduli spaces, together that fully defines peturbative string theory. Both aspects are rather completely understood abstracty and in principle and to a fair extent worked out. Specifically on the former point we have some contributions in the book collection ncatlab.org/schreiber/show/… . –  Urs Schreiber Jun 10 '13 at 11:56
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Even more is understood of course for the toy version of perturbative string theory given by the "topological string". Here 2d CFT is replaced by "non-compact" TQFT, which for historical reasons is known as "TCFT" ncatlab.org/nlab/show/TCFT As for perturbative string theory, this are solid definitions. –  Urs Schreiber Jun 10 '13 at 11:58
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The remaining problem of course is that the theory given by these solid definitions is immensely rich and so working out of all the consequences and phenomena takes time. Compare for instance vertex operator algebra theory, which is a small subsector of the formulation of full 2d CFT: the definition itself is, while non-trivial, fully precise, but deducing all the resulting phenomena is hard and at any given time will involve, among the actual theorems, open questions, conjectures, guesses and handwaving. This concerns all the sub-phenomena of string theory such as mirror symmetry etc. –  Urs Schreiber Jun 10 '13 at 12:01
    
@urs: i once heard that certain feynman path integrals involved subtracting infinities...has this issue been completely mathematically resolved? if so, a precise reference would be appreciated. thank you also for the comments, but i was looking for mathematical examples, not opinions on the state-of-the-art –  user34821 Jun 10 '13 at 16:07
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I didn't state any opinions (I could if you ask me for it... :-). Concerning finiteness: that's the point of string scattering amplitudes that they are finite, hence "already renormalized", see ncatlab.org/nlab/show/string+scatt for literature. That's why one says that perturbative string theory provides a "UV-completion" for gravity coupled to gauge theory (ncatlab.org/nlab/show/effective+quantum+field+theory). One can think of the higher massive string modes as naturally providing counterterms for the non-finite masslass scattering amplitudes of the effective low energy thory. –  Urs Schreiber Jun 10 '13 at 20:37
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