String theory is a perturbation theory of quantum gravity starting with perfectly linear Regge trajectories self-interacting in a consistent bootstrap. Bootstrap means that the interaction of the trajectories is only by exchange of other trajectories, so that the system is self-consistent, or, in 1960s terminology, that it pulls itself up by its own bootstraps.
The best way to learn what string theory is, is to get a copy of Gribov's "The Theory of Complex Angular Momentum", and learn the basic principles of Regge theory. You don't have to learn the Reggeon calculus covered later (although it is interesting), just the basic principles. The point of this theory is to understand spectral properties --- S-matrix states, not detailed microscopic field theory, which breaks down at the Planck scale. The S-matrix is valid at any scale, it is the fundamental observable object in relativistic quantum mechanics, when you don't have point probes.
In QCD, you can make little black holes and use them as point probes. You can even use electrons as point probes, without going to the trouble of making a black hole. This shows that the scale of QCD is not the appropriate scale for string theory, but nevertheless, string theory was discovered by trying to find a consistent bootstrap at this scale. This is very fortunate historically, and required vision and persistance.
Linear Regge trajectories can be understood as string-like excitations. They are the quantized states of an extended relativistic string described by a Nambu Goto action. But the string action by itself doesn't tell you anything about interactions. The interactions of these objects is only by exchanging states found in their own spectrum, with the condition that S-channel exchange is dual to T-channel exchange. They have no other interactions. This defines a dual string theory, the kind people study.
States and observables
String theory is a quantum mechanical S-matrix theory, so the state variables (in an asymptotically flat space) are the following:
- A classical configuration of fields at infinity, which defines the background.
- A finite number of incoming particles in a quantum superpostion of plane waves, which define the in-state at minus infinity.
The dynamical law is to produce the out-state, given these ingredients. There is only one observable,
- The S-matrix between in and out states.
That's it. Every other observable has to be extracted from this one, by some trick. This is extraordinarily difficult, but in practice, there are some simplifications.
- The classical field configuration is a background which must be consistent with the string theory itself. This gives classical equations which the background field must satisfy. These come from the condition that the string is still conformally invariant in the background, so that the $\beta$ function is zero. These equations define the classical allowed backgrounds, and from this you can extract the classical dynamics at infinity, which is what you use almost all the time, or the approximate quantum field theory description, which is what you use almost all the rest of the time.
- You can take a field theory limit, and start telling quantum field stories of particle propagation. Almost all the work in the 1980s was based on low-energy supergravity approximations, sometimes with higher order effective action corrections. Such a description can be thought of as an approximation scheme to string theory, by adding higher order string corrections to a quantum field theory, to get a string-corrected effective action. This way of dealing with strings is philosophically least challenging, but it doesn't seem to me to be a very convergent process. The string theory is not a quantum field theory, after all.
More dynamical formulations
There are more dynamical formulations than the S-matrix theory, and more honest formulations than the "effective-action" string-corrected quantum field theory. These honest formulations are due to Mandelstam, Kaku-Kikkawa, Banks-Fischler-Susskind-Shenker, and Maldacena-Witten-Gubser-Klebanov-Polyakov.
- When you absolutely need to use string theory itself, instead of quantum field theory with effective action corrections, you can move to a dynamical picture where the string tells a story which is local in a version of spacetime. In such a picture, the string theory can be thought of as a normal quantum theory, not an S-matrix theory. The states are defined by superpositions of configurations, jsut like any other quantum theory. The Mandelstam description of strings is one such picture, and because it exists, one could go to a second quantized string field theory, by defining creation and annihilation operators for the string states. So string field theory de-S-matrixes the S-matrix theory. But it is defined on a light cone, and it is technically complicated. But in such a picture, the basic states of string theory are quantum superpositions of light-cone configurations of strings.
- In 11-dimensional matrix theory, you have a point-particle description in which you can define the state space and evolution again like any other quantum field theory--- as superpositions of noncommutative matrix model configurations, with a normal quantum dynamics in 0+1 dimensions. This is the easiest state space to formulate.
- In asymptotically AdS backgrounds, instead of incoming/outgoing particles and an S-matrix, you have a full honest to goodness quantum field theory's worth of information at the boundary of space-time. This quantum field theory maps in a not-completely-understood way to the interior description, but the state-space and dynamics are obvious--- they are just like any other quantum field theory.
Notice that the descriptions of the state space is entirely different in the different formulations! This is important, both because their mutual self consistency is an insanely stringent consistency constraint, which has zero chance of being satisfied unless there is a consistent gravitational theory behind it, and also it gives completely different types of observables in different asymptotic backgrounds.
There is no unified way of defining the state space on all backgrounds at once. Each type of background has its own formulation. It requires physical intuition to move between the pictures, and there is no way to communicate the results to a mathematician without communicating the physical picture, because the theory isn't 100% complete.
Literature and Misconceptions
The best review for me was Mandelstam review from 1974. It is very important to learn these old-fasioned ideas, because otherwise you will have all sorts of nonsense in your head about what you can do to strings.
- Dual strings do not describe statistical polymer properties. It doesn't work, they aren't those types of strings. Polymers interact by self-intersection, strings don't.
- Dual strings don't work to describe vortex lines in a quantum field theory, although this idea was one of the ways in which they were discovered, by Nielsson. The vortex line picture was simultaneous with the flux-line picture, but the flux line picture is now known to be correct, while the vortex line picture, as far as I know, has no precise version beyond that the string is extended in 1d, like a vortex line. If you make an effective theory of vortex lines in a scalar/gauge field theory, they will interact in crazy non-string ways. In gauge theories with a gravitational dual, like N=4 gauge theory, you probably can make the string be a vortex line, so Neilsson's idea is not altogether wrong (I think), but then some duality will have to link up the vortex and flux line.
- Dual strings have no deformations: you can't make dual strings interact at collision points, you can't make them attach or detach other non-string objects. They are either a theory of everything or a theory of nothing.
- Dual strings only allow S-matrix probing. You can't calculate off-shell behavior, meaning you can't describe their detailed dynamics in space-time. You can formulate their world-sheet behavior in space and time. There is string field theory, which attempts to take strings off shell, but now we know the right way to do this is AdS/CFT, although string field theory is still very important.
- The only way to touch strings to classical objects is to fiddle with asymptotic values of fields. You can't probe them using local quantum fields, because they generate their own local fields. You can make them move in a classical background, but their dynamics determines the quantum properties of all the backgrounds.
- You can also make them interact with branes, but this is a surprise, and it only works because the branes are weak dual black holes, where strings can partially fall through. It doesn't work for arbitrary surfaces, and there are strong constraints on which branes are allowed.
If you learn the old-fasioned string theory of the 1960s and 1970s, you can understand the rest of the stuff. If you don't, you can't.
Saying that $E_6$ is "favored" over $E_8$ in GUT model building is a big understatement.
There can't be any grand unified theory with an $E_8$ gauge group because $E_8$ has no complex representations, i.e. representations that are inequivalent from their complex conjugates. The existence of complex representations is a necessary condition for the theory to contain chiral fermions, i.e. Dirac fermions whose left-handed components carry different quantum numbers (and interactions) than the right-handed components. One may also say that complex representations are needed for the violation of C, P, and CP.
$E_6$ is the only one among five exceptional groups that has any complex representations. It's related to the fundamental ${\bf 27}$ or antifundamental $\overline{{\bf 27}}$ representation of the group which are interchanged by an outer automorphism of $E_6$, a symmetry that boils down to the ${\mathbb Z}_2$ symmetry of its Dynkin diagram. $E_6$ is the only exceptional Lie group with a nontrivial symmetry of the Dynkin diagram.
All other exceptional Lie groups, namely $G_2, F_4, E_7, E_8$, only have real representations, a fact that can be seen by looking at their real fundamental representations, too. The spectrum of gauge theories using these groups would be inevitably left-right symmetric, and therefore experimentally excluded. Patterns about particles such as "neutrinos have to be left-handed" would be impossible.
Despite the comments above, $E_6$ is a subgroup of $E_8$. So in string theory, it is possible to break $E_8$ (a key group e.g. in heterotic string theory) to $E_6$ by stringy effects, e.g. by nontrivial configurations of the $E_8$ gauge field as a function of the extra (compact) dimensions in $E_8\times E_8$ heterotic string theory. Spontaneous breaking of $E_8$ by field-theoretical methods (Higgs mechanism) is no good because it would only produce real representations of $E_6$ again. In string theory, $E_6$, a viable GUT group, may emerge as a subgroup of $E_8$ (or $E_7$). $G_2$ and $F_4$ are too small to be relevant for GUT model building.
All papers that claim to build viable models of particle physics from an $E_8$ field theory are pseudoscientific gibberish, denying elementary features of the groups in which the known quantum fields transform. In the case of Garrett Lisi's paper, the absence of complex representations is the main point of the paper by Garibaldi and Distler.
Best Answer
One can disprove string theory by many observations that will almost certain not occur, for example:
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
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
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
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
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
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
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)
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
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.