Lubos Motl, in the context of a larger string theory discussion, recalls some useful "no go" theorems of String Theory in fairly accessible language.
1. There are no fields with spins of greater than 2 (the conventional expectation for gravity) that have masses of less than the string scale (i.e. huge), basically ruling out their existence in any low energy effective theory that could be tested experimentally.
2. Spin 2 fields (of masses of less than the string scale) must be "modes of the gravitational field" and the corresponding particles have to be
"gravitons of a sort". They don't have to be exactly like Einstein's theory of relativity on string theory grounds alone, but they have to be quite similar to it. Some string theory variants permit multiple kinds of gravitons.
3. Spin 3/2 fields (of masses of less than the string scale) "have to be accompanied by the local supersymmetry – they have to be
gravitino fields." So, if there are spin-3/2 fields in a given string theory, they have to act like the spin-3/2 fields of SUGRA (supergravity) theories.
He doesn't address at length, but implicitly assumes familiarity with, the principle that fundamental field spins have to come in discrete half-integer increments, meaning that all other fields of string theory have to be spin-1 (associated with photons, gluons, W bosons and Z bosons in the Standard Model), spin-1/2 (associated with Standard Model fermions) or spin-0 (associated with the Standard Model Higgs boson).
Thus, any newly discovered particles predicted by String Theory have to have some fundamental properties similar in many ways:
(1) to Standard Model fermions or bosons, or
(2) to the hypothetical spin-2 graviton, or
(3) to the hypothetical spin-3/2 supersymmetric gravitino. The superpartners of Standard Model fermions in Supersymmetry theories (i.e. squarks and sleptons) are spin-0 bosons.
While there are no fundamental particles with spin-3/2, there are exotic hadrons (three quark composite particles) with spin-3/2 that have been observed. so we can be fairly confident that we know how particles of this spin would behave if they existed.
The particle content of supersymmetry theories
The superpartners of both Standard Model spin-0 bosons (Higgsinos) and Standard Model spin-1 bosons (Winos, Binos and Gluinos) in Supersymmetry theories are spin-1/2 fermions. Linear combinations of these particles produce "Charginos" and "Neutralinos". Supersymmetric theories also include more than one Higgs boson (the simplest have five - two neutral parity even Higgs bosons, one neutral parity odd Higgs boson and two charged Higgs bosons that are each other's antiparticles). Non-minimal supersymmetric theories generally have more kinds of Higgs bosons in addition to other complexities.
In supersymmetric theories, the electromagnetic charge, weak force interactions, and color charge interactions of these particles are highly constrained, but the masses of supersymmetric particles and some of their other properties are experimentally fitted, rather than theoretically predicted.
Existing high energy physics experiements imply that Supersymmetric Higgs bosons must either be quite heavy, or must have interactions so subtle that previous searchers for them in lighter mass ranges (typically below the mass of the Higgs boson discovered already) would have been missed because something in the theory suppresses their interactions or creation. They also imply that other superpartners must be quite heavy. And, experimentally valid Supersymmetry theories with many very heavy superpartners also need a mechanism to suppress naively expected high rates of neutrinoless double beta decay which are not observed.
Discoveries that would falsify the Standard Model, SUSY and String Theory
String theories that have low energy effective field limits that don't look like either the Standard Model, or Supersymmetry (with Supergravity), in the low energy effective field limit are excluded.
The discovery of any spin-0 fundamental particles other than the Standard Model Higgs boson falsified the Standard Model. The discovery of any new spin-1/2 fermions falsifies the Standard Model. The discovery of any new fundamental spin-1 bosons falsifies the Standard Model. the discovery of any spin-3/2 or spin 5/2+ particles falsifies the Standard Model. Discovery of a spin-2 graviton would not falsify the Standard Model as this would be outside of the scope of the phenomena that it describes.
In Supersymmetric theories, searches for fundamental particles with properties different from these are ill motivated theoretically and would falsify supersymmetry, supergravity and string theory.
Experimental constraints on Supergravity and String Theory from gravitino properties
Astronomy observations strongly constrain many variants on the properties of a gravitino (and hence, string theory parameter space).
If the gravitino is too heavy and nearly stable, it gives rise to too much dark matter and doesn't solve the "hierarchy problem" that supersymmetry was devised to address. If it isn't stable enough, it produces a universe with no stars in it. In split supersymmetry models, a heavier gravitino is allowed by we need to be observing other superpartners at the LHC very soon. Limitations associated with the properties of the gravitino are one of the main reasons that R-parity conserving theories of supersymmetry are now strongly disfavored experimentally. In other words, it is experimentally necessary if supersymmetry exists at all, that almost all supersymmetric particles, like all second and third generation fermions in the Standard Model, are inherently unstable particles that with the exception of one or two dark matter candidates, do not exist in nature in the modern universe.
In the most plausible possibility for gravitino properties consistent with experimental evidence, the gravitino is the lighest supersymmetric particle (LSP) and the source of most dark matter: "R-parity is slightly violated and the gravitino is the lightest supersymmetric particle. This causes almost all supersymmetric particles in the early Universe to decay into Standard Model particles via R-parity violating interactions well before the synthesis of primordial nuclei; a small fraction however decay into gravitinos, whose half-life is orders of magnitude greater than the age of the Universe due to the suppression of the decay rate by the Planck scale and the small R-parity violating couplings."
Increasingly tight experimental constraints on dark matter properties, however, greatly narrow the available parameter space for gravitino dark matter. The gravitino is a much smaller target to search for that it would have been not so many years ago.
Yet, boundaries on the properties of the gravitino, since it is common to all supergravity theories by definition, impose limits all of extended supersymmetry theories whatever their particle physics content and impose limits on all string theories. So, these generic constraints of theory-space from the experimental limits on the properties of any particle with a gravitino's intrinsic spin are quite powerful.
If the gravitino parameter space is overconstrained, meaning that its existence with all of its theory dependent properties that make it such a small target are experimentally ruled out, than so are all supergravity theories.