The only finite mathematical framework that incorporates both the standard model of particle physics and gravity under one umbrella that I am aware of is string theory. I would like to know whether there are any other mathematical possibilities exist which do not depend on supersymmetry and still consistent with the standard model and gravity and produce finite answers. In a nutshell my question is: can there be any alternative to string theory? (Remember, I am not talking about only gravity. I am talking about gravity as well as other phenomena).
This is the proverbial sixty-four thousand dollar question for fundamental physics. It may be helpful to split it down into steps.
- What are the possible consistent theories of quantum gravity?
- Which of these can (or must) be extended to include matter and guage fields?
- Which of these can be made to include the standard model as a low energy limit?
Once we have answered these questions the theoretical program to understand the foundations of physics is essentially complete and the rest is stamp collecting and experiment. That is not going to happen today but let's see where we are.
String Theory works very well as a perturbative theory of gravitons that appears to be finite at all orders, but there is no full proof that it is a complete theory of quantum gravity. It requires matter and gauge fields with supersymmetry to avoid anomalies. The size of gauge groups suggests that it could potentially include the standard model. It is too strong a claim to say that it does incorporate the standard model. A popular view is that it has a vast landscape of solutions which is sufficiently diverse to suggest that the standard model is covered, but crucial elements such as supersymmetry breaking and the cosmological constant problem are not yet resolved.
Supergravity theories are potentially alternative non-string theories that could provide a perturbative theory of quantum gravity. Indications are that they are finite up to about seven loops due to hidden E7 symmetry but they are likely to have problems at higher loops unless there are further hidden symmetries. These theories have multiplets of gauge groups and matter. The 4D theories do not have sufficiently large gauge groups for the standard model but compactified higher dimensional supergravity does. A more subtle problem is to include the right chiral structure and this may be possible only with the methods of M-theory.
It has long been the conventional wisdom that supergravity theories can only be made complete by adding strings. Recent work using twistor methods on 4D supergravity seems to support this idea (e.g. Skinner etc.)
Loop Quantum Gravity is an attempt to quantise gravity using the canonical formualism and it leads to a description of quantum gravity in terms of loops and spin network states which evolve in time. Although this is regarded as an alternative to string theory and supergravity it does not give a picture of a purtabative limit which would make it possible to compare with these approaches. It is possible that ST/SUGRA and LQG are looking at similar things from a different angle. In fact the recent progress on N=8 supergravity as a twistor string theory has some features that are similar to LQG. Both involve 2D worldsheet objects and network like objects.
The main distinctions are that LQG does not have supersymmetry and N=8 SUGRA does not use knots. Even then there has been some progress on a supersymmetric version of LQG and the Yangian symmetries used in N=8 SUGRA should be amenable to a q-deformation that brings in knots. It remains to be seen if these theories can be unified.
It is worth saying that all these approaches involve trying to quantise gravity in different ways. Although quantisation is not a completely unique procedure it is normal to expect that different ways of quantising the same thing should lead to related results, If something like supersymmetry or strings or knots are needed to get consistency in one approach the chances are that they will be needed in another.
I have not mentioned other approaches to quantum gravity such as spin foams, group field theory, random graphs, causal sets, shape dynamics, non-commutative geometry, ultra-violet fixed points etc. Some of these are related to the other main approaches but are less well developed. It should also be mentioned that there are always attempts to unify gravity and the standard model classically e.g. Garrett's E8 TOE, Weinstein's Geometric Unity etc. These may tell us something interesting or not, but it is only when you try to quantise gravity that strong constraints apply so there is no reason to think they should be related to the attempts to quantise gravity.
So in conclusion all approaches that have made any kind if real progress with quantising gravity look like they may be related. Much more has been revealed so far from this need to quantise consistently than from directly trying to unify gravity with the standard model. This may not be so surprising when you consider the enormous difference in energy scales between the two.
So far, the answer seems to be no, but there is no mathematical proof. The main reason to believe that string theory is essentially unique is that it incorporates the holographic principle, the idea that the spacetime near and inside a black hole is emergent from the degrees of freedom of the black hole, and this idea is so difficult to imagine working, that it is hard to see some other solution.
Within string theory, the standard model emerges from either some matter, or from the Horava-Witten orbifold which produces an E8 gauge group in a circular compactification of M-theory. The E8 gauge group can naturally break to E6 and contains the standard model in a way as natural as SO(10) or U(5) (it is just a supergroup). So there is no difficulty embedding the standard model, but it is not predicted, just happens to work.
In other approaches, not only does the gravity not work well, the other stuff is not so natural as it is in string theory, where the total amount of stuff, like fields, gauge-groups, is constrained to be (of the right order but a few times bigger than) what we see.
The Loop Quantum Gravity folks have not been able to get elementary particle theory into their picture very well. They have been working with braids which have twists, which start to sound a bit like string to me. There seems to be a trend where all roads lead to string theory.