Such a proof would have to be nonperturbative, because of the likely perturbative renormalizability of maximal supergravity. Further, if you go to a matrix theory limit, you can describe the physics of strings with what are essentially point particles (although their mutual separations are noncommutative). So it is not clear what you would be proving exactly.
I think that it is better to ask for a compelling argument that the physics of gravity requires a string theory completion, rather than a mathematical proof, which would be full of implicit assumptions anyway. The arguments people give in the literature are not the same as the personal reasons that they believe the theory, they are usually just stories made up to sound persuasive to students or to the general public. They fall apart under scrutiny. The real reasons take the form of a conversion story, and are much more subjective, and much less persuasive to everyone except the story teller. Still, I think that a conversion story is the only honest way to explain why you believe something that is not conclusively experimentally established.
Some famous conversion stories are:
- Scherk and Schwarz (1974): They believed that the S-matrix bootstrap was a fundamental law of physics, and were persuaded that the bootstrap had a solution when they constructed proto-superstrings. An S-matrix theory doesn't really leave room for adding new interactions, as became clear in the early seventies with the stringent string consistency conditions, so if it were a fundamental theory of strong interactions only, how would you couple it to electromagnetism or to gravity? The only way is if gravitons and photons show up as certain string modes. Scherk understood how string theory reproduces field theory, so they understood that open strings easily give gauge fields. When they and Yoneya understood that the theory requires a perturbative graviton, they realized that it couldn't possibly be a theory of hadrons, but must include all interactions, and gravitational compactification gives meaning to the extra dimensions. Thankfully they realized this in 1974, just before S-matrix theory was banished from physics.
- Ed Witten(1984): At Princeton in 1984, and everywhere along the East Coast, the Chew bootstrap was as taboo as cold fusion. The bootstrap was tautological new-agey content-free Berkeley physics, and it was justifiably dead. But once Ed Witten understood that string theory cancels gravitational anomalies, this was sufficient to convince him that it was viable. He was aware that supergravity couldn't get chiral matter on a smooth compactification, and had a hard time fitting good grand-unification groups. Anomaly cancellation is a nontrivial constraint, it means that the theory works consistently in gravitational instantons, and it is hard to imagine a reason it should do that unless it is nonperturbatively consistent.
- Everyone else(1985): once they saw Ed Witten was on board, they decided it must be right.
I am exaggerating of course. The discovery of heterotic strings and Calabi Yau compactifications was important in convincing other people that string theory was phenomenologically viable, which was important. In the Soviet Union, I am pretty sure that Knizhnik believed string theory was the theory of everything, for some deep unknown reasons, although his collaborators weren't so sure. Polyakov liked strings because the link between the duality condition and the associativity of the OPE, which he and Kadanoff had shown should be enough to determines critical exponents in phase transitions, but I don't think he ever fully got on board with the "theory of everything" bandwagon. It gets uncomfortable trying to tell history of people who are still alive, so I will stop talking about other people.
My conversion story
It took place in 1999. I know exactly when, because I was watching a terrible movie: "Sleepy Hollow" with Johnny Depp, and I was bored, so I started thinking about physics.
I grew up near Syracuse University. The physics department there was partly put together by Peter Bergmann, one of Einstein's collaborators, and it was well respected for many things. It was where world-sheet supersymmetry was discovered by Ramond, and where the Ashtekar variables were worked out. I talked to Lee Smolin a bunch, and I bought into the idea that gravity could not be described by a perturbation series.
The reason is both more and less deep than is usually presented. The problem with a perturbative description is that it starts with empty space and works order by order. But all the interesting paradoxical things about gravity occur on backgrounds with horizons, which are an infinite number of gravitons away from being flat. It seemed obvious that such a description couldn't be completed nonperturbatively, because it starts out by over-counting the degrees of freedom at small scales, which are constrained by black-hole entropy bounds. (This argument is wrong, but some people still believe it.)
I liked the loop quantization program, because it was more in linw with what quantum gravity should look like, given classical gravity. I was impressed with the fact that geometric operators could have a discrete spectrum, and the theory looked like it gave a proper quantum geometry. I still believe loops are interesting, but the fundamental theory is string theory.
I knew about the bootstrap, and I hated it. The S-matrix is defined by the most annoying of limits. You have to take the incoming wave-packets far apart, and decompose them into plane waves, then you have to take the residual scattering from infinite area plane waves and extract the finite limit by multiplying by appropriate area factors. That's the fundamental quantity? The S-matrix in classical mechanics is computationally intractable, it can essentially solve the halting problem with enough particles, so there is no real way you can expect an S-matrix to have a simple description without a space-time picture of what is going on in intermediate steps. S-matrix theories do not allow you to describe local physics in any way, except by somehow mapping local states to S-matrix states that compose them by collisions in the far past. How could such a picture be complete?
I read through the early parts of Green Schwarz and Witten, and I was annoyed with the string picture. The string was moving in an unquantized space time, and it was clearly derived from Regge theory and bootstrap ideas. I thought that the existence of a spin-2 particle was a reason for rejecting string theory as a theory of gravity--- if it contains gravity it is only because of the accident that spin-2 implies gravity. There's no reason that the resummation of the series will respect black hole decay physics, or resolve the pressing semi-classical gravity paradoxes.
I decided to study string theory (but not to believe it) when I read 'tHooft's "Under the Spell of the Gauge Principle". The last paper describes a Schwartschild black hole in a very hokey model where it oscillates radially. But 'tHooft is convinced due to the information paradox that these oscillations completely describe the space-time around the hole, and the oscillations look exactly like a mode of a string theory (with an imaginary tension, and all sorts of nonsense too).
This was enough to convince me that there was a connection between black hole physics and strings, and that strings should be thought of as small oscillating black holes (this was during the duality revolution, but I wasn't following the string literature then). The picture, complemented by what Susskind was saying at around that time, was compelling--- string theory was a formulation of gravity because the description of any black hole is similar to string theory.
The correct description of black holes in string theory was worked out around this time, but I still was annoyed by the S-matrixy character of the theory. How could it be fundamental if it was just describing asymptotic states? (It is weird to me that I could believe the holographic principle and not the S-matrix principle for so long.) How could it describe black holes when you couldn't even quantize it on a deSitter background, or get it to be properly thermal on an Unruh background?
So I decided to kill string theory. This was in the late 1990s. I thought that I would come up with an elegant principle which could be seen to be true in semiclassical gravity and would rule out string theory completely. I thought about it for a long time without anything. I met a graduate student in California named Simeon Hellerman, and we went to a movie. I talked to him about Einstein, and thought experiments, and the like, and he said "Those things only give junk. Look at Hawking's information loss argument! (We all knew it was wrong for the same reason as 'tHooft and Susskind, even without an AdS/CFT demonstration)" But I said, how about charge quantization? Then I argued that charges must be quantized in a theory of quantum gravity, because you can get black holes to form containing the difference of any two charges.
But Simeon says, "Big deal. We already know charge is quantized for other reasons.“ And we saw sleepy hollow. So then I thought about how you can make a quantitative version of this argument. I thought about teeny tiny charges on black holes, and how they would polarize the surface, and that extremal black holes are just barely repulsive, and I realized that a charged black hole would get constipated if there was no charged particle with charge lighter than its mass. I told Simeon this, and I said, "This is a quantitative principle: every consistent quantum theory of gravity has to have a charged particle lighter than its charge! This will kill string theory, because there is no reason strings should obey this. It's a completely non-perturbative constraint."
I was very happy that I was going to kill string theory. Simeon said "I will find a string vacuum that violates this, and I will kill your principle." I thought it was funny, because I thought he would be killing string theory.
So he went home, and thought about it, and the next day he says "You can't violate it, because ...insert complicated stuff here... and ....insert complicated stuff here ... I now realize that our principle is true!" I was annoyed that his complicated stuff worked, because I wanted to kill string theory. It was only a year or two later that I figured out what complicated stuff he said (he was talking about T-duals and S-duals).
But in understanding why string theory doesn't violate this bound, I had an epiphany. The string is a black hole, so it is impossible to prove that it fails by black hole formation and evaporation. The laws of the world-sheet emission are just already-quantum laws of extremal Hawking radiation! Then all the dualities of string theory are required by holography, and the S-matrix theory is just the flat-space limit of all this.
This was enough to convince me that string theory was a consistent theory of quantum gravity. It is impossible that it obeys the mass/charge inequality if it is not. Once I understood this, I was depressed, because I felt that all of physics had just been essentially solved by Banks Fischler Susskind and Shenker, and Maldacena.
The holographic principle means that the same theory has to be mathematically expressible in ten thousand different ways, each way corresponding to some extremal black hole. This property is so constricting, that it is impossible to satisfy, except for the fact that string theory satisfies it. There is no chance that there is any other consistent theory of quantum gravity, it is just too implausible that you can satisfy the constraint that the same theory is dual to a bazillion other lower dimensional formulations.
So I became a string theory believer, for the simple reason that the world is holographic, and string theory is obviously the only consistent way to respect holography. I think my story is typical of my generation. This is the real reason why people are so adamant that it must be right, without experimental evidence (although that would be nice）。