The strings do not attach to the space-time manifold, they move around on it as a background. Option 1 is not right.
Option 2 is more like it, except that you are assuming that string theory as it is formulated in the string way builds up space-time from something more fundamental. This is not exactly true in the Polyakov formulation or in any of the string formulations (even string field theory). The string theory doesn't tell you how to build space-time from scratch, it is only designed to complete the positivist program of physics. It answers the question "if I throw a finite number of objects together at any given energy and momentum, what comes out?" This doesn't include every question of physics, since we can ask what happens to the universe as a whole, or ask what happens in when there are infinitely many particles around constantly scattering, but it's close enough for practical purposes, in that the answer to this question informs you of the right way to make a theory of everything too, but it requires further insight. The 1980s string theory formulations are essnetially incomplete in a greater way than the more modern formulations.
The only thing 1980s string theory really answers (within the domain of validity of perturbation theory, which unfortunately doesn't include strong gravity, like neutral black hole formation and evaporation) is what happens in a spacetime that is already asymptotically given to you, when you add a few perturbing strings coming in from infinity. It then tells you how these extra strings scatter, that is what comes out. The result is by doing the string perturbation theory on the background, and it is completely specified within string perturbation theory by the theory itself.
Option 3 is sort of the right qualitative picture, but I imagine you mean it as strings interacting with a quantum gravity field which is different from the strings, strings that deform space and then move in the deformed space. This is not correct, because the deformation is part of the string theory itself, the string excitations themselves include deformations of space-time.
This is the main point: if you start with the Polyakov action on a given background
$$ S = \int g_{\mu\nu} \partial_\alpha X^\mu \partial_\beta X^\nu h^{\alpha\beta} \sqrt{h} $$
Then you change the background infinitesimally, $g\rightarrow g+\delta g$, this has the effect of adding an infinitesimal perturbation to the action:
$$ \delta S = \int \delta g \partial X \partial X $$
with the obvious contractions. When you expand this out to lowest order, you see that the change in background is given by a superposition of insertion of vertex operators on the worldsheet at different propagation positions, and these insertions in the path-integral have the form
$$ \partial X^\nu \partial X^\mu$$
These vertex operators are space-time symmetric tensors, and these are the ones that create an on-shell graviton (when you smear them properly to put them on shell). So the changing background can be achieved in two identical ways in string theory:
- You can change the background metric explicitly
- You can keep the original background, and add a coherent superposition of gravitons as incoming states to the scattering which reproduce the infinitesimal change in background.
The fact that any operator deforming the world-sheet shows up as an on-shell particle in the theory, this is the operator state correspondence in string theory, tells you that every deformation of the background that can be long-range and slow deformation shows up as an allowed massless on-shell particle, which can coherently superpose to make this slow background change. Further, if you just do an infinitesimal coordinate transformation, the abstract path-integral for the string is unchanged, so these graviton vertex operators have to have the property that coordinate gravitons don't scatter, they don't exist as on-shell particles.
The reason this isn't quite "bulding space-time out of strings" is because the analysis is for infinitesimal deformations, it tells you how a change in background shows up perturbatively in terms of extra gravitons on that background. It doesn't tell you how the finite metric in space-time was built up out of a coherent condensation of strings. The question itself makes no sense within this formulation, because it is not fully self-consistent, it's only an S-matrix perturbative expansion. This is why the insights of the 1990s were so important.
But this is the way string theory includes the coordinate invariance of General Relativity. It is covered in detail in chapter 2 of Green Schwarz and Witten. The Ward identity was discovered by Yoneya, followed closely by Scherk and Schwarz.
The point is that the graviton is a string mode, a perturbation of the background is equivalent to a coherent superposition of gravitons, and graviton exchange in the theory includes the gravitational force you expect without adding anything by hand (you can't--- the theory doesn't admit any external deformations, since the world-sheet operator algebra determines the spectrum of the theory).
In the new formulations, AdS/CFT and matrix theory and related ideas, you can build up string theory spacetimes from various limits in such a way that you don't depend on perturbation theory, rather you depend on the asymptotic background being fixed during the process (so if it starts out flat, it stays mostly flat, if it starts out AdS, it stays AdS). This allows you to get a complete answer to the question of scattering on certain fixed backgrounds, and get different pictures of the same string-theory spectrum in terms of superficially completely unrelated gauge fields or matrix-models.
But you asked in the Polyakov string picture, and this is only consistent for small deformations away from a fixed background that satisfies the string equations of motion for the classical background.
This question has many aspects.
First, there is the fact that theories without closed strings are inconsistent. If we allow an open string to split into two open strings (line intervals) and vice versa (Hermiticity), the same interaction may also merge the end points of an open string and produce a closed string.
Alternatively, one-loop diagrams in a theory of open strings may also be read in the "intermediate closed string channel", thus proving that there are intermediate closed strings and there must be physical closed strings, too.
In type II-based string theories, it is also possible to produce D-branes out of closed strings only. D-branes automatically allow strings to terminate on them, so one produces the potential for open strings or the open strings themselves, too.
The reason is that a D-brane is physically the same object as a black $p$-brane which is a higher-dimensional generalization of a black hole. Just like one can prove that under certain circumstances, black holes will be formed (the singularity theorems by Penrose and Hawking), there will also be black $p$-branes formed out of closed string fields (the closed string Ramond-Ramond electromagnetic-like fields are around the charged black $p$-branes found in type II string theories).
It is possible to show that the minimum-energy configurations with the same charge must look like D-branes at weak coupling, and their excitations are therefore described by open strings attached to these D-branes.
Now, the D-branes localized in some circular dimensions are T-dual to D-branes that are wrapped around these directions. In this T-duality map (equivalence), the location of the localized D-brane is mapped to the value of the Wilson line of the wrapped brane. Choosing one of the positions of the D-branes from the list of a priori allowed many positions breaks the translational invariance, obviously. This is equivalent to a nonzero vacuum expectation of the Wilson loop around the circle.
Shiraz in particular has repeatedly studied a similar phenomenon in AdS/CFT in which this extra circle is the Euclideanized time in thermal calculations. The Wilson loops around this thermal circle are essentially called the Polyakov loops, at least in some non-Abelian generalizations, and these Polyakov loops can get a vev. This is a description of the deconfinement transition in the gauge theory.
It seems to me that Shiraz has been talking about many independent topics and each of them is discussed in different references and many of them are rather standard parts of the string theory basic education and textbooks (whose rest is sort of needed to understand these particular aspects, too, so it doesn't make much sense to try to isolate these aspects from the rest).
Best Answer
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:
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)。