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Why unification sits at the core of string theory

Previous articles about a similar topic:

Unity of strings I
Unity of strings II
Is AdS/QCD and AdS/CMT relevant for unification?
Two roads from N=8 SUGRA to strings
Most people, including those who have been interested in theoretical physics for years, still seem to be incapable to understand the inevitability of string theory and the unity and inseparability of its mathematical tools and the landscape of its solutions.

They would still like to cherry-pick pieces of string theory, while attempting to hide or deny the rest, and especially string theory as the/an ultimate theory that unifies the known forces.

But such cherry-picking is not possible. To say the least, it is mathematically inconsistent. Once we accept that a particular "corner" of the string-theoretical landscape or its toolkit is relevant for the explanation of the physical Universe, the rest of it inevitably follows.

If someone likes addition of integers and the prime integers 2,3,5,7, she inevitably has to add the remaining integers and accept that there are infinitely many additional primes such as 11,13,17,19 etc. The case of string theory is analogous.

This is a statement with many aspects and dimensions. In this text, my aim is to provide the dear reader with a couple of examples demonstrating that
  1. any approach to a physical problem that consistently uses string theory in any form inevitably links the spectrum of forces and particles species (or types of other objects) in such a way that it is impossible to remove them one by one; it means that all of them have to co-exist in a constrained way for the total structure to make sense: a weak form of unification
  2. any approach to any quantum theory based on string theory always implies that the "local" number of spacetime dimensions must be 10/11 for string/M-theory; they can either be counted directly, or it can be shown that there exists a quasi-geometric structure generating some degrees of freedom whose appropriately counted "effective dimensionality" agrees with the total dimension's being 10/11.
To summarize these two points in different words, I want to make it absolutely obvious that those who want to cherry-pick string theory, use some parts of it, but deny that it must be viewed as a unifying theory of all forces and mater, are incompetent kibitzers or liars.

Perturbative string theory

If one wants to cure the ultraviolet problems of quantized general relativity by introducing extended objects, he will find out that the internal "world volume" theory inside these extended objects is a gravitational theory, too: it must have a reparameterization symmetry. The only perturbative way how to "marginally" get rid of the infinities both in spacetime and in the world volume is to work with one-dimensional strings and two-dimensional world volumes i.e. world sheets.

Classically, i.e. without quantum mechanics, such a theory of strings propagating in spacetime will work in any spacetime dimension. However, when quantum mechanics is taken into account, the theory is only consistent in the critical dimension of the spacetime. For superstring theory, it's D=10. For pedagogical reasons, it's easier to derive such things in the simplified (and not quite stable) bosonic string theory where it is D=26. There are many ways to see that D=26 is special.

Joe Polchinski's Big Book of String is deriving the critical dimension of the bosonic string, D=26, in seven different ways if you count both the text and the homework exercises. This result is genuinely important. In some sense, it's the most basic, most important, and most universal result about string theory once you add quantum mechanics.

Because the particles - and the corresponding fields - in spacetime are obtained as energy eigenmodes of vibrating strings, it's not hard to see that gravity is a derived concept. The graviton multiplet is just one collection of mass eigenstates of a vibrating string. There are others that inevitably describe other fields - and their quanta, i.e. particles.

It should be obvious - and kind of tautological - that when you work with string theory in its oldest, perturbative formulation, you will be forced to accept all the consistency conditions and re-derive the same usual set of possible consistent classical solutions or "vacua". The critical dimension matters everywhere in perturbative string theory and all the possible fields and particles are "correlated with one another" because all of them arise from the quantization of one string. That's why I am going to look at less trivial things.

But for a while, let's continue with perturbative string theory.

Gepner models

The simplest "versions" of perturbative string theory - which we know should really be called "classical solutions" or "vacua" of one and unique underlying string theory - include free fields on the world sheet. As a result, the quantization is equivalent to the quantization of an infinite-dimensional harmonic oscillator with various frequencies. The number of bosons or fermions living on the world sheet has to be linked to the critical dimension, D=10. That looks too simple.

What if you try something completely different. Instead of these simple free CFTs, you may want to put non-trivial conformal field theories on the two-dimensional world sheet, such as the Ising model and its generalizations ("minimal models"). These models have central charges - a kind of generalization of the "spacetime dimension" - that can be fractional. They're not free. Their primary operators are pretty complicated. It's much harder to find the dimension of these operators than in the case of the free fields.

Imagine that you combine these minimal models in all conceivable ways that would lead to 4 large spacetime dimensions and spacetime supersymmetry. You might think that you will discover completely new types of string theory that have nothing to do with D=10. After all, you are combining some crazy fractional dimensions that are not flat at all.

However, you will be shocked when you're proved wrong. Even if you combine such crazy minimal models - seemingly equivalent to fractional numbers of bosons (i.e. spacetime dimensions) - you will find out that the resulting "string theories" are completely equivalent to string theories in 10 dimensions with 3+1 large dimensions and 6 additional dimensions compactified on a tiny Calabi-Yau manifold whose size is comparable to the string scale - the typical distance scale associated with the stringy vibrations.

You try to look at all conceivable ways how to combine the minimal models into a D=4 supersymmetric theory in spacetime and you will see that every single one of them is secretly a 10-dimensional theory. Even though the D=10 unifying stringy framework of the usual type may fail to be manifest when you look at your "new" theory for the first time, when you work for a while, and you have the mathematical capacity to crack such problems, you will be forced to see that there is nothing "new" about your theory at all. You're just expanding the same theory around a particular highly curved configuration of the extra dimensions.

Non-critical string theory may achieve consistency for the price of making the dilaton's gradient huge (stringy) but it's still true that one can calculate the central charge, and it works - it replaces the counting of the dimensions - and moreover, there are doubts whether the non-critical string theories can be fully consistent at the non-perturbative level.

Let's look at even more non-trivial examples of Polchinski's theorem that all roads lead to string theory.

AdS/CFT: seeing D=10

Many of the cherry-pickers often say that the AdS/CFT correspondence is OK but it has nothing to do with D=10, with string theory as a unifying theory of all interactions. The purpose of this section is to demonstrate that these cherry-pickers are fundamentally wrong. Using a more rigorous vocabulary, they are hopeless imbeciles.

Start with the most well-known example of the AdS/CFT holographic correspondence: the N=4 gauge theory. If you really want a conformal field theory, it must be scale-invariant and the beta-function has to be zero. For a generic theory, it's not zero but you may expect that a lot of supersymmetry helps you with the cancellations so that the total beta-function will vanish and the theory will obey the conformal symmetry.

Indeed, that's the case if you require 16 supercharge. Such a theory inevitably contains gauge fields - because the spin differences between the most right-handed and most left-handed components must be at least two. So the spin may go from -1 to +1 and such a multiplet contains a gauge field (2 physical components), Dirac fermions (8 polarizations), and 6 scalars (because 2+6 = 8 must confirm that the numbers of bosonic and fermionic polarizations coincide).

Such a gauge theory is fully determined by the gauge group. The simplest family of compact Lie groups are SU(N). If N is large, a better holographic picture emerges. Aside from the 3+1 dimensions you started with, you find one new emergent holographic or radial direction - encoded in the length scale of the 3+1-dimensional gauge theory. In total, you have AdS5.

However, the 6 bosons above carry the SO(6) rotational symmetry. It's the isometry of a five-dimensional sphere, i.e. the surface of a six-dimensional ball. And if you look at the geometry in which the emergent gravitons can propagate, you will see that it is "AdS5 x S5". It's a D=10 space, just like you expect in each string theory.

It's not just a random D=10 theory; in other words, it's not just the dimensionality that is stringy. If you analyze the interactions of the gravitons - dual to the stress energy tensor and similar operators in the gauge theory - you will find out that they obey the D=10 type IIB supergravity which is the low-energy limit of type IIB string theory.

Needless to say, if you look more accurately, you will be able to extract the whole type IIB string theory, including all kinds of wrapped branes, excited massive string modes (explicit in the BMN or Penrose's pp-wave limit), black holes with the right thermodynamic and other properties, and all other objects and interactions that you would derive in type IIB string theory in flat space, too. It's the same thing. The N=4 gauge theory with the SU(N) group - a different N - is fully equivalent to string theory. All of the string theory is there. There's no room for you to cherry-pick things. If you accept a supersymmetric gauge theory as a part of physics, you must accept type IIB string theory on a pretty ordinary space - that becomes nearly flat for many colors - too.

You might think that this has been a coincidence because we worked with a highly supersymmetric theory. What about a theory with fewer supercharges? You must still make it conformal, so that the emergent AdS space has the right isometry. Some supersymmetry is helpful. But let's take N=1 only (four supercharges). And the gauge group is SU(N+M) x SU(N). You may add some bi-fundamental fields under these non-simple gauge group.

The coupling constants of the two groups are running, and so on. Surely, such a random gauge theory which is different from the N=4 gauge theory will have nothing to do with string theory, you could think.

However, what you have just proposed is the CFT side of the Klebanov-Strassler theory. If you study how the large M,N limit behaves, you will see that it describes M fractional 3-branes on the tip of a conifold - a singular type of a Calabi-Yau manifold with a point whose vicinity looks like a generalized cone. Again, you will see that the total D=10 and there are all the objects that you could derive in string theory, using its usual, non-gauge-theoretical, spacetime tools to analyze.

In some sense, this theory is even more stringy because of the appearance of the singular Calabi-Yau manifolds.

There are many other examples of field theories and they are always dual to string theory, with all of its special technical features that you derive if you analyze string theory carefully. When supersymmetry is completely broken, it becomes really hard to analyze what's happening for a large number of colors. But all the circumstantial evidence suggests that all consistent and stable backgrounds you can derive in this way may also be described by a string theory with its usual consistency rules - such as the derivations of the critical dimensions.

After all, that shouldn't be so shocking. If your gauge theory contains objects that look like strings - like some electric fluxtubes - their dynamics and interactions simply has to be consistent, and string theory classifies what it can be.

It's important to realize that both the CFT and AdS descriptions of the physical system are physically meaningful and comparably useful. You can't say that one of them is real and one of them is fake. You might decide that the CFT is more well-defined because you may define quantum field theories non-perturbatively and accurately e.g. by their lattice versions - while the "full" definition using the stringy variables may be absent. But both sides are comparably useful to parameterize or calculate the physics.

String theory as a different term for a consistent theory of quantum gravity

It's not just the AdS/CFT correspondence. All quantum mechanical but consistent ways to end up with an emergent theory of gravity inevitably have to fit to the set of consistent solutions to string theory - because "string theory" is just a more explicitly looking way to describe the term "consistent quantum gravity".

Even today, string theory is no longer just a theory of strings and we say that M-theory in D=11 is a vacuum of string theory. So it's pretty general. It's pretty clear that if you include all possible deformations, defects, and other objects in this theory that can interact with others in a consistent fashion, your generalized string theory will be the same thing as a set of all "consistent quantum theories of gravity".

At this moment, it's not quite possible to prove this equivalence as a mathematical theorem because we don't have an independent definition of "string theory" in the most general way - we've been learning and we're still learning what can be done with it - and even the definition of a "consistent quantum gravity theory" could have some uncertainties about it - but as far as the "state-of-the-art" approximate definition of both sides go, the equivalence seems to be perfectly satisfied.

Matrix theory

To see a different example than the AdS/CFT correspondence, let's consider another ingenious method to define a consistent theory of quantum gravity.

Let's imagine that there are particles flying in the space. But adding direct local interactions in the spacetime is problematic - and leads to UV problems - while non-local interactions would lead to problems with the causality constraints. Is there another way how to allow the particles to interact?

Imagine that you're really clever and you find the following way. The states shouldn't be described just by the coordinates of the particles in space. The coordinates X1 of the K particles in space should be just eigenvalues of a "K by K" matrix X1. In the basis of its eigenvectors, the diagonal elements - the eigenvalues - remember the positions of the particles. The off-diagonal elements can still be there in a general basis and they can fluctuate. The fluctuations of the (i,j) and (j,i) positions of the (Hermitean) matrix X1 will induce some interactions between the i-th and j-th particle.

So you can write a matrix model. The eigenvalues or diagonal entries of the matrices will remember the positions while the off-diagonal entries will produce some interactions. You may write some nice Hamiltonians for the matrices and hope that the result may be a consistent theory in spacetime - which may moreover reproduce the Lorentz symmetry even though this symmetry is not manifest from the beginning.

When you analyze this idea in some detail, you will find out that it can really work. And yes, because of some miracles, the theory may be exactly Lorentz-invariant for a large size of the matrices even though this symmetry is not manifest.

However, you have to satisfy many consistency criteria. For example, the off-diagonal entries of the matrices will behave like harmonic oscillators that will bring you some zero-point energies, as quantum mechanics dictates. These zero-point energies are proportional to the frequency. And you may find out that the frequency will depend on the distance between the eigenvalues in space - the distance between the particles.

Because you want these particles-eigenvalues to move freely in space, these zero-point energies have to cancel. Fermions give negative contributions which may compensate bosons' positive contributions. If you begin with the most supersymmetric theories with matrices (16 supercharges), you will find out that 8 components of the fermions may compensate against 8 bosons, which must be transverse to the vector between the two positions of the eigenvalues in space.

Consequently, there must be 9 bosonic matrices - one of them is along the vector, 8 of them are orthogonal to it and their off-diagonal entries cancel the fermions. It means that there are 9 spatial dimensions, one time, but you also find another spatial (actually null) dimension which is the Fourier dual to the size of the matrices N.

Once you combine everything, you will see that the maximally supersymmetric matrix model, known as the BFSS matrix model, actually describes physics in D=11. At low energies, it is completely equivalent to the eleven-dimensional supergravity. If you analyze it at more detail, you will find out that at all energies, the model is actually equivalent to M-theory in D=11. It was the first known definition of M-theory in D=11 that went beyond the supergravity approximation.

Again, I could repeat the story from the AdS/CFT correspondence. You may look for more complicated matrix models that are consistent and describe a flat or nearly flat space where objects can move pretty freely. You will always see that the resulting theory is exactly equivalent to one of the possible solutions of string/M-theory that more or less secretly knows about the right dimension. It contains all the fields, particles, and extended objects that you may derive in string theory, together with all the calculable interactions between them and all the additional information about the neighborhood of the relevant region of the "landscape".

In particular, all these descriptions always know that the total dimension is D=11 if it is a membrane-based M-theory or D=10 if it is a theory with strings. There's no way to escape these things. String/M-theory is the only possible consistent quantum theory of gravity and it restricts your attempts to generalize or deform it: only the deformations and changes that correspond to legitimate solutions of the same theory - that you could actually obtain by physically modifying any starting point - are physically allowed.

Summary

There doesn't exist any known counterexample to the statement that whenever gravity with physical modes of gravitons - i.e. with quanta of gravitational waves (which have been observed in reality) - emerge from a quantum mechanical theory, and if you study this theory in detail and require that it is consistent, the theory must admit a description that shows that it is "just another solution of the same and unique string/M-theory".

In particular, all of the emergent stable theories of gravity know that at some underlying level, the total spacetime dimension is D=10 or D=11, depending on whether or not there exist light one-dimensional excitations in the given background. And all the spectra of infinitely many mass eigenstates and all the interaction constants between these objects are fully determined by much more modest data describing the relevant solution of string/M-theory.

To consistently separate the AdS/CFT correspondence or Matrix theory or any other result of string theory from string theory is just impossible and whoever is trying to lead you into believing that you can do it, or that you can do quantum gravity, unification of forces, proper detailed AdS/CFT, or anything else while denying the fact that the underlying theory is a unique D=10 or D=11 string/M-theory with the conventional rules, is either an imbecile or a liar.

And that's the memo.

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