I would have liked to see the gauge symmetries of standard model emerging purely from the 10-dim geometry.In other words, he would have liked if string theory were nothing more than the Kaluza-Klein theory (general relativity with extra dimensions) from the 1920s. Well, the bad news for him is that string theory also contains... strings, for example, and they actually have physical consequences. It is not just Kaluza-Klein theory and the word "string" isn't just a marketing trick. ;-)
The quote above is just one example of the widespread prejudices and excuses by which people who don't like to learn new things about science try to justify their frozen closed minds.
The most generic group of those folks would say: "I would have liked string theory if it just said that everything is made out of five elements, earth, air, water, fire, and aether." A more sophisticated subgroup – one that has already figured out that the world is more subtle than just five elements – would have liked the current physics if it said that everything were just classical mechanics; or classical field theory; or non-relativistic quantum mechanics; or a simple quantum field theory, and so on.
But those folks are just stuck in various points of the history of science. They are trying to squeeze Nature and science into assorted straitjackets. They are claiming that their ideas are beautiful and convincing; in reality, they are obsolete and demonstrably wrong. They could have looked beautiful in the past but their old beauty is no longer competitive today.
Different people from those groups want to be frozen at different points of the history.
But I want to return to the particular topic introduced by the quote above: the emergence of gauge symmetries in string theory. String theory is a theory of everything so the gauge symmetries are not "assumptions we have to insert" which is the status of gauge symmetries in quantum field theories (at least if we "construct them" in the beautified ways).
In string theory, gauge symmetries, much like everything else, emerge from a more fundamental substrate. And the emergence of these gauge symmetries depends on several very original "tricks" that Nature and mathematics knew from their birth but that humans had to gradually learn. Let me start with the faces of gauge symmetries that string theory shares with previous frameworks of physics.
Pre-history 1: Chan-Paton factors
In quantum field theory, fields like the quark fields carry various color indices and similar indices, \(\psi_i\), \(i=1,2,3\). If you realize that these fields may create particles (or annihilate their antiparticles), the index literally means that there are point-like particles called quarks with three different colors. You may visualize them as red, green, blue points although the actual quantum number known as "color" has fundamentally nothing to do with the "colors" associated with various wavelengths of visible light, of course.
For many decades, people have realized that there exist objects in string theory that carry the same "colors" or discrete indices as the index \(i\) of the quark above. It's because string theory has been known to contain point-like-particle-like objects, namely the endpoints of the open strings.
An open string is on the left side, a closed string is on the right side.
Note that one may ban open strings in a string theory; but one may never ban closed strings. Why? To make open strings dynamical, we must allow them to break and join and change the number of endpoints by 2; otherwise such open strings would be indestructible but nothing in the world is indestructible. However, once we allow two open strings to merge their endpoints, the same local interaction is also capable of merging the endpoints of a single open string, thus creating a closed one. The endpoints can't know whom they belong to, by locality, so it is always possible to produce closed strings by string interactions. It's good news because the gravitational field is always carried by closed strings – so gravity is always a part of string theory.
On the other hand, string theory may "ban" all open strings. In type II string theory, the open strings exist if there are D-branes around. In heterotic string theory, there can't be open strings at all.
If we allow open-string-merging interactions at the top, we also allow open-to-closed interactions in the middle because they're locally the same processes. On the other hand, there can be string theories with closed string only. The bottom part of the diagram is the only allowed "crossover-type" interaction in these theories.
But let me return to the colorful quarks. The open strings' endpoints look like quarks or antiquarks so they may also be marked by colors; they may carry additional discrete information about the color, an index. If an open string is orientable, we may invariantly distinguish its beginning, the beginning of the arrow, and its end, the end of the arrow. We may call the former and the latter "quark" and "antiquark", respectively. The open string field creating such an open string therefore carries two color indices, one from a quark and one from an antiquark,\[
\Psi_i^j.
\] You see that because of the two indices of the opposite type, the open strings transform as the adjoint representation of a \(U(N)\) group. That's for the same reason as the reason why the quarks in QCD – which have a single index – transform in the fundamental representation (and antiquarks in the antifundamental representation).
The open-string-splitting-or-joining interaction annihilates a quark and antiquark of the same color only; or it creates them in a color-blind way. At any rate, the interactions between such open strings are proportional to \(\delta_i^j\). That's why the spectrum as well as the interactions are automatically invariant under the \(U(N)\) symmetry transformations
In fact, it's straightforward to see that this \(U(N)\) is actually a gauge symmetry in the spacetime. Much like closed strings inevitably contain a massless mode, the graviton, the open strings inevitably contain a massless mode, the gauge boson. If the open strings have the minimal (or next-to-minimal, in theories with tachyons) amount of internal vibrational energy, the field \(\Psi_i^j\) actually carries an extra Lorentz index \(\mu\) and it creates or annihilates a gauge boson of a \(U(N)\) gauge symmetry. One may show that this symmetry is local in spacetime or, equivalently, the time-like and longitudinal polarizations of the gauge field decouple. The latter property boils down to the conformal symmetry on the world sheet – which is a "more elementary" reason that produces gauge symmetries as well as the diffeomorphism symmetry of general relativity and many other things we know from the spacetime. But I won't offer you the proof here.
If strings are unorientable (if that holds for closed strings, it must hold for open strings and vice versa because the bulk of the strings is always made of the same "material" and you must know whether it carries a preferred arrow or not without asking whether the string ends up as an open one or a closed one), you can't distinguish the beginning and the end. Consequently, the endpoints are quarks and antiquarks at the same moment. The projection needed to identify the oppositely oriented strings has the effect of reducing the \(U(N)\) symmetry either to \(O(N)\) or \(USp(N)=USp(2k)\); the unitary group gets reduced to the orthogonal one or the symplectic one. Note that the orthogonal and symplectic groups don't distinguish the fundamental and the anti-fundamental representation, unlike the unitary groups. That's why the corresponding strings are unorientable.
So when we have open strings, we may make them either orientable or not and we may assign their endpoints with labels that distinguish colors. The number of colors is a priori variable. For bosonic string theory, there is no number of colors that would yield a more consistent or interesting theory than other numbers of colors. Well, that's not quite the case as Steven Weinberg has explained why the open bosonic string theory with the \(SO(8192)\) gauge group cancels some tadpoles and one-loop divergences.
For the superstring theory, the choice of the right number of colors is much more important. In 1984, Green and Schwarz realized that almost all choices led to anomalous theories but the unorientable \(SO(32)\) open "type I" strings miraculously cancel all world sheet and spacetime anomalies. Quite unexpectedly, a half-dozen of coefficients which are a priori integers comparable to a thousand get simultaneously cancelled due to contributions from a half-dozen of sources. All of them seem to conspire and vanish at the end! This miracle – slightly demystified in a few following years – had sparked the first superstring revolution in the mid 1980s.
If you think about Weinberg's \(SO(8192)\) gauge group in an IQ-test-style way, you might figure out that the superstrings' preferred gauge group is \(SO(32)\). How? Well, \(8192=2^{13}\) and \(13\) is not only an unlucky number but also one-half of \(D=26\) which is the critical spacetime dimension of bosonic string theory. If you repeat the same exercise with the \(D=10\) critical dimension of the superstring, divide it by two, obtain five, and calculate the fifth power of two, you get \(SO(32)\). This is not a coincidence but a genuine caricature of the calculation of the preferred gauge group.
In type I string theory, the Green-Schwarz-sponsored \(SO(32)\) group arises from colored endpoints of the open strings. In some sense, the colorful labels are added in the same ad hoc way as they are in quantum field theories. In the context of open strings, we call these colorful labels "Chan-Paton factors".
Pre-history 2: Old-fashioned Kaluza-Klein theory
Our second source of gauge symmetries, the Kaluza-Klein mechanism involving extra dimensions, is also non-stringy in character (the source may be incorporated into theories with pointlike particles only) but because extra dimensions are often associated with string theory, we could say that the Kaluza-Klein theory is "more stringy" than the Chan-Paton factors.
It started in 1919 when unknown German Silesian mathematician Theodor Kaluza realized that the general theory of relativity in five dimensions provides us with a nice surprise. He considered the five-dimensional metric tensor, \[
g_{\mu\nu},\quad \mu,\nu=0,1,2,3,4,
\] which contains some additional components aside from those we know, i.e. those with \(\mu,\nu=0,1,2,3\). In particular, the components of the metric tensor may be split to\[
g_{\mu\nu},\quad g_{\mu 5}, \quad g_{55}, \quad \mu,\nu=0,1,2,3.
\] I switched the "unusual" value of the index from "4" to "5" to emphasize that it is the fifth dimension. We see that the five-dimensional symmetric tensor splits into a four-dimensional symmetric tensor, a vector, and a scalar. Einstein's equations in five dimensions could have been rewritten in terms of these decomposed fields, too.
He found out that they reduced to Einstein's equations in four dimensions, with some extra sources improving the stress-energy tensor (the equations obtained by varying the four-dimensional tensor components); and some new equations. It turned out that the equations for the vector looked just like Maxwell's equations. We may identify \(g_{\mu 5}\equiv A_\mu\), the electromagnetic potential.
Meanwhile, \(g_{55}\) is a new scalar field, the Kaluza-Klein dilaton, and it may be seen to obey a Klein-Gordon-like equation if it is obtained from the metric tensor obeying Einstein's equations in this way.
Kaluza didn't really explain why we don't see the fifth dimension. He worked with a "dimensionally reduced" theory in which the fields are required to be independent of the fifth coordinate (without a convincing explanation). This hole was fixed by Oskar Klein a few years later. He argued that there may be a natural reason why the fields are independent of \(x^5\), the fifth coordinate: the coordinate may be periodic:\[
x^5\sim x^5+2\pi.
\] Of course, that's just a shortcut to demand the periodicity of the fields:\[
g_{\mu\nu}(x^\mu,x^5+2\pi) = g_{\mu\nu}(x^\mu,x^5)\dots
\] By a scaling or coordinate transformation, we have enough freedom to choose the periodicity of \(x^5\) equal to \(2\pi\) or anything else. Now, the metric tensor defines proper distances in the five-dimensional space:\[
ds^2 = g_{\mu\nu} dx^\mu dx^\nu + 2 g_{\mu 5} dx^\mu dx^5 + g_{55} dx^5 dx^5.
\] You see that the component \(g_{55}\) determines the actual proper length of the fifth dimension: the circumference is \(2\pi\sqrt{g_{55}}\) i.e. \(\sqrt{g_{55}}\) may be called the proper radius. The middle, mixed term is more interesting than this scalar. In fact, the angular i.e. periodic fifth dimension \(x^5\) may be redefined by an Einstein-like coordinate transformation that depends on the ordinary four dimensions:\[
x^5\to x^{\prime 5} = x^5 + \lambda(x^0,x^1,x^2,x^3).
\] The gauge transformation parameter \(\lambda\) itself is an angle; if you change it by \(\lambda\to\lambda+2\pi m\), nothing changes about the physical meaning of the transformation. We may use the usual general relativistic rules to figure out how the values of the tensors transform under this coordinate transformation. And we find out that \[
A_\mu = g_{\mu 5}\to A_\mu + \partial_\mu \lambda.
\] Approximately. Factors that depend on \(g_{55}\) and others have been omitted. That's cool because it's nothing else than the usual \(U(1)\) electromagnetic (or similar) gauge transformation. At each point of the four-dimensional spacetime, there is a circle that is attached. We may rotate it and that's interpreted as the electromagnetic gauge transformation. The gauge field remembers the "twisting" so gets corrected by the gradient.
Because of this transformation, we may also talk about charged fields. Recall that in electromagnetism, the charged fields transform as \[
\psi \to \psi \cdot \exp(iQ\lambda).
\] But that may be geometrically interpreted if \(\psi\) is just the \(Q\)-th Fourier mode in the Fourier decomposition of a field in five dimensions! It's because the Fourier component scales like \(\exp(iQ x^5)\) and the additive shift of \(x^5\) that we have identified with the gauge transformation multiplies the whole Fourier component – a field in four dimensions – by the appropriate phase.
Albert Einstein ultimately became the most prominent champion of the Kaluza-Klein theory and wrote several papers on it which apparently haven't added much. This theory was perfectly compatible with Einstein's big-picture goal, the unification of electromagnetism and gravity within a framework similar to his general relativity. Einstein didn't try to explain the strong and weak nuclear forces because he didn't believe they were real or fundamental. He ignored emerging particle physics. After all, he didn't even take quantum mechanics seriously.
The original Kaluza-Klein theory was obviously intriguing but had some crucial bugs. The additional scalar, the dilaton, was unobserved and it shouldn't really be there because new massless scalars would cause new long-range forces and destroy the equivalence principle along the way. Also, the charged particles had masses that were inseparably linked to \(1/R\), the inverse radius of the fifth dimension. Because the radius was apparently close to the Planck scale, the only natural length scale in physics of gravity and electromagnetism, the theory predicted that even the electron should be about as heavy as the Planck scale. It is almost 20 orders of magnitude lighter.
But sensitive physicists realized that the new "geometrized" way of describing the gauge symmetry is too cool an idea that would likely reemerge in more viable reincarnations. And it did. String theory uses all these wonderful ideas but it also modifies the physical phenomena by certain new robust, reliably provable effects that circumvent the undesirable properties of the old Kaluza-Klein theory.
Before I get there, let me mention that there exists a simple old-fashioned extension of the Kaluza-Klein theory to non-Abelian groups. If the extra dimensions are not a circle but a manifold whose isometry (geometric symmetry) group is \(G\), then it will be possible to reparameterize the coordinates labeling these extra dimensions at each point of the usual four dimensions independently. Consequently, \(G\) will be the gauge symmetry and the mixed components of the metric tensor will remember the non-Abelian gauge fields.
To obtain larger groups, you need a larger number of extra dimensions. For example, you could try to get an \(SU(2)\sim SO(3)\) gauge symmetry by adding two extra dimensions shaped as a two-sphere whose isometry group is \(SO(3)\). A problem with this shape is that the two-sphere isn't Ricci-flat so it doesn't solve the higher-dimensional Einstein's equations.
Well, if there are extra sources, it actually does. For example, you may have the famous \(AdS_5\times S^5\) compactification of type IIB string theory, the most frequently studied example of the AdS/CFT correspondence. The isometry of the five-sphere, \(SO(6)\), is indeed a gauge symmetry of the five-dimensional effective AdS gravitational theory. You may want to remember that the same \(SO(6)\) is actually just a global symmetry, the R-symmetry, of the dual holographic conformal field theory. In that description, it is not a gauge symmetry.
The Standard Model gauge group \(SU(3)\times SU(2)\times U(1)\) has the rank (the maximum number of independent yet mutually commuting generators) equal to \(2+1+1=4\). Because you need at least a pair of dimension for each rotation in the rank, you would need at least 8 compactified dimensions to geometrize the Standard Model gauge group as the isometry of some Kaluza-Klein dimensions. Well, the actual minimum number of dimensions is even higher because the non-Abelian group is much larger than its commuting generators. Even if it were just eight, it is easy to see that no such geometry fits the string/M/F-theoretical spacetimes. F-theory allows up to 8 hidden dimensions but 2 of them are too constrained, giving you at most new \(U(1)\) groups.
If string theory is producing the Standard Model group, it is using its new features rather than simply reducing the problem to the old-fashioned Kaluza-Klein theory. And indeed, string theory manages to give solutions to this problem. In fact, it gives us several qualitatively different solutions. More precisely, they look qualitatively different but one may show that they're related by exact yet surprising physical equivalences, the so-called dualities. The different solutions look qualitatively different but they may be continuously transformed into each other.
String theory's added value 1: \(p\)-forms
String theory extends the old ideas of gauge theory in various ways. First of all, it seems to contain some fields similar to the electromagnetic fields from scratch. All fields in weakly-coupled string theory emerge from particular energy eigenstates of a vibrating string and that's true for these electromagnetic and related fields, too.
But sometimes these fields obtained from strings carry several Lorentz indices. The differential forms – completely antisymmetric tensors with \(p\) indices – are an important subgroup.
String vacua with orientable strings always contain the so-called \(B\)-field, \(B_{\mu\nu}=-B_{\nu\mu}\). It generalizes the electromagnetic potential \(A_{\mu}\) which has 1 index and may be considered a 1-form (the antisymmetry condition doesn't affect the tensor because there are no nontrivial ways to permute indices if you only have one index). Note that electromagnetism allows you the coupling of the electromagnetic field to charged particles via the term in the action:\[
S = e\cdot \int \dd x^\mu A_\mu.
\] The electromagnetic field is simply integrated over the world line of the charged particles with the most natural contraction of the Lorentz index. A funny thing is that this formula has a straightforward and completely natural generalization to charged objects with an arbitrary number of dimensions. For example, we may have a 2-form \(B\)-field which may be contracted with an antisymmetric tensor representing an infinitesimal "two-dimensional surface" embedded into the spacetime,\[
S = \rho_{\rm charge}\cdot \int \dd \Sigma^{\mu\nu} B_{\mu\nu}.
\] That's great because the integral may be interpreted as the two-dimensional integral over the world sheets of all strings. Whenever the term above exists, the strings are charged under the \(B\)-field. If we vary the term above with respect to the \(B\)-field, we see that the strings add delta-function sources to the right hand side of Maxwell-like equations for the \(B\)-field. That's why they are charged.
In type II and heterotic string theories, there is a \(B\)-field and the strings are charged. This fact depends on the orientability of the strings. In type I string theory, strings are unorientable so they can't be distinguished from their antimatter (oppositely oriented string) and you wouldn't know how to choose the right sign of the charge. It's because the charge is really zero and there's no \(B\)-field that would be sourced by this non-existent charge. Equivalently, the antisymmetric \(B\)-field is filtered out of the spectrum by the unorientability condition for the strings; only the symmetric metric tensor and the stringy dilaton survive among the massless closed string (bosonic) modes.
If there is an extra circular hidden dimension of spacetime, strings may be wound around this circle \(w\) times. The integer \(w\) is known as the winding number. From the four-dimensional viewpoint, \(w\) will look like a new kind of an electric charge. The components \(B_{\mu 5}\) of the \(B\)-field will act as the corresponding new electromagnetic potential for this new \(U(1)\) gauge symmetry.
In the Kaluza-Klein theory, we saw that the electric charge was given by the label of the Fourier component - in other words, by the momentum \(n\). In this \(B\)-field case, the electric charge is given by the winding number \(w\). These two constructions look completely different. And so do the electromagnetic potentials \(g_{\mu 5}\) and \(B_{\mu 5}\), for example because they arise from symmetric and antisymmetric tensors, respectively. However, in the next section, I will argue that string theory nevertheless contains exact symmetries that mix these two different \(U(1)\) groups into each other and it even allows them to be incorporated into larger non-Abelian gauge groups that join them (and something else).
Before I get there, however, I must tell you that in type I/II string theory, there are many more massless fields that are differential forms (completely antisymmetric tensor). They may have any number of indices (well, any odd number of indices in type IIA and any even number in type IIB or type I which picks a subset). These \(p\)-form fields arise from the Ramond-Ramond (RR) sector and they have been known for a long time.
While these RR fields generalize electromagnetism, it was believed for quite some time that nothing in string theory was charged under them. Joe Polchinski revolutionized this question in the mid 1990s when he realized that string theory contains a new class of heavy objects, the D-branes, which are charged to these RR fields and produce the integral in the effective action that resembles the term \(\int \dd \Sigma^{\mu\nu}B_{\mu\nu}\) that I have mentioned in the case of the fundamental strings. However, in the RR case, the charged objects are much heavier (and therefore seemingly "less important") than the fundamental strings. And they may have any even or any odd number of spatial dimensions in type IIA and type IIB, respectively.
String theory's added value 2: enhanced non-Abelian symmetries at self-dual radii and abstract CFTs with current algebras
I want to return to the hidden relation between the seemingly different ways to obtain the electromagnetic field from something more fundamental, either from \(g_{\mu 5}\) in the Kaluza-Klein theory or from \(B_{\mu 5}\) where the charge arises from the winding number of strings around a circular dimension.
Squirrels look different from humans – nevertheless, Darwin's theory in biology tells us that they have common ancestors. In the same way, string theory in physics shows that the two strategies for string theory to produce electromagnetic fields and electric charges secretly arise from the same physical framework, too. There can even be an exact symmetry between them, a T-duality. How does it work?
In the Kaluza-Klein theory, we associated fields of charge \(n\) with the Fourier modes scaling like \(\exp(inx^5)\). That's nothing else than the wave function of a particle in quantum mechanics that has \(n\) units of momentum in the direction of the fifth (extra) coordinate. Don't forget that the momentum along a periodic dimension is quantized because the wavefunction has to be single-valued. In dimensionful units, the momentum component is\[
p^5 = \frac{n}{R}
\] where \(R\) is the proper radius of the circle, previously represented by \(R=\sqrt{g_{55}}\). Because the energy in the five-dimensional space can't be smaller than any momentum component, you see that the momentum of \(n\) units is giving us a lower bound \(n/R\) for the mass of a particle with the charge \(n\) units.
There is a similar story for winding strings. If we take a string and wrap it \(w\) times around the same circle, its minimum mass will be \(2\pi R w\cdot T\) because \(2\pi R\) is the circumference of the circular dimension of our spacetime, it has to be multiplied by \(w\) to get the minimum length of the string, and \(T=1/2\pi \alpha'\) is the string tension i.e. the linear density (mass per unit length) of the string.
What's funny is that \(n/R\) is inversely proportional to the radius while \(2\pi R w T\) is directly proportional. We may actually map these two expressions into each other if we make the following replacement:\[
n\leftrightarrow w, \quad R \leftrightarrow \frac{1}{2\pi T R}=\frac{\alpha'}{R}.
\] Please check that if the momentum and winding integers are interchanged and the radius is inverted in the \(\alpha'=1\) units, we also get the exchange\[
\frac{n}{R} \leftrightarrow 2\pi R w T.
\] So the momentum-winding exchange combined with an inverted radius is the symmetry of the spectrum, at least of the simple \(n,w,R\)-dependent terms for the mass. You might think it's just some coincidence resulting from overly simplistic formulae for the masses and it won't be a symmetry of the interactions. However, you would be wrong. This exchange, the T-duality, is an exact symmetry of string theory including all of its interactions. (Well, I should discuss which string theories and whether it changes type IIA to IIB and vice versa, and so on, but let me avoid these details now.)
Note that the thing I am just telling you is true – because it is I who is informing you – but it sounds crazy, too. String theory tells you that the winding number is physically the same thing as the momentum. How can it be true?
The symmetry boils down to the factorization of two-dimensional conformal field theories (governing the stringy world sheets) into the left-movers and the right-movers or, using the Euclidean signature, into the holomorphic and antiholomorphic sectors. They can be dealt with – and transformed by symmetries – almost separately from one another!
The conformal field theory contains spacetime coordinates \(X^\mu\) which are scalar fields depending on the temporal and spatial coordinates on the world sheet, \(\tau\) and \(\sigma\). It is useful to talk about left-moving and right-moving derivatives of the coordinates, \[
\partial_\pm X^\mu = \partial_\tau X^\mu \pm \partial_\sigma X^\mu.
\] Sorry for having omitted factors of two (or its square root) and for using reverted signs relatively to some conventions. You must be careful when you jump to serious accurate calculations but the right (reverted) signs and extra factors would be distractions (points refocusing you from the essence to some irrelevant technicalities) in this basic introduction.
A funny thing is that the variables \(\partial_+ X^\mu\) and \(\partial_- X^\mu\) behave almost independently from each other – much like left-moving and right-moving waves on a rope that simply penetrate through one another. That also means that we can transform the left-moving and right-moving parts by symmetries separately.
T-duality is nothing else than the reflection of one coordinate, the circular \(X^{25}\) coordinate (I raised the index from 5 to 25 again, so that it may emulate the last dimension of the 26-dimensional bosonic string), but a reflection that is only applied to the right-movers (or left-movers: these two choices only differ by an addition of a normal spacetime reflection that acts both on left-movers and right-movers).
The map \(X^{25}\to -X^{25}\) would be a symmetry, an ordinary mirror reflection. But let's apply it to the \(\partial_- X^{25}\) degrees of freedom only while keeping \(\partial_+ X^{25}\) fixed. What does this map mean?\[
\eq{
(\partial_\tau - \partial_\sigma) X^{25} &\leftrightarrow -(\partial_\tau - \partial_\sigma) X^{25}\\
\partial_\tau X^{25}&\leftrightarrow \partial_\sigma X^{25}.
}
\] You see that the first line may be derived from the second one: the right-movers-only reflection of \(X^{25}\) is nothing else than the exchange of the temporal world sheet derivative of \(X^{25}\) with the spatial world sheet derivative. What a bizarre transformation but we have discussed why such a thing should be a symmetry: the left-movers and right-movers are mostly independent on the world sheet.
But such an exchange of tau- and sigma- derivatives has another funny consequence we already know. The integral of the tau-derivative of \(X^{25}\) is interchanged with the integral of the sigma-derivative of the same thing. But the former is the integral of a momentum density i.e. the total \(p^{25}\) while the latter is just the overall jump of \(X^{25}\) between the beginning and end of the closed string and that's nothing else than the winding number!
So the same transformation also exchanges the momentum with the winding, something we have declared to be a characteristic property of the T-duality at the very beginning. This transformation may be derived from a mirror reflection of a (circular) coordinate of space that is only applied to right-moving excitations on the world sheet: the T-duality is a "chiral parity", if you wish.
As we described it, the T-duality changes the radius. It inverts it with some coefficient. However, there exists a self-dual value of the radius\[
R = \sqrt{\alpha'} \equiv l_{\rm string},
\] the string length, which is invariant under T-duality. When a circular dimension in string theory has this radius, the momentum and winding contribute two exactly analogous, confusable pieces to the squared mass of a closed string:\[
m^2 = \frac{n^2+w^2+2(N_L+N_R-2)}{\alpha'}
\] I've added the contribution from \(N_L,N_R\), the internal excitations of the string that don't change the overall momentum or the total winding. Note that for bosonic string theory, there is also the \((-2)\) piece that guarantees that the ground state of the string produces a tachyonic particle in the spacetime. Such negative terms arise from contributions proportional to the famous sum\[
1+2+3+4+5+\dots = -\frac{1}{12}.
\] This negative, "tachyonic" shift of the ground state, has many welcome consequences. For example, it provides us with the loophole from the previously inevitable conclusion of the old Kaluza-Klein theory that "the electrons should be as heavy as the Planck scale".
At any rate, you see that for \(R^2=\alpha'\), the squared masses of all string states are integer multiples of \(1/\alpha'\) as long as we neglect the string interactions (if the coupling is very weak, it's OK). There are huge degeneracies at each energy level and that's why we shouldn't be shocked if we find new symmetries.
And we do find them and they're shocking even if we're ready for surprises.
Consider the 26-dimensional bosonic string theory on the self-dual radius. We said that this had a \(U(1)\) symmetry from the Kaluza-Klein isometry-induced gauge symmetry; and another \(U(1)\) symmetry whose electric charge is the winding number \(w\). In total, we have a \(U(1)\times U(1)\).
It's convenient to rotate the basis of these \(U(1)\) groups by 45 degrees and describe the gauge group as \(U(1)_L\times U(1)_R\) where the left- and right- factors in the gauge group are generated by the combinations of electric charges \(n+w\) and \(n-w\), respectively. Fine, we still have two \(U(1)\) groups.
A stunning thing is that at the self-dual radius, the symmetry actually gets enhanced to\[
U(1)_L\times U(1)_R \to SU(2)_L \times SU(2)_R.
\] Both the gauge group carried by the left-moving excitations on the string and its right-wing counterpart get promoted from \(U(1)\) to a non-Abelian group, \(SU(2)\). This non-Abelian group is an exact symmetry of the full spectrum and one may actually prove that it extends to a symmetry of the interactions, too.
One may do lots of consistency checks, verify that at each level, we get nice representations of both \(SU(2)\) groups, and so on. We also get the massless gauge fields for the whole non-Abelian groups at the appropriate, massless level. But what's the compact and comprehensive proof that the symmetry is there?
Well, continuous groups are generated by generators which are integrals of the charge density over the whole string. The charge densities for \(n\) and \(w\), the \(U(1)\) groups, were simply \(\partial_\tau X^{25}\) and \(\partial_\sigma X^{25}\). It was kind of easy to see why they were generating \(U(1)\) symmetries.
The new thing is that there are new generators that complete the symmetry to \(SU(2)_L\times SU(2)_R\). Recall the generators of an \(SU(2)\) from the basic QM lectures on the angular momentum. We need new generators \(J_\pm\) for each \(SU(2)\), if I choose a clever basis. The sign is linked to the sign in \(J_x\pm i J_y\).
In string theory on the self-dual radius, however, the density of the charge(s) \(J_\pm\) may be written simply as\[
\rho_\pm = {:\exp(i X^5):}
\] The colons mean some ordering; I don't want to go into these QM technicalities here. In essence, we just exponentiate the \(i\)-multiple of the circular coordinate whose radius is self-dual. To verify that this combines with the old \(U(1)\) generators to \(SU(2)\) symmetries, it's enough to calculate the so-called OPEs, operator product expansions. They have the nice simple form reproducing the commutators in the \(SU(2)\) Lie algebra and that's all we need because the transformations of all operators under the \(SU(2)\) may be obtained from the OPEs, too. Moreover, we also know how to transform the states because there exists a state-operator correspondence (a dictionary) in these world sheet CFTs. Finally, an OPE of our \(SU(2)\) generators with the world sheet Hamiltonian may be shown to de facto vanish, proving that the Hamiltonian is invariant under the symmetry, too.
Note that the operator \(:\exp(i X^5):\) carries a nonzero momentum and a nonzero winding number, too. The \(SU(2)\) rotations generated by this non-Abelian generator mix a given state with states that have different values of momenta and windings.
The fact that we could discover such unexpected non-Abelian symmetries is related to another fact, namely that two-dimensional conformal field theories may often be expressed in many different equivalent ways. One free boson is equivalent to two free fermions or to a "current algebra". There exist CFTs whose operator spectrum is completely calculable and determined by symmetries, the minimal models, and so on. There exist many other non-Abelian symmetries you may get from other radii differing from the self-dual radius by simple numerical constants. It's a cool set of insights. This rich and calculable structure is related to the conformal symmetry – the fact that on the two-dimensional plane i.e. complex plane, the angle-preserving diffeomorphisms form an infinite-dimensional group, the group of all holomorphic transformations of a complex variable. Only in two real dimensions, the conformal group is infinite-dimensional and that's why you can't build an equally robust theory from objects with higher dimensions than strings, at least not along the very same lines we follow when we construct perturbative string theory.
If you want a theory constructed in this way, strings are special and unique.
I must emphasize that the extended or "enhanced" symmetry \(SU(2)\) is exactly as robust, natural, and provable as the simpler symmetries such as the \(U(1)\) symmetries we may identify with the isometries or the winding number. The generators obtained in the new, stringy way have the same status as the old generators. After all, they are related by symmetries. Any suggestion that the \(SU(2)\) enhanced symmetry is less justified, less natural, less pretty, less satisfactory, or less anything-positive always boils down to the mathematical stupidity or at least ignorance of the critic. Such criticisms are completely idiotic. A sane person can't treat two generators related by exact symmetries in "completely different ways"; that would be an intolerable and unjustifiable "discrimination" of (or, equivalently, "affirmative action" for) physically isomorphic objects.
The heterotic case
A variation of the enhanced symmetry story is particularly cool in the case of the heterotic string. Note that heterosis means "hybrid vigor" and describes some advantages of hybrid organisms in which they exceed both inequivalent parents. The heterotic string theories are daughters of the \(D=10\) superstring used for the right-moving excitations on the string; and the \(D=26\) bosonic string theory used for the left-movers. Yes, left-moving and right-moving excitations may think that the spacetime dimension is different; that's because of the de facto segregation of these two sectors of the degrees of freedom on the string.
Concerning the "hybrid vigor", the right-wing superstring parent gives the heterotic daughter its consistency, tolerably low spacetime dimension, absence of tachyons, and spacetime supersymmetry (which may be broken); the left-wing bosonic string parent gives it some dirt, potential for conflicts, desire to behave in manners incompatible with the conservative values. You could ask how it could be a good thing to have these or any other left-wing characteristic. However, the superstring may be too ordered and too constrained so having sex with a left-wing bosonic string may be a good idea. The right-wing parent is enough to guarantee some spacetime supersymmetry and the absence of tachyons.
Fine. So how does the hybrid work? The left-moving excitations seem to see \(26-10=16\) spacetime dimensions that are not seen by the right-movers. What does it imply? If you denote these sixteen left-moving-only coordinates \(X^{10}\dots X^{25}\), they should be literally absent on the right-moving side. It means that\[
(\partial_\tau - \partial_\sigma) X^{10\dots 25} = 0.
\] These coordinates are not allowed to produce waves moving on the string to the right side. Note that if you integrate the equations above over the string, \(\int \dd \sigma\), you will also be able to derive that\[
n^{10\dots 25} = w^{10\dots 25}
\] i.e. the momenta are equal to the windings for all these 16 "chiral" spacetime dimensions. However, momenta as well as windings live in lattices (set closed under integer linear combinations) but these two lattices are dual to each other. That's the direct generalization of our previous observation that the momentum was proportional to \(1/R\) but the winding scaled like \(R\).
If the radius were an arbitrary number, like \(R=22/7\pi\) in the string units \(\alpha'=1\), the inverse radius would be \(7\pi/22\) and the generic integer combinations of these two radii just couldn't be equal to each other. The only obvious enough way to satisfy the constraint that "momenta and winding live in dual lattices but their values are equal" is to actually make the lattice self-dual i.e. dual to itself. It is a direct generalization of the self-dual radius.
For technical reasons I won't discuss here, the lattice also has to be even. One may prove that in the 16-dimensional space, there are only two inequivalent even self-dual lattices. One gives the \(SO(32)\) gauge group; the other produces \(E_8\times E_8\). In both cases, the \(U(1)^{16}\) commuting subgroup emerges in the Kaluza-Klein or \(B\)-field way. All the remaining 480 generators of these non-Abelian groups arise in the same way as our enhanced symmetry \(SU(2)_L\) a few minutes ago. In the heterotic case, these gauge groups only arise from the left-movers because the right-movers don't contain these 16 degrees of freedom at all. We are treating the left-movers and right-movers asymmetrically. The possibility to obtain a gauge symmetry is one of the virtues of the ugly non-supersymmetric left-wing bosonic string parent of the heterotic string. ;-)
The two solutions may also be obtained in another description in which the 16 left-moving bosons are converted to 32 left-moving real fermions and one studies what groups of allowed boundary conditions (and corresponding GSO-like projections) lead to consistent theories. The \(SO(32)\) gauge group emerges if all 32 real fermions are either periodic, or all of them are antiperiodic. The \(E_8\times E_8\) emerges if they are divided to two groups of 16 real fermions which may be periodic or antiperiodic independently of the other group. The naive \(SO(16)\times SO(16)\) symmetry of this construction is enhanced to \(E_8\times E_8\) because of some extra generators analogous to the enhanced \(SU(2)\) above – but now coming from fermionic degrees of freedom and transforming as Weyl spinors under the \(SO(16)\) groups.
Both groups, \(SO(32)\) and \(E_8\times E_8\) have the rank equal to 16 and the dimension equal to \(496 = 31\times 16 = 2\times 248\). The \(SO(32)\) heterotic string theory turns out to have the same supergravity-super-Yang-Mills effective field theory in the 10-dimensional spacetime as the Green-Schwarz type I string theory with the \(SO(32)\) Chan-Paton factors from the beginning of this article. It's not a coincidence. These theories are actually completely equivalent, S-dual, to one another. One of them with coupling \(g\) is the same as the other one with the coupling \(1/g\). [This is a schematic description. Sometimes you need to identify one \(g\) with another \(g^2\) or add a multiplicative constant to make the previous sentence accurate and strictly valid.]
The strong coupling, \(g\to\infty\), fate of the other heterotic string theory is completely different. Instead of producing a 10-dimensional theory with open and unorientable strings, it develops a new, 11th dimension whose shape is a line interval: at strong coupling the heterotic \(E_8\times E_8\) string theory yields the Hořava-Witten M-theory with two ends of the world.
A piece of the Hořava-Witten world. Well, the hole is only possible if the boundaries are oppositely oriented and it is an unstable Hořava-Fabinger world instead. Click the image for some context.
String theory's added value 3: boundaries of M-theoretical worlds
And that's another surprising mechanism by which string/M-theory develops gauge symmetries and gauge bosons (and also gauginos because there is spacetime supersymmetry). M-theory is the stringy, fully consistent completion of the effective theory known as the eleven-dimensional supergravity. You may ask whether this 11-dimensional spacetime is allowed to have boundaries.
Edward Witten and Petr Hořava demonstrated that the answer was "Yes" at the end of 1995. Their discovery was pretty much the final step needed to understand the strong coupling limit of all major "string theories" – if I use the plural and now deprecated terminology because we only have one string/M-theory today.
They showed that if there is a boundary in an 11-dimensional spacetime, the gravitinos must be projected in a left-right-asymmetric way near the 10-dimensional boundary. This produces gravitational anomalies. Those must be cancelled by the addition of new chiral objects that only live on the boundary. They must be gauginos because the 16 supercharges preserved by the boundary make a vector boson component an unavoidable part of every supermultiplet. So the boundary must carry some gauge group which might make things even worse. Aside from gravitational anomalies, we also create a potential for gauge anomalies and mixed anomalies.
However, a strip of the 11-dimensional spacetime has 2 otherwise identical boundaries. Each of them carries one-half of the overall 10-dimensional anomaly. If we find a way to cancel all anomalies by a gauge group that may be divided to two equal factors, one factor per boundary, we're finished. And indeed, it's been known from the 1980s that the \(E_8\times E_8\) gauge group miraculously cancels all the gravitational, gauge, and mixed anomalies much like the \(SO(32)\) gauge group. In 1985, this fact materialized in the heterotic string, a particular vacuum of string theory that realizes this beautiful anomaly-cancelling possibility. And the \(E_8\times E_8\) heterotic string indeed does have two equal pieces of the gauge group.
It follows that if M-theory has boundaries, anomaly cancellation forces it to carry a single \(E_8\) gauge multiplet on each boundary. There's actually a lot of other evidence that this gauge symmetry is automatically attached to every boundary in M-theory – for example the matrix constuctions of M-theory with boundaries, something I've made non-negligible "pioneering" contributions to, before I worked on matrix string theory.
But it's still true that I would love to see a much more explicit and constructive method to derive that M-theory produces the \(E_8\) gauge symmetry on its boundaries. On the other hand, I feel that unlike perturbative string constructions, objects in M-theory are much less constructive and much more bootstrap-like, determined "recursively" by the internal consistency conditions. On the third hand, and I hope it's the right number of hands ;-), the ABJM membrane minirevolution indicated that many things that looked intrinsically bootstrap-like may be described by constructive explicit equations, after all.
Chan-Paton factors in the D-brane language
In the 1980s, especially in the revolutionary papers by Green and Schwarz sparking the first superstring revolution, people would only consider "string theories" with open strings that are allowed to end anywhere in space. So the coordinates \(X^\mu\) at the stringy endpoints had to obey the Neumann boundary conditions: the normal (sigma) derivative has to vanish. Anything else would violate the Lorentz or translational symmetry which is holy.
However, in the mid 1990s, Joe Polchinski – building on some previous hints by himself, Hořava, Dei, Leigh, and others from the late 1980s – realized that while the Lorentz and translational symmetries are holy, it makes a complete sense to consider strings with the other, Dirichlet boundary conditions for some coordinates (the coordinate is required to have a specific value at the world sheet's boundary), too. Such Dirichlet boundary conditions indeed pick preferred submanifolds of the spacetime – the Dirichlet branes or D-branes for short – but it's OK because (and that's Joe's remarkable statement) these D-branes are dynamical objects, much like string themselves. They're just heavier so it's harder (it requires more energy) to create "big waves" on them. However, waves of modest energy propagating on the D-branes exist and their quanta are physically identical to open strings in various vibration modes that are attached to these D-branes!
But it turned out that one must allow to mix any number of Dirichlet boundary conditions for some coordinates with the Neumann boundary conditions for the rest. These conditions geometrically mean that the open strings are required to terminate on loci of any dimensions, the D-branes. The old-fashioned, purely Neumann picture from the 1980s (with the Chan-Paton factors) was suddenly interpreted as open strings ending on spacetime-filling D-branes, the so-called D9-branes (the number only counts the spatial dimensions which is also why the similar object localized at a point of the Euclidean spacetime has to be called the D\((-1)\)-brane).
Vacua with D\(p\)-branes of different values of \(p\) are actually T-dual to each other. T-duality is an exact symmetry of string theory that exchanges the momentum with winding. To do so, it must also exchange open strings with a nonzero momentum with open strings that "wind around" and they may only be wound an integer number of times if the D-brane on which they're obliged to end is localized, e.g. at \(X^{25}=0\). In that case, both \(X^{25}\) at the beginning and end of the open string is a multiple of \(2\pi R\) or \(2\pi\), depending on the convention for the radius, and their difference is \(2\pi R w\).
A stack of coincident identical branes naturally produces a \(U(N)\) gauge group – or its orthogonal and symplectic cousins if the branes are placed on the "orientifold plane" (a hyperplane in space allowing strings to be unorientable, more precisely relating a string with one orientation somewhere to the string with the opposite orientation at the locus that is mirror-reflected relatively to the orientifold plane's dimensions).
Separation of D-branes in the transverse dimension corresponds to the Higgs mechanism breaking \(U(N)\) to \(U(N_1)\times U(N_2)\), and so on. There are too many things that should be mentioned here. At any rate, the D-branes are "almost fundamental" objects in weakly coupled string theory. Open strings are only allowed to terminate on D-branes and the information about which D-brane an open string is ending upon also gives a colorful Chan-Paton label or index to the open string.
String theory's added value 4: groups from ADE-like singularities
I have already discussed ADE singularities and ADE classification six years ago, in the robustly pre-\(\rm\LaTeX\) epoch of The Reference Frame – and this fact discourages me from writing many of the things again.
But there is a mechanism to produce gauge symmetries in string theory out of singularities. This mechanism is related to pretty much all the previous ways to obtain gauge symmetries in string theory by very interesting limiting procedures and dualities.
As the 2006 article discusses, an ADE singularity is a space of the form of the quotient\[
\CC^2 / \Gamma
\] where \(\Gamma\) is a finite subgroup of \(SO(3)\) – the group of symmetries of an orientable or unorientable polygon or a Platonic polyhedron – translated into a subgroup of \(SU(2)\) via the usual \(SO(3)=SU(2)/ \ZZ_2\) isomorphism so that we know how the elements of \(\Gamma\) act on two complex coordinates of \(\CC^2\). Those two complex coordinates may be interpreted as four real coordinates which means that we get a four-dimensional space that is ordinary and flat everywhere except near the "orbifold singularity" i.e. in the vicinity of\[
(x_1,x_2,x_3,x_4)=(0,0,0,0).
\] In general relativity, we would have to say that the geometry is singular near that point and we don't know what happens. In some sense, the Riemann curvature tensor contains delta-function-like pieces and things get singular, divergent, and ill-defined.
However, string theory allows us to predict what happens exactly and consistently. At this point, I need to emphasize one detail. People often think that a valid theory of quantum gravity has to "smoothen all singularities" so that they are replaced by ordinary smooth shapes with a finite curvature. But that ain't the case. The task for a quantum theory of gravity is to be predictive in arbitrary extreme situations that may be encountered, at least in principle. In some cases, such a theory may say that the space is actually smooth. But in others, it may say that the space remains as singular as before – if we want to describe its geometry only – but physics on this space is non-singular and well-defined, anyway.
The latter is what string/M/F-theory often does, especially on ADE and similar spaces.
Nevertheless, an ADE singularity may be resolved i.e. made smooth by a small deformation. (Again, even the point at which it is not resolved is a legitimate point on the configuration space of possible shapes of spacetime – in fact, a completely non-singular point on the configuration space even though the spacetime geometry may look singular.) And when we do so, we see various 2-cycles on which M2-branes (in M-theory) or D2-branes (in type IIA string theory) may wrap. When we return the shape of the spacetime to the orbifold singularity point, these membranes get infinitely light, i.e. massless, and they provide us with new degrees of freedom, new non-Abelian gauge bosons, new symmetries.
The mathematics leading to the enhanced non-Abelian groups is fully analogous to the enhanced gauge symmetries at the "self-dual radius"; recall the discussion about T-dualities as well as heterotic strings. We have to deal with the "Cartan subalgebras" and "root lattices" in both cases. That's the relation to the heterotic origin of the gauge group.
However, the ADE singularities may also be identified with D-branes which gave us another method to discover gauge symmetries in string theory. In particular, D7-branes in type IIB string theory may be reinterpreted in F-theory, the non-perturbative "completion" of type IIB string theory, and they become ADE-like singularities (singular fibers etc.) i.e. purely geometric configurations without "non-geometric matter" on them. Because string/M/F-theory unifies gravity with all other forces and matter, it makes the previously sharp boundary between "geometry" and "matter propagating on this geometry" fuzzy and the geometric interpretation of 7-branes (which used to be clear "matter", namely D7-branes, in type IIB string theory) in F-theory is a clear example of this fact.
Finally, even the \(E_8\) at the end point of the M-theoretical world may be interpreted as a gauge group arising from a singularity which needed to absorb some branes to cancel its pathologies. Well, this picture is very manifest in type I' string theory (a T-dual of type I string theory with D8-branes and O8-planes).
Summary: unity in its diversity
To summarize, gauge symmetries emerge from string/M/F-theory in many ways that we used to consider very different. However, all these mechanisms are connected to one framework in which they may "melt" into each other or be proven equivalent i.e. dual to each other. The understanding of all these diverse origins of gauge symmetries as well as the relationships between them brings us unusual feelings of intellectual satisfaction – and it has become as necessary for a theoretical physicist's proper understanding of these gauge degrees of freedom as some basic introductions to gauge theory.
So if you have some prejudices against any of these things, I recommend you to be humble and learn, learn, learn for a while. After some time, some of you will hopefully be able to see that all these things make a perfect sense.
And that's the memo.
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