Basic scenarios of string theory
Gordon has assured me that (almost) no non-expert has understood advanced basics of string phenomenology, despite dozens if not hundreds of blog entries about these topics that have been written on this blog during the years.
So I would like to be a little bit (but not too much) more comprehensible and address this text to some of the readers who have never studied any string theory at a technical level but who have some idea about quantum field theory and the concept of extra dimensions. I will review the basic vacua of string/M-theory in 10-11 spacetime dimensions and their basic relationships.
It turns out that almost each of them may give rise to a particular, idiosyncratic class of realistic universes with 3+1 large dimensions that we may inhabit.
So what kinds of string theory are there?
First, I must say that this very question is obsolete if it is phrased in this way. In the 1980s, people would be talking about "different string theories" (and non-experts are doing so even today). But in the mid 1990s, string theorists have understood that all the "different string theories" are actually just environments in a single theory.
You should imagine that string/M-theory is a single theory with many "fields" and similar objects and if you tune these fields (think about scalar fields) to various values, you will obtain a universe with properties that are described by what used to be called "a particular string theory". And string theory dictates how these points in the configuration space or "landscape" are connected, too. For example, the number and types of low-energy fields depend on the point in the configuration space, too.
Since the 1990s, we know that there is just one string theory and not many.
Higher-dimensional vacua
Fine. But we may still use the vocabulary of the 1980s for a little while. What string theories do we have if we don't allow any compactification? There are six of them: all of them live in spacetime whose dimension is either 10 (string theory) or 11 (M-theory).
String theory was originally born in the late 1960s and within a few years, people understood that the right spacetime had 26 dimensions. But this was a different, older, not quite healthy string theory, the so-called "bosonic string theory". This theory predicted that there were no fermions which is a bad starting point to describe our reality with lots of fermions. Even more seriously, bosonic string theory did include a (bosonic) tachyon, a particle that naively moves faster than light (but it's surely not a neutrino) and that makes the spacetime of bosonic theory unstable (much like the "h=0" point of the Higgs field is unstable).
So I will not treat bosonic string theory as a part of the genuine, fully consistent string theory (although there are interesting papers that describe hypothetical dynamical processes that may change a 26-dimensional spacetime to a 10-dimensional one or vice versa). We will only focus on the string theories in 10-11 dimensions and assume that the 26-dimensional "theory" is just a toy model, not a fully consistent one, to learn the actual theory that matters and works, namely superstring/M-theory.
The six theories in the maximum dimension
The list of uncompactified string theories is short: it only contains 6 entries:
Those five string theories had been found by the mid 1980s; M-theory was added in the mid 1990s and during the last 15 years, the list of uncompactified string/M-theories hasn't changed. (F-theory may also be added there, as a formally 12-dimensional theory; but I will treat it as another description of type IIB string theory.)
It's plausible that we actually know the full list of uncompactified "master" theories; it's equally plausible that we're still missing something. At any rate, it's pretty clear that the best theoretical physicists in the year 3000 will know about this basic structure of the world. The maximally decompactified string theories belong among the most fundamental discoveries that are close to the origin of the universe that people have ever found and they will stay with us.
Only hopeless crackpots may have fantasies that a good physicist may totally forget about these key structures of theoretical physics sometime in the future. The entries on the list above may look like complicated theories; but in some invariant sense, they're the simplest or most symmetric "cousins of the real world" one may think of.
Differences and relationships between the 5+1 guys
The old 26-dimensional bosonic string theory had to be upgraded to a "supersymmetric" version in the 1970s. This event, leading the physicists to construct an entirely new theory, incorporated new fermions on the world sheet – which is the name for the 2-dimensional history or propagating strings. It also added a new symmetry, the world sheet supersymmetry, into the world sheet. When those things are done properly, the right theory predicts a 10-dimensional spacetime.
There are various technicalities such as the GSO projections which eliminate tachyons (and other equally "odd") states from the spectrum and that force us to allow the fermions on the world sheet to be either periodic or antiperiodic (which is a step that gives birth to fermions in the spacetime, too). A simple GSO projection may be done in two ways because it picks the chirality of a spacetime spinor. There are two supersymmetric ways to impose the GSO projection: you either require the opposite chiralities for the left-moving excitations along the string and the right-moving ones, or the same chiralities. In these two cases, you get two inequivalent theories, type IIA and type IIB string theory, respectively.
In 1985, the Princeton String Quartet realized a cute thing: one can give a birth to a new hybrid string theory that uses the old bosonic 26-dimensional string maths for the left-moving excitations and the 10-dimensional spacetime for the right-moving ones (or vice versa). Imagine that you're a string and your thoughts are going around your head either in the clockwise or counter-clockwise direction. In the first case, the thoughts think that you live in 26 dimensions; in the latter case, you think that you live in 10 dimensions. ;-) Remarkably enough, this is totally possible and leads to a fully consistent theory (largely because the left-moving and right-moving degrees of freedom on a string are almost decoupled from each other, except for some "global" conditions that correlate their properties).
The difference between the dimensionalities is 16 and these 16 "unmatched" dimensions have to be treated consistently. It turns out that they must be compactified on a 16-dimensional torus associated with an "even self-dual lattice". (It's because the dimensions have to be "purely left-moving" which means that the winding number has to be equal to the momentum; but the momenta and windings generally live in lattices that are dual to each other: here, they must coincide.) There are two such lattices in 16 dimensions – the weight lattices of the gauge groups spin(32)/Z_2 and E_8 x E_8, respectively. So two new "heterotic string theories" may be discovered in this way; an alternative, fermionic description (and perhaps others) may also be given for the 16 unmatched dimensions.
We have described 2+2 string theories so far – two type II theories and two heterotic string theories. The last fifth theory is type I string theory. It is related to type IIB string theory but the strings are made unoriented. This is not possible in type IIA string theory, at least without introducing special places in the spacetime, because the string looks different in the two directions (different spacetime chirality) so it automatically carries an arrow. (Type IIA strings may be made unoriented if we include odd-co-dimensional orientifold planes.)
In type IIB string theory, however, you can make closed strings unoriented. It turns out that to make the theory consistent, you also have to add unoriented open strings. Moreover, the endpoints of these new open strings must carry one of 32 possible self-adjoint "colors" or "colorful quarks". The gauge group is SO(32), at least at the level of Lie algebra; SO(32) is what almost all physicists "sloppily" say instead of spin(32)/Z_2. Note that the dimension of this group is 32 x 31/2 = 496, much like the dimension of E_8 x E_8 which is also 248+248 = 496. The dimension of each group is one of the numerous invariants that happen to have exactly the right miraculous value for all the anomalies to vanish (this coincidence would really look like a miracle if there were no string theory; the existence and internal logic of string theory "explains" this miracle).
Type I string theory with the SO(32) gauge group was found to be super duper consistent (and viable for semi-realistic model building) by Green and Schwarz in 1984 and this discovery sparked the first superstring revolution.
The last entry, M-theory (born at the beginning of 1995), is 11-dimensional and most closely linked to type IIA string theory and to an effective field theory originally found independently of strings, the 11-dimensional supergravity (from the late 1970s), which is the long-distance limit of M-theory.
Basic relationship between the six theories
In the text above, I sketched the reasons why you found exactly this collection of six maximally decompactified theories. And that's how people originally found them. But one may actually find each of them by "squeezing and transforming the Universe" described by any of its siblings. There are many relationships between the theories. In fact, the theories at various values of "string couplings" and various "compactification manifolds" are all equivalent to each other. We say that they are "dual" and the highly surprising equivalences are known as "dualities".
What are they? The following basic dualities are enough to connect all 6 uncompactified string/M-theories into a single structure but they are far from being the only dualities that exist.
M-theory compactified on a circle gives you type IIA string theory. If one of the 11 spacetime dimensions of M-theory is made periodic, the remaining 10 dimensions form a 10-dimensional spacetime and physics in this 10-dimensional spacetime is identical to type IIA string theory with the "string coupling" (strength of interactions between the strings) equal to a positive power of the radius of the circle.
So if the circle is infinitesimal, you get type IIA string theory at a tiny coupling – free type IIA string theory.
Also, if you compactify type IIA string theory on another circle of radius R, you will get the same thing as if you compactify type IIB string theory on a circle of radius 1/R, assuming you choose the "string units". We say that type IIA and type IIB are T-dual to one another. The class of theories obtained by compactifying either of them on a circle (or a torus) is the same class, regardless of the A/B letter. Of course, it's almost always better to use the description for which the radius R or 1/R is greater than 1 (times the string length): the idea is that the compactification on a very tiny circle (or another manifold), much smaller than 1, produces difficult new physics analogous to "strongly coupled physics".
So far, we have connected M-theory, type IIA, and type IIB into a single network. See Dualities vs singularities to check how these three descriptions exactly cover the whole moduli space of maximally supersymmetric vacua (M-theory or type II theory compactified on tori: type II means type IIA or type IIB). I forgot to say, all these three theories have a maximal amount of supersymmetry, 32 supercharges.
The remaining three uncompactified theories (type I and two heterotic ones) only have 16 supercharges, i.e. one-half of the maximum number, and they may only exist in 10 infinite dimensions (because spinors in 10 dimensions have at least 16 real components while in 11, they start at 32). They're connected to the maximally supersymmetric theories, too.
If you want a connection in the maximum dimension, take M-theory and choose one of its dimensions (the 11th dimension, to be clear) to be a line interval. Physicists use a clever notation S^1/Z_2 for a line interval. Why it is the same thing? Take a circle and define a Z_2 symmetry, the left-right reflection, to derive the "equivalence classes" under this reflection. This reflection has two fixed points (the upper and lower "corner" ;-) of the circle) which become the endpoints of the line interval; the left and right half of the circle get identified with each other by the Z_2 orbifold operation so there's only one interval (half a circle) left.
This is a helpful way to describe a line interval because theories defined on such a line interval may be (at least formally) obtained as some Z_2 operation applied on theories defined on a circle.
What kind of physics you get if you compactify M-theory on a line interval? It was actually one of the last relationships that were understood in 1995. Edward Witten and Petr Hořava realized that some special dynamics has to take place on the two endpoints of the line interval (which become two 9+1-dimensional boundaries of the 10+1-dimensional world as long as you Cartesian-multiply the line interval by the 10-dimensional spacetime). They figured out that anomaly cancellation implies that an E_8 gauge supermultiplet must live on such a boundary. There are two boundaries, so there are two E_8 groups. Two independent E_8 gauge groups per point in 9+1 dimensions means that there is an E_8 x E_8 gauge group in 10 dimensions, just like for one of the heterotic strings!
It's not just the gauge group that comes out correctly: all features of the spectrum and interactions works, too. So by compactifying M-theory on a line interval, you get the 10-dimensional E_8 x E_8 heterotic string theory in the very same way as you got type IIA string theory when the line interval was replaced by a circle. Also, the length of the line interval is an increasing function of the heterotic string coupling once again. In fact, the power law is the same in some clever units.
Recall that when we got from M-theory to type IIA string theory, we continued with a T-duality to go from type IIA to type IIB. We can use a T-duality again: heterotic E_8 x E_8 string theory on a circle is equivalent (T-dual) to heterotic spin(32)/Z_2 string theory on a circle. Both theories have lots of "Wilson lines" (monodromies of the gauge field around the circular dimensions) that generically break the original gauge group (any of them) to U(1)^{16}: note that the rank is 16. When you choose the Wilson lines on both sides of the T-duality correctly, you get physically identical vacua: it's really related to the fact that there is a unique 17+1-dimensional Minkowskian even self-dual lattice: you get the same lattice by adding Gamma(1,1) to the lattice of E_8 x E_8; or to the lattice of spin(32)/Z_2.
So we have already connected five of the six string theories: only type I string theory is left. But note that type I has the gauge group SO(32) which really should be spin(32)/Z_2 if you were more accurate and it's the same gauge group as the gauge group of the heterotic spin(32)/Z_2 string. It's no coincidence: as Polchinski and Witten pointed out, these two theories are actually S-dual to each other. It means that one of them with the string coupling "g" is the same theory as the other one with the string coupling "1/g".
By the way, type IIB string theory is S-dual to itself: you may revert its "g" to "1/g" and get the same theory.
Fine. At any rate, I have connected all six theories. There are actually additional ways how to connect them. The next important example that requires one to understand some complex geometry is the K3/M-theory duality. If you compactify M-theory on a four-real-dimensional K3-manifold, which is the second simplest four-dimensional manifold solving Einstein's equations, you get the same thing as heterotic string theory on a three-dimensional torus! This duality may be compactified on additional circular dimensions.
You're invited to study string theory textbooks if you're intrigued by this duality, by the previous ones, or by additional dualities that I haven't even mentioned. ;-)
Compactifying to get realistic models
Now, let's get closer to the reality. How do you obtain a 3+1-dimensional Universe from these 6 maximally dimensional vacua of string/M-theory? Well, even though the progress with type I string theory in 1984 (which contained gravity as well as gauge fields and fermions) was what started the first superstring revolution, type I string theory is actually the only one whose realistic vacua haven't been constructed (and probably don't exist).
The gauge group, SO(32), is just too awkward a starting point for realistic Yang-Mills model building. But you may actually find attempts in this direction in literature, too.
When I said this disappointing thing about type I string theory, let me mention that type I isn't really independent of the remaining 5. And I don't mean just the S-duality with the heterotic string. Type I string theory is related to another theory, type IIB string theory, because type I may be interpreted as "type IIB with extra objects". The objects are a spacetime-filling orientifold (a kind of mirror for strings reverting their orientation), 16 spacetime-filling D9-branes, and their mirror images.
The spin(32)/Z_2 heterotic strings are also bad for phenomenology, because of the same unrealistic gauge group. But they're close to the other heterotic string theory whose success compensates the failure of spin(32)/Z_2.
Good. So let's focus on the 4 remaining string/M-theories. Each of them may produce realistic compactifications; M-theory can do so in 2 very different ways. One of the compactifications could be hosting our Universe. Because of the dualities are other relationships and transitions, our Universe could actually be described by a stringy compactification in more than one way. The more dimensions you compactify, the higher number of connections, equivalences, and transitions you get that make the "landscape" of string/M-theory interconnected.
M-theory: G_2 holonomy manifolds
M-theory has 11 dimensions and we need 4. It means that 7 of them have to be compactified. However, 7 is an odd number and physics in odd-dimensional spaces is notoriously left-right symmetric. For example, there is no way to distinguish left-handed and right-handed spinors. (Note that O(2n+1) is just SO(2n+1) times Z_2 because the minus unit matrix has a negative determinant and gets you to the other component of O(2n+1).) Consequently, if you compactify odd-dimensional M-theory on an odd-dimensional manifold, the resulting even-dimensional (four-dimensional) theory won't be able to break the CP-symmetry (and maybe not even P-symmetry: I could refine this comment within a few minutes but I suspect no one cares haha).
So this seemingly kills M-theory as a mother of realistic vacua. However, there is a loophole. If the 7-dimensional manifold is singular, the P- and CP-violating objects may happily live at the singularities. This is what people such as Bobby Acharya and Edward Witten realized around the beginning of this millennium.
Those 7 compact dimensions of M-theory produce a nice shape, the so-called G_2 holonomy manifold. It is Ricci-flat (satisfies Einstein's vacuum equations) and preserves 4 supercharges out of 32 (which is what you need for nice phenomenology) which is 1/8 as a relative fraction. Here, G_2 is the symmetry group ("automorphism group", if you like fancy names) of the octonions (which have 7 different imaginary units: those 7 units are identified with 7 tangential directions in the G_2 holonomy manifold): it's the smallest exceptional Lie group (unlike E_8 which is the largest one).
So one (modern) class of string/M-theory realistic compactifications involves M-theory compactified on a 7-dimensional manifolds which must be singular and all the chiral fermions etc. must live at the singularities. It's likely that all things except for gravity live there: only supergravity lives in the bulk of M-theory, after all. But in principle, there may be several singularities (and e.g. different generations of fermions may live at different places of the extra dimensions) and also singularities that are not point-like but that are interpolating between point-like singularities, and so on.
Some phenomenologists such as Gordon Kane became enthusiastic believers in the M-theory scenario. Various "split" spectra of superpartners that are consistent with the null LHC observations so far follow from those models, much like "axiverse" (lots of axions with various masses) and other things, aside from the Standard-Model-like physics coupled to gravity.
Heterotic string vacua and heterotic M-theory
However, the first realistic vacua obtained from string theory in the mid 1980s were different: all Standard Model (and similar) fields live in all the 10 dimensions, much like gravity. Nothing is localized. Heterotic E_8 x E_8 string theory is left-right asymmetric already in 10 dimensions. When you compactify it on a 6-dimensional Calabi-Yau manifold, which is the 6-dimensional counterpart of the G_2 holonomy manifold above, you preserve the left-right asymmetry (chirality).
I have mentioned that M-theory on a line interval is equivalent to heterotic E_8 x E_8 string theory, as shown by Hořava and Witten: the more strongly coupled the heterotic string is, the longer the line interval becomes. So the heterotic E_8 x E_8 string vacua may also be reinterpreted as "heterotic (or Hořava-Witten) M-theory" vacua with two end-of-the-world branes. In this case, gravity lives in all spacetime dimensions (it always does) but all the other fields live in 10 dimensions at the boundaries of the world only.
Typically, the whole Standard Model lives in one of the E_8 groups only, on one endpoint of the world. The other "evil" E_8 is usually broken differently and may be responsible for SUSY breaking by gluino condensation and other things. These heterotic models are the oldest class and they're most naturally compatible with "grand unification" and "field-theoretical models of physics beyond the Standard Model" (because nothing is really localized, except for being at the co-dimension-one end of the world, so you may get rid of the extra dimensions at the very beginning, if you wish).
My ordering of the classes is chaotic. Note that in the M-theory on G_2 holonomy manifold case, the Standard Model fermions live on co-dimension-seven loci of the spacetime: in the heterotic vacua, it's co-dimension zero or one. In this sense, the first two classes I started with are "opposite extremes".
Note that out of the promised 4 maximally dimensional starting points for realistic vacua, I have exhausted 2: M-theory and heterotic E_8 x E_8 strings. (M-theory was used twice: either on G_2 holonomy singular manifolds or on line interval times a six-dimensional Calabi-Yau manifold.)
There are 2 scenarios left: type IIA and type IIB string theory.
Type IIA braneworlds
Type IIA (and also type IIB) has too much supersymmetry. Even if you compactify it on a Calabi-Yau manifold, you would get 8 supercharges which is too much: N=2 supersymmetry in four dimensions. This amount of residual supersymmetry couldn't be broken, wouldn't allow any P/CP-violation, and would lead to huge and experimentally ruled out multiplets, anyway. So one has to reduce the supercharges by some "extra objects" to one-half, too.
The relevant objects are D-branes and orientifold planes. Most typically, realistic type IIA vacua are "braneworlds" with numerous D6-branes stretched in various (triple) directions of the compactified 6 dimensions. Unoriented strings are helpful so the compactification also often includes orientifold planes.
One may get pretty realistic models even if you start from toroidal compactifications. These models have also been studied for a decade or so. Barton Zwiebach's textbook allows undergrads to build realistic models of this sort. It's kind of amazing that undergrads may already learn everything that is needed to work with a theory of everything according to all early 21st century requirements.
These "type IIA intersecting braneworlds" are usually not compatible with "grand unification". Instead, each stack of N D-branes produces its own U(N)-like group (it becomes symplectic or orthogonal if the stack sits on an orientifold) which is independent from the U(N)-like groups of other stacks. So the "braneworlds" are on the opposite extreme side from heterotic strings when it comes to grand unification. Heterotic strings are as unified as you can get: brane worlds allow you to make the factors in your gauge groups as independent as you want. No one knows which of the two paradigms is actually preferred by Nature Herself (gauge coupling unification is a piece of circumstantial evidence that the idea of grand unification in general could be right).
Such braneworld models have various favorable properties. For example, they may naturally explain why the electron mass is so much smaller than the top quark mass: the Yukawa couplings behind the low masses may be produced by "world sheet instantons" (of a disk shape, actually a triangle) which may explain why they're exponentially small. Of course, ideally, one would like to find a stringy model that has all the great properties. ;-)
If you reinterpret type IIA as M-theory on a circle, you may get a dual G_2-holonomy manifold out of some type IIA vacua. The D6-brane of type IIA itself becomes a "Kaluza-Klein monopole" – a particular purely gravitational solution of 11-dimensional general relativity whose metric tensor encodes the topologically non-trivial electromagnetic field of a magnetic monopole (one that needs a Dirac string), using a dictionary between electromagnetism and compactified gravity from the Kaluza-Klein theory. So an intersection of several D6-branes produces a very particular 7-dimensional G_2-holonomy geometry, and so on.
There are other relationships between the vacua from different classes. But I already want to get to the last scenario:
F-theory: a fancy type IIB toolkit
If you enjoy fancy overpopulation of dimensions, F-theory may be interpreted as a 12-dimensional theory, the Father theory, where "F" stands for "father", "fiber", "frankenstein" (because of geometric engineering), "fafa", or "vaFa", according to your taste. ;-) F-theory is related to type IIB string theory in a similar but not identical way to the way how M-theory is related to type IIA. It's important that the two cases are not identical or isomorphic (otherwise we wouldn't invent so many redundant terms): much like the males and females (mothers and fathers: note that the F/M get interchanged), M-theory and F-theory structurally differ. They have very different roles and detailed logic.
The growing dimension of type IIA may become infinite, producing an independently viable "mother" theory in 11 dimensions. This is not the case of F-theory.
F-theory has 12 dimensions but 2 of them have to be compactified on a two-dimensional torus (whose size is unphysical), otherwise the father theory doesn't make any sense. Once 2 dimensions are compactified, you're down to a 10-dimensional theory, type IIB string theory, and the shape of the torus knows about the two scalar fields of type IIB string theory, the axion-dilaton field, which are symmetric under the SL(2,Z) S-duality group (which includes the "g" goes to "1/g" map mentioned previously).
To get realistic 4-dimensional vacua, you formally need to compactify F-theory on an 8-dimensional manifold: 12-8 = 4. However, out of this 8-dimensional manifold, 2 dimensions must be reserved for the toroidal "fiber" because F-theory must always compactify two of its dimensions on a 2-torus. It follows that the 8-dimensional manifold used for F-theory must be "elliptically fibered", which is just a different way of saying that it is a 6-dimensional manifold with a 2-dimensional torus attached to each point.
However, what's f(ancy) about F-theory relatively to "ordinary type IIB string theory" is that the 2-torus may have a shape that depends on the location on the 6-dimensional "base space". And in fact, if you go around special paths in this 6-dimensional base space, the 2-torus may be translated to itself in a nontrivial way, labeled by an SL(2,Z) transformation!
In some sense, we're talking about special 8-dimensional manifolds here (even though only 6 of them may be "large enough to be seen"). 8 compactified dimensions is the maximum number you find in stringy phenomenology. A higher number of dimensions means "more complex geometry". So the people who study F-theory compactifications are the best mathematicians (geometers) among the string phenomenologists (even though a comparable amount of mathematics is needed for the 6-dimensional and 7-dimensional manifolds, too).
In recent 5 years, F-theory phenomenology began to thrive as a very healthy subject because of Cumrun Vafa (the Father of F-theory), his collaborators, and several other bold physicists. In this case, the Standard Model fields also live on singular places of the compactification manifold. But in this case it is really a rule rather than exception that there are different loci where different fields like to live. Sometimes they intersect, and so on. Particles are associated with loci of "intermediate dimensions" and there are just many ways how to stretch such loci and how to make them intersect.
What's the difference between singularities within the F-theory compactification manifold and e.g. D6-branes in type IIA intersecting braneworlds? The difference is that D-branes, while inducing some potentially singular values of the fields, are still "objects inserted into a pre-existing space" and the pre-existing space may be non-singular. On the other hand, the singularities in the 8-dimensional Calabi-Yau manifolds of F-theory or 7-dimensional singular G_2 holonomy manifolds of M-theory are linked to the very shape of the whole manifold: the singularity makes the space look singular even outside the location of the singularity; D-branes may be viewed as strictly localized singular intruders. (Nevertheless, D-branes, especially those with a low co-dimension, distort the surrounding space "qualitatively" as well, so there's another relationship between singularities and D-branes, especially D7-branes.)
Under Vafa, Heckman, Marsano, Saulina, and others, F-theory model building became a hunt for very special models that have welcome properties (that may agree with observations or some of the qualitative patterns found in them) and lots of things about phenomenologically attractive properties of these vacua (and some potential problems) has been written.
However, F-theory model building has yet another, evil face. F-theory on 8-dimensional manifolds is also the setup that was used to produce the infamous 10^{500} or so vacua in the "landscape". So F-theory became the main propaganda tool of the anthropic ideologues. Because F-theory allows us to play with 8 dimensions, kind of, which is the highest number of dimension, we find the maximized number of possible ways to compactify them, too. That's why the landscape of the (generic messy) F-theory compactifications is so large (especially when you include the fluxes etc.).
Nevertheless, the anthropic ideologues have never justified their claim that just because this class of vacua (F-theory KKLT-like vacua) has the highest number of elements, it is the most likely one to be realized in Nature. I am convinced that this implication is invalid. One must look for promising vacua in all the groups mentioned above. Because F-theory vacua have also helped to promote the pathological meme that it makes no sense to continue to search for the right compactification, you may also say that "F" stands for "ucked-up". (Its non-anthropic father Vafa would violently disagree.)
F-theory vacua (but not only them) are also a natural starting point for the string theory incarnations of Randall-Sundrum warped geometries and other things.
It's important to note that in various classes of vacua that have been discussed, different fields may be confined to particular branes or singularities whose dimensionalities may differ. It is not true that all non-gravitational fields and particles have to live "at the same place" in extra dimensions. In fact, the separation (or "segregation") of some particle species along extra dimensions may explain why some of their interactions are weak, and so on. Especially when you're near "intermediate dimensions" in the compactification manifold, there are many choices.
Summary
I've tried to cover all four or five main classes of realistic string theory compactifications, some of their characteristic features, their links to the 10- or 11-dimensional master theories, the relationships between the master theories, and other things. I don't believe that too many readers who were previously unaware of these fascinating achievements will understand everything I wrote but it is still conceivable that the concepts explained above will help some people to get further in their understanding where the technical properties of the Cosmos surrounding us come from.
When you're a layman and you happen to attend a string phenomenology talk, you may at least ask more sophisticated would-be technical questions. Instead of "is the string really vibrating faster than light, proving that 42 is the answer to everything?", you may ask "can you find a G_2 holonomy manifold associated with uplifting these D6-branes in type IIA"? :-)
Gordon has assured me that (almost) no non-expert has understood advanced basics of string phenomenology, despite dozens if not hundreds of blog entries about these topics that have been written on this blog during the years.
So I would like to be a little bit (but not too much) more comprehensible and address this text to some of the readers who have never studied any string theory at a technical level but who have some idea about quantum field theory and the concept of extra dimensions. I will review the basic vacua of string/M-theory in 10-11 spacetime dimensions and their basic relationships.
It turns out that almost each of them may give rise to a particular, idiosyncratic class of realistic universes with 3+1 large dimensions that we may inhabit.
So what kinds of string theory are there?
First, I must say that this very question is obsolete if it is phrased in this way. In the 1980s, people would be talking about "different string theories" (and non-experts are doing so even today). But in the mid 1990s, string theorists have understood that all the "different string theories" are actually just environments in a single theory.
You should imagine that string/M-theory is a single theory with many "fields" and similar objects and if you tune these fields (think about scalar fields) to various values, you will obtain a universe with properties that are described by what used to be called "a particular string theory". And string theory dictates how these points in the configuration space or "landscape" are connected, too. For example, the number and types of low-energy fields depend on the point in the configuration space, too.
Since the 1990s, we know that there is just one string theory and not many.
Higher-dimensional vacua
Fine. But we may still use the vocabulary of the 1980s for a little while. What string theories do we have if we don't allow any compactification? There are six of them: all of them live in spacetime whose dimension is either 10 (string theory) or 11 (M-theory).
String theory was originally born in the late 1960s and within a few years, people understood that the right spacetime had 26 dimensions. But this was a different, older, not quite healthy string theory, the so-called "bosonic string theory". This theory predicted that there were no fermions which is a bad starting point to describe our reality with lots of fermions. Even more seriously, bosonic string theory did include a (bosonic) tachyon, a particle that naively moves faster than light (but it's surely not a neutrino) and that makes the spacetime of bosonic theory unstable (much like the "h=0" point of the Higgs field is unstable).
So I will not treat bosonic string theory as a part of the genuine, fully consistent string theory (although there are interesting papers that describe hypothetical dynamical processes that may change a 26-dimensional spacetime to a 10-dimensional one or vice versa). We will only focus on the string theories in 10-11 dimensions and assume that the 26-dimensional "theory" is just a toy model, not a fully consistent one, to learn the actual theory that matters and works, namely superstring/M-theory.
The six theories in the maximum dimension
The list of uncompactified string theories is short: it only contains 6 entries:
- type I string theory with spin(32)/Z_2 gauge group
- type IIA string theory
- type IIB string theory
- heterotic E_8 x E_8 string theory
- heterotic spin(32)/Z_2 string theory
- M-theory
Those five string theories had been found by the mid 1980s; M-theory was added in the mid 1990s and during the last 15 years, the list of uncompactified string/M-theories hasn't changed. (F-theory may also be added there, as a formally 12-dimensional theory; but I will treat it as another description of type IIB string theory.)
It's plausible that we actually know the full list of uncompactified "master" theories; it's equally plausible that we're still missing something. At any rate, it's pretty clear that the best theoretical physicists in the year 3000 will know about this basic structure of the world. The maximally decompactified string theories belong among the most fundamental discoveries that are close to the origin of the universe that people have ever found and they will stay with us.
Only hopeless crackpots may have fantasies that a good physicist may totally forget about these key structures of theoretical physics sometime in the future. The entries on the list above may look like complicated theories; but in some invariant sense, they're the simplest or most symmetric "cousins of the real world" one may think of.
Differences and relationships between the 5+1 guys
The old 26-dimensional bosonic string theory had to be upgraded to a "supersymmetric" version in the 1970s. This event, leading the physicists to construct an entirely new theory, incorporated new fermions on the world sheet – which is the name for the 2-dimensional history or propagating strings. It also added a new symmetry, the world sheet supersymmetry, into the world sheet. When those things are done properly, the right theory predicts a 10-dimensional spacetime.
There are various technicalities such as the GSO projections which eliminate tachyons (and other equally "odd") states from the spectrum and that force us to allow the fermions on the world sheet to be either periodic or antiperiodic (which is a step that gives birth to fermions in the spacetime, too). A simple GSO projection may be done in two ways because it picks the chirality of a spacetime spinor. There are two supersymmetric ways to impose the GSO projection: you either require the opposite chiralities for the left-moving excitations along the string and the right-moving ones, or the same chiralities. In these two cases, you get two inequivalent theories, type IIA and type IIB string theory, respectively.
In 1985, the Princeton String Quartet realized a cute thing: one can give a birth to a new hybrid string theory that uses the old bosonic 26-dimensional string maths for the left-moving excitations and the 10-dimensional spacetime for the right-moving ones (or vice versa). Imagine that you're a string and your thoughts are going around your head either in the clockwise or counter-clockwise direction. In the first case, the thoughts think that you live in 26 dimensions; in the latter case, you think that you live in 10 dimensions. ;-) Remarkably enough, this is totally possible and leads to a fully consistent theory (largely because the left-moving and right-moving degrees of freedom on a string are almost decoupled from each other, except for some "global" conditions that correlate their properties).
The difference between the dimensionalities is 16 and these 16 "unmatched" dimensions have to be treated consistently. It turns out that they must be compactified on a 16-dimensional torus associated with an "even self-dual lattice". (It's because the dimensions have to be "purely left-moving" which means that the winding number has to be equal to the momentum; but the momenta and windings generally live in lattices that are dual to each other: here, they must coincide.) There are two such lattices in 16 dimensions – the weight lattices of the gauge groups spin(32)/Z_2 and E_8 x E_8, respectively. So two new "heterotic string theories" may be discovered in this way; an alternative, fermionic description (and perhaps others) may also be given for the 16 unmatched dimensions.
We have described 2+2 string theories so far – two type II theories and two heterotic string theories. The last fifth theory is type I string theory. It is related to type IIB string theory but the strings are made unoriented. This is not possible in type IIA string theory, at least without introducing special places in the spacetime, because the string looks different in the two directions (different spacetime chirality) so it automatically carries an arrow. (Type IIA strings may be made unoriented if we include odd-co-dimensional orientifold planes.)
In type IIB string theory, however, you can make closed strings unoriented. It turns out that to make the theory consistent, you also have to add unoriented open strings. Moreover, the endpoints of these new open strings must carry one of 32 possible self-adjoint "colors" or "colorful quarks". The gauge group is SO(32), at least at the level of Lie algebra; SO(32) is what almost all physicists "sloppily" say instead of spin(32)/Z_2. Note that the dimension of this group is 32 x 31/2 = 496, much like the dimension of E_8 x E_8 which is also 248+248 = 496. The dimension of each group is one of the numerous invariants that happen to have exactly the right miraculous value for all the anomalies to vanish (this coincidence would really look like a miracle if there were no string theory; the existence and internal logic of string theory "explains" this miracle).
Type I string theory with the SO(32) gauge group was found to be super duper consistent (and viable for semi-realistic model building) by Green and Schwarz in 1984 and this discovery sparked the first superstring revolution.
The last entry, M-theory (born at the beginning of 1995), is 11-dimensional and most closely linked to type IIA string theory and to an effective field theory originally found independently of strings, the 11-dimensional supergravity (from the late 1970s), which is the long-distance limit of M-theory.
Basic relationship between the six theories
In the text above, I sketched the reasons why you found exactly this collection of six maximally decompactified theories. And that's how people originally found them. But one may actually find each of them by "squeezing and transforming the Universe" described by any of its siblings. There are many relationships between the theories. In fact, the theories at various values of "string couplings" and various "compactification manifolds" are all equivalent to each other. We say that they are "dual" and the highly surprising equivalences are known as "dualities".
What are they? The following basic dualities are enough to connect all 6 uncompactified string/M-theories into a single structure but they are far from being the only dualities that exist.
M-theory compactified on a circle gives you type IIA string theory. If one of the 11 spacetime dimensions of M-theory is made periodic, the remaining 10 dimensions form a 10-dimensional spacetime and physics in this 10-dimensional spacetime is identical to type IIA string theory with the "string coupling" (strength of interactions between the strings) equal to a positive power of the radius of the circle.
So if the circle is infinitesimal, you get type IIA string theory at a tiny coupling – free type IIA string theory.
Also, if you compactify type IIA string theory on another circle of radius R, you will get the same thing as if you compactify type IIB string theory on a circle of radius 1/R, assuming you choose the "string units". We say that type IIA and type IIB are T-dual to one another. The class of theories obtained by compactifying either of them on a circle (or a torus) is the same class, regardless of the A/B letter. Of course, it's almost always better to use the description for which the radius R or 1/R is greater than 1 (times the string length): the idea is that the compactification on a very tiny circle (or another manifold), much smaller than 1, produces difficult new physics analogous to "strongly coupled physics".
So far, we have connected M-theory, type IIA, and type IIB into a single network. See Dualities vs singularities to check how these three descriptions exactly cover the whole moduli space of maximally supersymmetric vacua (M-theory or type II theory compactified on tori: type II means type IIA or type IIB). I forgot to say, all these three theories have a maximal amount of supersymmetry, 32 supercharges.
The remaining three uncompactified theories (type I and two heterotic ones) only have 16 supercharges, i.e. one-half of the maximum number, and they may only exist in 10 infinite dimensions (because spinors in 10 dimensions have at least 16 real components while in 11, they start at 32). They're connected to the maximally supersymmetric theories, too.
If you want a connection in the maximum dimension, take M-theory and choose one of its dimensions (the 11th dimension, to be clear) to be a line interval. Physicists use a clever notation S^1/Z_2 for a line interval. Why it is the same thing? Take a circle and define a Z_2 symmetry, the left-right reflection, to derive the "equivalence classes" under this reflection. This reflection has two fixed points (the upper and lower "corner" ;-) of the circle) which become the endpoints of the line interval; the left and right half of the circle get identified with each other by the Z_2 orbifold operation so there's only one interval (half a circle) left.
This is a helpful way to describe a line interval because theories defined on such a line interval may be (at least formally) obtained as some Z_2 operation applied on theories defined on a circle.
What kind of physics you get if you compactify M-theory on a line interval? It was actually one of the last relationships that were understood in 1995. Edward Witten and Petr Hořava realized that some special dynamics has to take place on the two endpoints of the line interval (which become two 9+1-dimensional boundaries of the 10+1-dimensional world as long as you Cartesian-multiply the line interval by the 10-dimensional spacetime). They figured out that anomaly cancellation implies that an E_8 gauge supermultiplet must live on such a boundary. There are two boundaries, so there are two E_8 groups. Two independent E_8 gauge groups per point in 9+1 dimensions means that there is an E_8 x E_8 gauge group in 10 dimensions, just like for one of the heterotic strings!
It's not just the gauge group that comes out correctly: all features of the spectrum and interactions works, too. So by compactifying M-theory on a line interval, you get the 10-dimensional E_8 x E_8 heterotic string theory in the very same way as you got type IIA string theory when the line interval was replaced by a circle. Also, the length of the line interval is an increasing function of the heterotic string coupling once again. In fact, the power law is the same in some clever units.
Recall that when we got from M-theory to type IIA string theory, we continued with a T-duality to go from type IIA to type IIB. We can use a T-duality again: heterotic E_8 x E_8 string theory on a circle is equivalent (T-dual) to heterotic spin(32)/Z_2 string theory on a circle. Both theories have lots of "Wilson lines" (monodromies of the gauge field around the circular dimensions) that generically break the original gauge group (any of them) to U(1)^{16}: note that the rank is 16. When you choose the Wilson lines on both sides of the T-duality correctly, you get physically identical vacua: it's really related to the fact that there is a unique 17+1-dimensional Minkowskian even self-dual lattice: you get the same lattice by adding Gamma(1,1) to the lattice of E_8 x E_8; or to the lattice of spin(32)/Z_2.
So we have already connected five of the six string theories: only type I string theory is left. But note that type I has the gauge group SO(32) which really should be spin(32)/Z_2 if you were more accurate and it's the same gauge group as the gauge group of the heterotic spin(32)/Z_2 string. It's no coincidence: as Polchinski and Witten pointed out, these two theories are actually S-dual to each other. It means that one of them with the string coupling "g" is the same theory as the other one with the string coupling "1/g".
By the way, type IIB string theory is S-dual to itself: you may revert its "g" to "1/g" and get the same theory.
Fine. At any rate, I have connected all six theories. There are actually additional ways how to connect them. The next important example that requires one to understand some complex geometry is the K3/M-theory duality. If you compactify M-theory on a four-real-dimensional K3-manifold, which is the second simplest four-dimensional manifold solving Einstein's equations, you get the same thing as heterotic string theory on a three-dimensional torus! This duality may be compactified on additional circular dimensions.
You're invited to study string theory textbooks if you're intrigued by this duality, by the previous ones, or by additional dualities that I haven't even mentioned. ;-)
Compactifying to get realistic models
Now, let's get closer to the reality. How do you obtain a 3+1-dimensional Universe from these 6 maximally dimensional vacua of string/M-theory? Well, even though the progress with type I string theory in 1984 (which contained gravity as well as gauge fields and fermions) was what started the first superstring revolution, type I string theory is actually the only one whose realistic vacua haven't been constructed (and probably don't exist).
The gauge group, SO(32), is just too awkward a starting point for realistic Yang-Mills model building. But you may actually find attempts in this direction in literature, too.
When I said this disappointing thing about type I string theory, let me mention that type I isn't really independent of the remaining 5. And I don't mean just the S-duality with the heterotic string. Type I string theory is related to another theory, type IIB string theory, because type I may be interpreted as "type IIB with extra objects". The objects are a spacetime-filling orientifold (a kind of mirror for strings reverting their orientation), 16 spacetime-filling D9-branes, and their mirror images.
The spin(32)/Z_2 heterotic strings are also bad for phenomenology, because of the same unrealistic gauge group. But they're close to the other heterotic string theory whose success compensates the failure of spin(32)/Z_2.
Good. So let's focus on the 4 remaining string/M-theories. Each of them may produce realistic compactifications; M-theory can do so in 2 very different ways. One of the compactifications could be hosting our Universe. Because of the dualities are other relationships and transitions, our Universe could actually be described by a stringy compactification in more than one way. The more dimensions you compactify, the higher number of connections, equivalences, and transitions you get that make the "landscape" of string/M-theory interconnected.
M-theory: G_2 holonomy manifolds
M-theory has 11 dimensions and we need 4. It means that 7 of them have to be compactified. However, 7 is an odd number and physics in odd-dimensional spaces is notoriously left-right symmetric. For example, there is no way to distinguish left-handed and right-handed spinors. (Note that O(2n+1) is just SO(2n+1) times Z_2 because the minus unit matrix has a negative determinant and gets you to the other component of O(2n+1).) Consequently, if you compactify odd-dimensional M-theory on an odd-dimensional manifold, the resulting even-dimensional (four-dimensional) theory won't be able to break the CP-symmetry (and maybe not even P-symmetry: I could refine this comment within a few minutes but I suspect no one cares haha).
So this seemingly kills M-theory as a mother of realistic vacua. However, there is a loophole. If the 7-dimensional manifold is singular, the P- and CP-violating objects may happily live at the singularities. This is what people such as Bobby Acharya and Edward Witten realized around the beginning of this millennium.
Those 7 compact dimensions of M-theory produce a nice shape, the so-called G_2 holonomy manifold. It is Ricci-flat (satisfies Einstein's vacuum equations) and preserves 4 supercharges out of 32 (which is what you need for nice phenomenology) which is 1/8 as a relative fraction. Here, G_2 is the symmetry group ("automorphism group", if you like fancy names) of the octonions (which have 7 different imaginary units: those 7 units are identified with 7 tangential directions in the G_2 holonomy manifold): it's the smallest exceptional Lie group (unlike E_8 which is the largest one).
So one (modern) class of string/M-theory realistic compactifications involves M-theory compactified on a 7-dimensional manifolds which must be singular and all the chiral fermions etc. must live at the singularities. It's likely that all things except for gravity live there: only supergravity lives in the bulk of M-theory, after all. But in principle, there may be several singularities (and e.g. different generations of fermions may live at different places of the extra dimensions) and also singularities that are not point-like but that are interpolating between point-like singularities, and so on.
Some phenomenologists such as Gordon Kane became enthusiastic believers in the M-theory scenario. Various "split" spectra of superpartners that are consistent with the null LHC observations so far follow from those models, much like "axiverse" (lots of axions with various masses) and other things, aside from the Standard-Model-like physics coupled to gravity.
Heterotic string vacua and heterotic M-theory
However, the first realistic vacua obtained from string theory in the mid 1980s were different: all Standard Model (and similar) fields live in all the 10 dimensions, much like gravity. Nothing is localized. Heterotic E_8 x E_8 string theory is left-right asymmetric already in 10 dimensions. When you compactify it on a 6-dimensional Calabi-Yau manifold, which is the 6-dimensional counterpart of the G_2 holonomy manifold above, you preserve the left-right asymmetry (chirality).
I have mentioned that M-theory on a line interval is equivalent to heterotic E_8 x E_8 string theory, as shown by Hořava and Witten: the more strongly coupled the heterotic string is, the longer the line interval becomes. So the heterotic E_8 x E_8 string vacua may also be reinterpreted as "heterotic (or Hořava-Witten) M-theory" vacua with two end-of-the-world branes. In this case, gravity lives in all spacetime dimensions (it always does) but all the other fields live in 10 dimensions at the boundaries of the world only.
Typically, the whole Standard Model lives in one of the E_8 groups only, on one endpoint of the world. The other "evil" E_8 is usually broken differently and may be responsible for SUSY breaking by gluino condensation and other things. These heterotic models are the oldest class and they're most naturally compatible with "grand unification" and "field-theoretical models of physics beyond the Standard Model" (because nothing is really localized, except for being at the co-dimension-one end of the world, so you may get rid of the extra dimensions at the very beginning, if you wish).
My ordering of the classes is chaotic. Note that in the M-theory on G_2 holonomy manifold case, the Standard Model fermions live on co-dimension-seven loci of the spacetime: in the heterotic vacua, it's co-dimension zero or one. In this sense, the first two classes I started with are "opposite extremes".
Note that out of the promised 4 maximally dimensional starting points for realistic vacua, I have exhausted 2: M-theory and heterotic E_8 x E_8 strings. (M-theory was used twice: either on G_2 holonomy singular manifolds or on line interval times a six-dimensional Calabi-Yau manifold.)
There are 2 scenarios left: type IIA and type IIB string theory.
Type IIA braneworlds
Type IIA (and also type IIB) has too much supersymmetry. Even if you compactify it on a Calabi-Yau manifold, you would get 8 supercharges which is too much: N=2 supersymmetry in four dimensions. This amount of residual supersymmetry couldn't be broken, wouldn't allow any P/CP-violation, and would lead to huge and experimentally ruled out multiplets, anyway. So one has to reduce the supercharges by some "extra objects" to one-half, too.
The relevant objects are D-branes and orientifold planes. Most typically, realistic type IIA vacua are "braneworlds" with numerous D6-branes stretched in various (triple) directions of the compactified 6 dimensions. Unoriented strings are helpful so the compactification also often includes orientifold planes.
One may get pretty realistic models even if you start from toroidal compactifications. These models have also been studied for a decade or so. Barton Zwiebach's textbook allows undergrads to build realistic models of this sort. It's kind of amazing that undergrads may already learn everything that is needed to work with a theory of everything according to all early 21st century requirements.
These "type IIA intersecting braneworlds" are usually not compatible with "grand unification". Instead, each stack of N D-branes produces its own U(N)-like group (it becomes symplectic or orthogonal if the stack sits on an orientifold) which is independent from the U(N)-like groups of other stacks. So the "braneworlds" are on the opposite extreme side from heterotic strings when it comes to grand unification. Heterotic strings are as unified as you can get: brane worlds allow you to make the factors in your gauge groups as independent as you want. No one knows which of the two paradigms is actually preferred by Nature Herself (gauge coupling unification is a piece of circumstantial evidence that the idea of grand unification in general could be right).
Such braneworld models have various favorable properties. For example, they may naturally explain why the electron mass is so much smaller than the top quark mass: the Yukawa couplings behind the low masses may be produced by "world sheet instantons" (of a disk shape, actually a triangle) which may explain why they're exponentially small. Of course, ideally, one would like to find a stringy model that has all the great properties. ;-)
If you reinterpret type IIA as M-theory on a circle, you may get a dual G_2-holonomy manifold out of some type IIA vacua. The D6-brane of type IIA itself becomes a "Kaluza-Klein monopole" – a particular purely gravitational solution of 11-dimensional general relativity whose metric tensor encodes the topologically non-trivial electromagnetic field of a magnetic monopole (one that needs a Dirac string), using a dictionary between electromagnetism and compactified gravity from the Kaluza-Klein theory. So an intersection of several D6-branes produces a very particular 7-dimensional G_2-holonomy geometry, and so on.
There are other relationships between the vacua from different classes. But I already want to get to the last scenario:
F-theory: a fancy type IIB toolkit
If you enjoy fancy overpopulation of dimensions, F-theory may be interpreted as a 12-dimensional theory, the Father theory, where "F" stands for "father", "fiber", "frankenstein" (because of geometric engineering), "fafa", or "vaFa", according to your taste. ;-) F-theory is related to type IIB string theory in a similar but not identical way to the way how M-theory is related to type IIA. It's important that the two cases are not identical or isomorphic (otherwise we wouldn't invent so many redundant terms): much like the males and females (mothers and fathers: note that the F/M get interchanged), M-theory and F-theory structurally differ. They have very different roles and detailed logic.
The growing dimension of type IIA may become infinite, producing an independently viable "mother" theory in 11 dimensions. This is not the case of F-theory.
F-theory has 12 dimensions but 2 of them have to be compactified on a two-dimensional torus (whose size is unphysical), otherwise the father theory doesn't make any sense. Once 2 dimensions are compactified, you're down to a 10-dimensional theory, type IIB string theory, and the shape of the torus knows about the two scalar fields of type IIB string theory, the axion-dilaton field, which are symmetric under the SL(2,Z) S-duality group (which includes the "g" goes to "1/g" map mentioned previously).
To get realistic 4-dimensional vacua, you formally need to compactify F-theory on an 8-dimensional manifold: 12-8 = 4. However, out of this 8-dimensional manifold, 2 dimensions must be reserved for the toroidal "fiber" because F-theory must always compactify two of its dimensions on a 2-torus. It follows that the 8-dimensional manifold used for F-theory must be "elliptically fibered", which is just a different way of saying that it is a 6-dimensional manifold with a 2-dimensional torus attached to each point.
However, what's f(ancy) about F-theory relatively to "ordinary type IIB string theory" is that the 2-torus may have a shape that depends on the location on the 6-dimensional "base space". And in fact, if you go around special paths in this 6-dimensional base space, the 2-torus may be translated to itself in a nontrivial way, labeled by an SL(2,Z) transformation!
In some sense, we're talking about special 8-dimensional manifolds here (even though only 6 of them may be "large enough to be seen"). 8 compactified dimensions is the maximum number you find in stringy phenomenology. A higher number of dimensions means "more complex geometry". So the people who study F-theory compactifications are the best mathematicians (geometers) among the string phenomenologists (even though a comparable amount of mathematics is needed for the 6-dimensional and 7-dimensional manifolds, too).
In recent 5 years, F-theory phenomenology began to thrive as a very healthy subject because of Cumrun Vafa (the Father of F-theory), his collaborators, and several other bold physicists. In this case, the Standard Model fields also live on singular places of the compactification manifold. But in this case it is really a rule rather than exception that there are different loci where different fields like to live. Sometimes they intersect, and so on. Particles are associated with loci of "intermediate dimensions" and there are just many ways how to stretch such loci and how to make them intersect.
What's the difference between singularities within the F-theory compactification manifold and e.g. D6-branes in type IIA intersecting braneworlds? The difference is that D-branes, while inducing some potentially singular values of the fields, are still "objects inserted into a pre-existing space" and the pre-existing space may be non-singular. On the other hand, the singularities in the 8-dimensional Calabi-Yau manifolds of F-theory or 7-dimensional singular G_2 holonomy manifolds of M-theory are linked to the very shape of the whole manifold: the singularity makes the space look singular even outside the location of the singularity; D-branes may be viewed as strictly localized singular intruders. (Nevertheless, D-branes, especially those with a low co-dimension, distort the surrounding space "qualitatively" as well, so there's another relationship between singularities and D-branes, especially D7-branes.)
Under Vafa, Heckman, Marsano, Saulina, and others, F-theory model building became a hunt for very special models that have welcome properties (that may agree with observations or some of the qualitative patterns found in them) and lots of things about phenomenologically attractive properties of these vacua (and some potential problems) has been written.
However, F-theory model building has yet another, evil face. F-theory on 8-dimensional manifolds is also the setup that was used to produce the infamous 10^{500} or so vacua in the "landscape". So F-theory became the main propaganda tool of the anthropic ideologues. Because F-theory allows us to play with 8 dimensions, kind of, which is the highest number of dimension, we find the maximized number of possible ways to compactify them, too. That's why the landscape of the (generic messy) F-theory compactifications is so large (especially when you include the fluxes etc.).
Nevertheless, the anthropic ideologues have never justified their claim that just because this class of vacua (F-theory KKLT-like vacua) has the highest number of elements, it is the most likely one to be realized in Nature. I am convinced that this implication is invalid. One must look for promising vacua in all the groups mentioned above. Because F-theory vacua have also helped to promote the pathological meme that it makes no sense to continue to search for the right compactification, you may also say that "F" stands for "ucked-up". (Its non-anthropic father Vafa would violently disagree.)
F-theory vacua (but not only them) are also a natural starting point for the string theory incarnations of Randall-Sundrum warped geometries and other things.
It's important to note that in various classes of vacua that have been discussed, different fields may be confined to particular branes or singularities whose dimensionalities may differ. It is not true that all non-gravitational fields and particles have to live "at the same place" in extra dimensions. In fact, the separation (or "segregation") of some particle species along extra dimensions may explain why some of their interactions are weak, and so on. Especially when you're near "intermediate dimensions" in the compactification manifold, there are many choices.
Summary
I've tried to cover all four or five main classes of realistic string theory compactifications, some of their characteristic features, their links to the 10- or 11-dimensional master theories, the relationships between the master theories, and other things. I don't believe that too many readers who were previously unaware of these fascinating achievements will understand everything I wrote but it is still conceivable that the concepts explained above will help some people to get further in their understanding where the technical properties of the Cosmos surrounding us come from.
When you're a layman and you happen to attend a string phenomenology talk, you may at least ask more sophisticated would-be technical questions. Instead of "is the string really vibrating faster than light, proving that 42 is the answer to everything?", you may ask "can you find a G_2 holonomy manifold associated with uplifting these D6-branes in type IIA"? :-)
Where are we in extra dimensions?
![Where are we in extra dimensions?]()
Reviewed by MCH
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September 27, 2011
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