In the previous post, we looked at the construction of superstring theory, but to our surprise, found two slightly different methods of constructing it! Now we will explore what the difference between the two is, and whether they are consistent. Be prepared for a couple of surprises along the way.
Superstring theories!?
But first, a detour into spin. To this date, spin remains the most misleading piece of terminology since nothing is actually spinning in a fundamental particle! Rather, it consitutes an intrinsic angular momentum of the particle, and is not ad hoc, but a direct consequence of unifying quantum mechanics with special relativity. Like any vector, the spin of a particle can point in a particular direction and, for example, the relative direction between the spin and an external magnetic field determines the strength of the interaction and hence the magnitude of the force experienced by the particle.
For massless particles, we can define a related concept called helicity. A massless particle is left-handed if its spin and momentum are antiparallel, and right-handed if they are parallel. If it seems peculiar to ascribe momentum to a massless particle, recall that $p=\frac Ec$ according to special relativity. The idea of helicity coincides with a related but distinct property called chirality for massless particles, which will be of interest here.
Now theorists have been continually interested in symmetries of nature. In particular, there were three eminent discrete symmetries that were regarded as sacrosanct: charge (C) symmetry, parity (P) symmetry and time-reversal (T) symmetry. Roughly, C-symmetry flips all the internal quantum numbers of every particle (e.g. electrical charge), P-symmetry replaces left with right, and T-symmetry reverses the direction of time. Electromagnetism, the strong force and gravity all preserve each of these independently. Unfortunately, the pesky weak force violated both C-symmetry and P-symmetry. It was subsequently hoped that CP-symmetry (flipping charges and direction simultaneously) would be respected, but further experiments involving decays of neutral kaons discredited this, and indirectly destroyed T-symmetry as well. The only discrete symmetry which the Standard Model assumes to hold exactly is CPT-symmetry, a combination of all three inversions simultaneously. Theories like the Standard Model which do not respect CP-symmetry are said to be chiral, and any master-theory that subsumes it must provide a mechanism to generate this asymmetry.
Back to strings! Once we perform a GSO projection on the left-movers, we can choose to perform the same projection, or the other projection on the right-movers. Naturally, choosing identical projections results in the massless fermions in the theory having the same chirality, resulting in Type IIB string theory, while choosing opposite projections makes the theory non-chiral, forming Type IIA string theory. The “II” in both cases refers to the $\mathcal N=2$ supersymmetry that they display, with each boson having two partner fermions and vice versa. To ensure that all the states are physical, the dimension of spacetime is forced to be 10 - much less than the 26 required for bosonic string theory! The fact that type IIB string theory is chiral makes it very useful for string phenomenology and realistic model building.
This should be the end of the story, but string theory continues to yield new surprises. The type IIB theory has a worldsheet parity symmetry, where the left-moving modes and right-moving modes are exchanged. Since both sectors have the same chirality, this renders the theory invariant. This does not mean that each state is parity invariant: considering the function ${1, 2, 3, 4}\to{1, 3, 2, 4}$, the set (the “theory”) remains invariant, but only the subset ${1, 4}$ (the “parity-invariant states”) are mapped to themselves. Naturally the situation is a little more complicated (different parts of each sector can “decompose” into variant and invariant pieces), but the general idea persists. What if were to “gauge” this symmetry, which would amount to projecting type IIB string theory down to only those components which were left-right symmetric?
First success - the graviton is symmetric, so it survives the projection. We still have a theory of quantum gravity! The massless gravitino also survives, forcing spacetime supersymmetry for consistency. We breathe a sigh of relief when we see that the number of degrees of freedom match in the fermionic and bosonic sectors: we have constructed a new, consistent string theory. This has some interesting features, as we shall see.
Firstly, the parity projection destroys half the amount of supersymmetry, so there is only one superpartner for each particle. This is $\mathcal N=1$ SUSY, so we have obtained Type I string theory. Another curious aspect is the presence of open strings in the theory. Imagine a -multi-coloured loop (a “closed string”) which is left-right symmetric. The projection operator “squashes” this circle vertically into a symmetric string with disjoint ends (the open string!). The once oriented fundamental string now yields both unoriented closed strings and unoriented open strings in type I string theory.
Now the ends of these strings can have different internal charges associated with them. Determining how these charges should transform under the parity operator is initially mystifying, but experience tells us to search for and impose consistency requirements. This time we must explore the low-energy behaviour of string theory. Eliding all the mathematical details, string theory at low energies looks like a supersymmetric theory of gravity with the same value of $\mathcal N$, in the same number of spacetime dimensions. This is akin to how molecular interactions are constantly fizzling inside water, but we sweep over these microscopic details while building an accurate description of the “big picture”, as in fluid dynamics. In the case of string theory this correspondence is arguably stronger since the relation holds on purely mathematical grounds rather than having to appeal to observation. As it turns out, $\mathcal N=1$ supergravity in 10D is inconsistent on its own due to so-called “anomalies”.
The only way to remedy this to fuse supergravity with a particular type of Yang-Mills theory, which is an analogue of electromagnetism in which charges can interact with themselves - for instance, in QCD, gluons are self-interacting. Such a Yang-Mills theory is uniquely specified by a choice of dimension and “gauge group”, a deep mathematical object whose mathematical beauty I cannot overstate, but also cannot explain due to a lack of space here - one may regard them merely as “data” here. The only gauge groups which work are $\mathrm{SO}(32)$ and $E_8\times E_8$, but the latter is not compatible with the open type I string. So the gauge group of the low-energy effective theory, and by extension the open string, is mathematically and uniquely determined to be $\mathrm{SO}(32)$, and this in turn fixes the transformation properties of the internal charges on the ends of the string.
The final (and my personal favourite) feature is the existence of D-branes. Briefly, a D-brane is a hypersurface (a high-dimensional object) on which the end of an open string can lie. If the ends of the string are unconstrained, one obtains a D9-brane which fills all of spacetime (the number after the “D” indicated the number of spatial dimensions only. So a D9-brane for instance has 9 spatial dimensions and one time dimension, as does spacetime). In type I superstring theory, there also exist D1 and D5 branes. The idea initially seems rather unhelpful, and possibly downright useless. However, Polchinski in 1995 discovered that the branes themselves carry charges, and so, to supplement the fundamental string, should be regarded as dynamical objects in their own right! Furthermore, they are adored by phenomenologists because branes intersecting along different dimensions give rise to lively, interacting gauge theories on their surface, and specific brane constructions and backgrounds can be used to create realistic models of the universe!
That concludes this whirlwind tour of string theory. Haha, no, there’s more. It sounds ridiculous, but we can use the bosonic string construction for the left-movers, and the superstring construction on the right-movers in order to create a new string theory! This should trigger instant revulsion - isn’t there a dimensional mismatch between the two sides? Naïvely yes, but we can introduce a 16-dimensional toroid with a particular gauge symmetry to offset this. We obtain another consistent $\mathcal N=1$ string theory, and we know from above that we should use the $\mathrm{SO}(32)$ gauge group with this toroid. We have constructed $\mathrm{SO}(32)$ heterotic string theory! But we’re still not done. This time, the $E_8\times E_8$ group is also compatible, so we can create (or rather, discover) the $E_8\times E_8$ string theory as well. This beast is well suited for model building, since a unified fundamental force embeds very nicely within this gauge group.
And that, ladies and gentlemen, is how the five superstring theories (and their little brother, the bosonic string) were discovered. As an afterword, I would like to ease your likely persistent discomfort at how string theory could ever explain the universe. After all, we see only 4 dimensions. This is solved rather straightforwardly through the process of compactification.
Now string theory has no preference to the geometry of spacetime. As such, we can posit that the 10D background is really a product of ordinary 4D macroscopic spacetime, and a curled-up 6D surface which at ordinary energies, yields microscopic directions that cannot be probed. If this also sounds ad hoc, it is imperative to understand the fundamental nature of string theory: it is not a “theory” in the conventional sense. It is a framework, much like quantum field theory. Just as QFT serves a language in which the Standard Model (a true theory) can be written, string theory can be used to construct models with whatever features we desire.
As mentioned in the previous blog post, supersymmetry would solve a lot of problems in the Standard Model and cosmology, and it is generally expected to exist at high energies on empirical grounds. So we would like this string compactification to preserve $\mathcal N=1$ SUSY, the simplest, most constraining form of supersymmetry. A beautiful result here is that the geometry of this 6D “internal” manifold determines the amount of supersymmetry preserved, in addition to other predictive features like the number of generations of fundamental particles (the Standard Model has three generations). If, for example, we start with $\mathcal N=2$ type IIB string theory, compactifying on a Calabi-Yau manifold creates a $\mathcal N=1$ field theory. As I also mentioned previously, we can fiddle around with D-brane backgrounds and fluxes to generate models. A key geometrical result is that the number of realistic compactifications is on the order $10^{500}$, a humongous but number. This is just one of the reasons string theory is held in high regard by many people.
However, if this five-fold disjunction of string theories smacks to you of inelegance, you are not alone. This puzzled superstrings long past the superstring revolution, but it led to the most beautiful result in superstring theory. Stick around to find out.