# Does string theory explain the existence of 3 generations of quarks/leptons?

I am wondering whether string theory explains the existence of 3 families of quarks/leptons or not. I have a very limited understanding of string theory, as of now, and I have a mathematical background, so I am asking this question here, so that people with better knowledge of string theory might answer it.

There is a related discussion: Origin of lepton/quark generations? and one of users, Andrew Holzner, quoting wikipedia, gave an explanation that CP violation requires at least 3 generations (and there are in that discussion a number of other explanations). This sounds like a reasonable explanation, but my question is more from the point of view of string theory.

• As of today string theory doesn't explain anything, or, more precisely, it explains too much. The "string landscape" allows something like $10^{500}$ different models of reality, but it doesn't give us any reason (apart from an intellectually hollow anthropic argument) to select one over the other. – CuriousOne Jul 2 '16 at 14:57
• Related: physics.stackexchange.com/q/2051/2451 and links therein. – Qmechanic Jul 2 '16 at 15:01
• @CuriousOne thank you for the information on the status quo. User heather wrote something similar. You both answered my question. – Malkoun Jul 2 '16 at 15:08
• String theory explains anything, as far as I can see! – tfb Jul 3 '16 at 13:36

## 3 Answers

Part 1:

The branch of string theory which actually tries to match experiment is called string phenomenology. The state of the art in string phenomenology is that, starting from different forms of string theory (heterotic string theory, M-theory, F-theory...), it is possible to define space-time geometries, arrangements of branes, background fluxes... such that strings in the defined environment will behave qualitatively like the particles of the standard model.

The underlying reason why there are three generations in such a model really depends on the nature of its construction.

In an M-theory model such as those championed by Gordon Kane, the particles in a given generation correspond to states of M2-branes located at specific singular points in the compactification manifold, so the number of such generations is just the number of such singular points.

In a heterotic model such as those that Brian Greene has written about, it's more complicated. The topology of the compactification manifold permits a specific number of light left-handed fermionic states, and another number of light right-handed fermionic states; then left and right combine to make heavy states; and the generations correspond to the light handed fermionic states that are left over, that didn't pair up with anything. The original numbers of handed light states equal two of the "Hodge numbers" characterizing the topology, so in this case, there are three (or however many) generations because the difference between those two numbers equals three.

In still other models, the reason for there being three generations would be something else again.

Part 2:

Since the state of the art in string phenomenology is still just at the level of searching the vast "landscape" of possibilities for models that match experiment, any current explanation for "why three generations?" is going to lead back to contingent properties of the model that happens to be successful, like those that I sketched in Part 1 of this answer.

In evolutionary biology, they speak of proximate causes and ultimate causes. Why does a flower bend to follow the sun? The proximate cause is the set of molecules that it happens to be made of. The ultimate cause is natural selection - that's the reason why it's made of molecules that react like that, and not in some other way.

We can look at explanations like those from Part 1 as proximate causes of there being three generations. What are the possible ultimate causes?

One possibility is anthropic. Maybe we live in an eternally inflating universe where different string vacua are realized in different regions, and maybe e.g. the cosmological consequences of the CP violation that requires at least three generations in order to occur, helps make life, or even just stars, possible.

Another possibility is that it is just random. In genomic evolution, there's a lot of neutral evolution, features of the genome which are just contingent, which don't help the organism survive, but also don't hinder it, so those features aren't eliminated by natural selection. Anthropics can't determine everything, and maybe three generations is just a brute fact about how our corner of reality turned out.

Still another possibility is that it's the product of the natural dynamics of string theory. String phenomenology fixes the geometry of the extra dimensions (etc) and studies the results, but in fact you can have quantum tunneling between different geometries, and there may have been a lot of that in the early universe. The 2007 paper "Triadophilia" speculates that three-generation heterotic manifolds may be favored in this way.

I will address the title question:

does string theory explain the existence of 3 generations of quarks leptons

because of the word "explain".

Physics is about measurements and observations and mathematical models which not only fit the measurements and observations but also have predictive power. Otherwise the model is just a map, not a physics theory.

Newtonian gravitational theory assumes the 1/r^2 behavior and using classical mechanics with its laws generates the very successful gravitational model which can predict most astronomical data within errors. Deviations from Newtonian mechanics were predicted by the theory of General Relativity, and the validation of the predictions established GR as an undelying theory from which Newtonian gravity emerges.

The Standard Model of particle physics with its Lagrangian formulation is the analogue of the Newtonian gravitational theory: a large number of measurements and observations went into the SM to build up the structure, and its predictions have been mostly validated up to now. Candidates for string theory models are where GR was before its validation by not before seen data.

String theories can accommodate the group structure in the Lagrangian of the standard model, so there is no problem in envisaging a string theory model, also, and very important, string theories are the only candidate theories that can have quantization of gravity naturally. They also demand supersymmetry to do their magic. In this sense super symmetry is predicted by candidate string theory models, and if supersymmetry is found at LHC it will be like the validation of GR by predicting the anomalous perihelion advance of the planet Mercury without any arbitrary parameters

As was stated in the comments there are too many possible theories, and nature/data have to choose for us which is the one that fits the data, in the same way that nature chose for us the standard model lagrangian.

So "explain" is not a good verb, a physical theory fits the data and makes predictions for future measurements.

Now if physics ever reaches the point to have a mathematical Theory of Everything, from a few postulates and few measurement input for constants, then it might be legitimate to say that the TOE "explains" everything. Certainly no physics theory is at that point , more so String Theories which are at the research level.

• Thank you anna v for your thoughts on the philosophy of Physics (if you don't mind my describing them like that). I enjoyed your answer. I just meant by "explain" that it postulates some principles, which are basic, and from them it derives the existence of 3 generations of quarks. In this sense, it would have "explained" the 3 generations in terms of more fundamental principles. But yes, as you and others have written, string theory, as of the time of writing, does not do that. – Malkoun Jul 2 '16 at 18:59

The official string theory website says this:

Theoretical physics has not explained why there are three generations of particles that make up matter. Maybe string theory will come up with an answer for this.

That's really where it stands. In fact, there's another question on physics SE here, where one of the answers says

The question as to why there are exactly three generations is still an open problem, even in string theory.

Another answer at the same question says

All in all, to answer questions like the number of generations in the SM, the masses of the fermions, etc you need a candidate for a complete theory at high energies and the answers will depend on this candidate. String theory is considered to be the most successful framework for this job and string models do indeed make concrete predictions about (among other things) the number of generations we should be observing. Unfortunately, there are too many models (vaccua) to choose from and no obvious way to make the choice.

To sum up: we have absolutely no idea why there are three generations, even in string theory. Like @CuriousOne said in the comments, string theory has too many possible models of reality with no way to select which one to use for string theory to really explain anything.

Note:

In the comments below this answer, @CuriousOne made a good point: all of the quoted statements make it sound like this is one of the only problems with string theory. This is not true. There is no experimental evidence for string theory. Describing why there are 3 generations of quarks/leptons is the least of string theory's problems.

• I would group these statements under "false advertising". String theory hasn't explained anything about the structure of the standard model, so it's more than just a little frivolous to say "We haven't explained a particular statement...", which leaves the reader with the impression that they have explained some or maybe even most others. :-( – CuriousOne Jul 2 '16 at 15:08
• String theory explains the complete structure of the universe as oscillating strings. QFT and GR follows from string theory, so it explains our most succesful theories. – user122089 Jul 2 '16 at 15:11
• ok, I see where things stand. Too many possibilities for the "curled up dimensions". Too many possibilities for the CY 3-fold, for 10-dimensional string theories. Hmm, this brings up a question in my head. How does one make concrete predictions in string theory, given a choice of a CY 3-fold? Can someone recommend something to read which is string theoretic and of a more phenomenological approach? (Technically, this is a different discussion, though related, so maybe I should move it to another post?) – Malkoun Jul 2 '16 at 15:13
• @CuriousOne, I agree that the first statement is a little ridiculous; I was including it to show that even the official website doesn't claim an answer. – heather Jul 2 '16 at 15:17
• @Malkoun I'm not sure specifically what sort of book you are looking for, but Brian Greene's books discuss string theory and some of it's predictions, etc. I've personally read parts of The Hidden Reality and there was a section discussing string theory and its 'status', so that might help you. – heather Jul 2 '16 at 15:20