# How many string theories with more than one supersymmetry?

1. Aug 18, 2009

### MTd2

With one, we have 5, what about more than one? What are they? What are the importance of these, phenomenologicaly saying?

2. Aug 18, 2009

### suprised

This is an ambiguous question, the answer depends on the dimension. in 10d there are theories with 2,1 and 0 supersymmetries. That's all what can exist due to Nahm's classification of supersymmetries. In four dimensions one can have everything vom 0 to 8 supersymmetries (well, N=7 is equivalent to N=8).

Phenomenologically, only theories with N=1 and 0 can be relevant because only for those the theory can have chiral fermions.

3. Aug 18, 2009

### MTd2

This is not ambiguous, I just didn't put it well, sorry. I mean string theory by a sigma model whose worldsheet has no conformal anomaly.With 1 supersymmetry, there are 5 theories, all of them shows that 10 dimension is a condition needed to cencel the anomaly. What if that sigma model, instead of 1 supersymmetry, had 2 or more? How many anomaly-free free would one get for each case? How many dimensions each one of them would have?

4. Aug 19, 2009

### suprised

Ahh you mean world-sheet supersymmetries. Well there is still something not quite accurate in what you are saying, namely one can have world sheet susy in the right- and leftmoving sectors independently and thus there are various combinations possible. So the two type-II strings have (N_L,N_R)=(1,1) susies and the two heterotic strings have (1,0) (or (0,1), which is of course equivalent). Then there is also the open type-I string, which has N=1 supersymmetry.

Moreover, which is something that is not often mentioned, there are various theories in 10d which are not supersymmetric in space-time, but only on the world-sheet, those come again in type-II and heterotic versions.

What you are asking for are theories with more world sheet supersymmetries, yes they do exist, most famous is the so-called N=2 string for which the supersymmetries as written above are doubled. Its critical dimension is four, but this does not refer to Minkowski spacetime with signature (time,space)=(-,+++) but to a spacetime with signature (--,++); you may call that a two-time theory if you like... So the theory is physically not very relevant but has some nice mathematical properties which is why it was studied. Then there are N=4 strings, I think, but I can't recall out of my mind any of their properties.

Be aware that these considerations are based on the world-sheet, which is not a fundamental concept but rather, by its very definition, a concept of perturbation theory. Thus non-perturbative relationships between those various strings are not visible in this framework; and the 11-dimensional membrane theory and the 12-dim F-theory are not describable in this language at all. So questions concerning the nature of string theory should not be formulated in this language in the first place.

Also, the concept of dimensions can be quite misleading. For example, in the heterotic string the "non-supersymmetric chiral half" (ie, the "0" in (N_L,N_R)=(1,0)) formally corresponds to the bosonic string which has critical central charge c equal to 26. So for the heterotic string the critical central charges are formally (c_L,c_R)=(15,26), but this does not mean that there are 26 spacetime dimensions somewhere in the theory. A better viewpoint is that the central charges simply label internal degrees of freedom of the string, which may or may not have an interpretation in terms of compactified dimensions.

Last edited: Aug 19, 2009
5. Aug 19, 2009

### MTd2

Sounds like what I see sometimes when I see Berkovitz talking about twistor strings. I see everywhere N=2 and (--++) signature. Anyway, couldn't one rotate (2,0) to (1,1)?

But you have to to start counting dimensions somewhere, right? So the 10d strings spacetime is fundamental in this aspect. Also, isn't F-Theory a mathematical device to understand Type II strings?

6. Aug 19, 2009

### suprised

Well formally one can rotate the backround metric in the effective theory but then the theory is, I guess, not consistent any more. The theory in (--,++) is very special, has very few degrees of freedom, is integrable, etc, and probably not many of these special properties would survive this "wick" rotation. There is no reason why they should.

Actually, counting dimensions in this way has, in my opinion, historically created much confusion. It sounds like as if the 10d theories would be more "fundamental", and all the lower dimensional theories are "compactified". But it is well known that many, say 4d, theories are not compactifications in the sense that there is no 10d lorentz symmetry restored no matter how high we go up in energy. There is, in fact, no absolute notion of compact dimensions, and an example is the heterotic string alluded to above.

Another is AdS/CFT duality....is it just 4d N=4 Yang-Mills theory, or is it compactified 10d type-II strings? Either description is good in its own regime of validity. More generally it seems that almost any strongly coupled 4d theory has some higher dimensional holographic properties. Non-perturbative quantum effects in the 4d theory can be described in terms of classical background gemetry of some compactified higher dimensional dual gravity or string theory. Coordinates of some higher dimensions turn into coupling constants in the 4d theory in this way.

So there is in general no unambiguous notion of compactified dimensions to start with; sometimes they simply play the role of internal degrees of freedom (and sometimes they enocde, and in a sense emerge from, non-perturbative effects). All what matters that the Virasoro central charge is cancelled, and this requires a certain amount and structure of extra internal degrees of freedom. Whether these extra degrees of freedom have an interpretation in terms of higher dimensions or not, does not matter at all; see the "extra" 16 dimensions of the heterotic string that can be rewritten in terms of 2d fermions for which the extra-dimensional interpretation disappears. All what counts is that they provide the E8xE8 Kac-Moody symmetry which must be there for consistency.

Unfortunately, historically, too much emphasis was put on literal compactifications of the 10d theories and so many people get preoccupied with misleading questions; I have seen here threads discussing what the meaning of those extra dimensions is and so on. Well, again, they just provide internal degrees of freedom, using the KK idea that compactified momenta are nothing but charges. I would even go as far and provocatively say that only in special circumstances those internal degrees of freedom also happen to be interpretable in terms of compactifications of higher dimensional theories.

As for F-theory, yes, this is another example of theory where extra dimensions (at least formally) emerge from non-pertubative dynamics, here the type IIB strings. Everybody is free to choose her viewpoint, namely either whether these extra dimensions are "real" and are tied to some hypothetical F-theory, or whether these extra dimensions are simply the coupling constants of the type IIB string which miraculously behave like a torus; it does not matter. At any rate, my point was that these non-pertubatve considerations cannot be captured by world-sheet considerations, so those are not fundamental.

So you asked.. "But you have to to start counting dimensions somewhere, right?" I hope to have argued that this is not necessarily a good question to ask. Probably you have meant: "what is the most fundamental string (or whatever) theory"? But as said, this may also not be a good question.

Last edited: Aug 19, 2009
7. Aug 19, 2009

### MTd2

But is that Berkovitz Twistor Theory? Also, when I said rotate, I didn't refer to rotate the metric, but rotate the chiralities, if that makes any sense. For example, like Left would be the x-axis and Right the y-axis. So, (2,0) or (1,1) would be a kind of coordinate.

"All what matters that the Virasoro central charge is cancelled, and this requires a certain amount and structure of extra internal degrees of freedom."

That means string theory is a theory in without ghosts in surfaces?

8. Aug 19, 2009

### suprised

Rotating chiralities doesn't make sense IMHO. Chiralities involve differerent (spinor) representations and (perhaps apart from SO(8) triality) changing reps relative to each other does change phyiscs.

Not sure whether I understand the question. The point is that in perturbative string constructions based on CFT, the relevant ghost central charge must be cancelled by something; one may choose spacetime coordinates all the way up, so this then gives strings in d=26 or d=10, but one may also choose a subset, say 4, spacetime coordinates. The missing central charge must then be compensated by something else, and in fact any suitable CFT can be used. The degrees of freedom arising from that are then by definition internal and not space-time degrees of freedom. Sometimes but not always these extra degrees of freedom look like compactified coordinates of some higher dimensional spacetime, and then it is often useful to visualize them in terms of higher dimensions.

9. Aug 19, 2009

### MTd2

Alright. But I was thinking just it the world sheet, like, operators that exchanged the chirality of supercharges, so that the count of supercharges could go left right.

Not always, how?

10. Aug 19, 2009

### suprised

There is no reason why the "internal" part of the CFT would correspond to a compactifaction of a higher dimensional theory on some manifold. There are lots of "non-geometric" compactifications. A geometrical interpretation is sometimes/often(?) possible but this is not important. And this is in fact often even ambiguous... due to dualities, one and the same theory can have different interpretations, namely as different world sheet theories compactified on different manifolds (like type-II strings on K3 being equivalent to heterotic ones on T4). This shows that there is in general no absolute meaning of a compactification manifold.

I am just saying this because often people are too closely tied to notions like compactification manifold, dimension, D-brane etc. All these are just different parametrizations and approximations of one and the same, probably unique, largely unknown theory one may call M-theory.

11. Aug 19, 2009

### MTd2

I see, so the different string theories are like generalized fourier transforms. In the usual Quantum Mechanics, you just transform between momentum and position, in string theory you transform between several different types of basis, which are the different string theories. Right?

Besides that, is there anything that forbids an operator on the worldsheet that exchange chiralities?

12. Aug 20, 2009

### suprised

Vaguely yes. Some of the dualities act like a Fourier transformation, for example between momentum and string winding states, or between electric and magnetic degrees of freedom.

Hm.. probably not. But such an operation will in general change the physical theory, and the question is what properties survive it.

However sometimes the theory stays invariant and then this operation can be very interesting. For example, flipping relative chiralities map the type IIA and type IIB strings into each other, which are a priori completely different theories. However, if one theory (say Type IIA) is compactified on a Calabi-Yau space and is transformed into the other (ie, type IIB), then the theory stays invariant provided the Calabi-Yau is replaced by a specific other one (the "mirror dual"). This has extremely non-trivial mathematical consequences.

13. Aug 20, 2009

### MTd2

What is missing to define an M-Theory nowadays?

14. Aug 20, 2009

### suprised

hmm this is a hard one... one just knows various approximations to it, in various corners of its moduli space. The feat would be to formulate something that gives rise to all those field, string and membrane theories when expanded around the relevant regions of the vacuum parameter space, in terms of the appropriate degrees of freedom. Clearly almost everyone can say a lot of vague and naive things about this, but to come up with something concrete still seems very hard.

15. Aug 20, 2009

### MTd2

Would you mind citing some things that are known not to be reproduced "when expanded around the relevant regions of the vacuum parameter space" ?

16. Aug 20, 2009

### suprised

Well for example sick QFT with gauge anomalies.

17. Aug 20, 2009

### MTd2

Isn't that a good thing? Why should M-Theory include them?

18. Aug 20, 2009

### suprised

Yes it is a good thing... M-Theory supposedly is the mother of all consistent theories....and if I may say something provocative, IMHO it is defined by this property.

19. Aug 20, 2009

### MTd2

OK, then. I thought you said that there were consistent theories that were not reproduced "when expanded around the relevant regions of the vacuum parameter space". Hmm, but what I actually mean, if it is that they can be reproduced right now? If there is any, can you tell me which one is?

20. Aug 20, 2009

### suprised

.. don't understand the question...;-(