This article involves links to papers with a very complicated formalism. Although many of us think that the topic is pretty fascinating, I am less sure that the physically oriented Nobel prize winners among the readers of this blog will appreciate the depth of the ideas behind these calculations. Consequently, they should stop reading at this point, otherwise I am not responsible for their good mood! Thanks for your understanding.
Nathan Berkovits - Green & Schwarz done right
Today, perturbative string theory is just a part of our knowledge about string/M-theory but it is still the most "stringy" - and in some sense, the most mathematically solid of the insights about string/M-theory that we have. The ultimate physical observable computed in perturbative string theory is the S-matrix. In superstring theory, the graviton supermultiplet is the entire list of states that are stable at non-zero coupling. Therefore we calculate the scattering amplitudes between the gravitons and their superpartners. It's only them that appear in the initial and the final state, and the S-matrix for them is unitary.
Bosonic string theory contains loop infrared divergences induced by the tachyon, and it's not the truly interesting case to study at the quantum (loop) level. We want to focus on the string theory without the tachyon and with fermions - namely superstring theory. Its amplitudes are finite, and free of the IR divergences. They're calculated in some kind of conformal field theory. From now on, let's focus exclusively on type II string theories - those 10-dimensional theories that have the maximum of 32 real supercharges.
There are two well-known formalisms to calculate the amplitudes:
- The RNS formalism that describes the fields X^m and their superpartners psi^m that are fermions and worldsheet spinors (and the superpartners of X^m under the worldsheet supersymmetry), but they are spacetime vectors. One imposes the GSO projections that remove the tachyon, as well as to preserve the spin-statistics relation. One can calculate the amplitudes covariantly - i.e. with Lorentz symmetry being manifest. Spacetime supersymmetry is harder to see. One must add the Fadeev-Popov ghosts b,c.
- The light cone gauge. This is the formalism that naturally appears from Matrix (string) theory and the pp-wave limit of the AdS/CFT correspondence. The light cone gauge makes a part of the Lorentz symmetry harder to see - in fact, the closure of it determines the critical dimension in the light cone gauge. However, one can choose the Green-Schwarz (GS) formalism to make the spacetime supersymmetry manifest. In the Green-Schwarz formalism, the fermionic fields on the worldsheet are spacetime spinors, not vectors.
OK, you may ask: is not there a formalism that makes the full super Poincaré symmetry manifest? Something like a (Lorentz) covariant Green-Schwarz formalism? The answer is: Yes, there is a covariant GS formalism, but it's hard to quantize it. It's hard, but it can be done - as long as you have one of the world's strongest CFT hackers if not the strongest one - namely Nathan Berkovits - nearby.
So what does the Berkovits' approach to superstring theory look like? You have a conformal field theory with the following basic GS physical fields:
- Ten fields X^m, the usual spacetime coordinates that live on the string
- Thirty-two fields theta^a, transforming as a worldsheet scalar (so that they're just superpartners of X^m under the spacetime supersymmetry which is manifest).
- Add Berkovits' pure spinor ghosts \lambda^a (a boson)
What is a pure spinor? It is a spinor - imagine a complex Weyl spinor lambda^a with 16 real components - such that most of the bilinears constructed out of it vanish. In this case:
- \lambda^a . (\gamma^mu)_{ab} . \lambda^b = 0
In the case of a 10-dimensional Weyl complex spinor, the only a priori nonzero bilinears are the self-dual five-form and the one-form. The middle-dimensional bilinear (the five-form) is always allowed for a pure spinor, so in ten dimensions we just impose the condition that the vector (Dirac current, so to say) constructed out of the spinor vanishes.
Let me cheat a little bit to count the central charge. We had 32 real components of the spinor \lambda^a to start with, and we imposed 10 conditions (don't ask me about the reality conditions!), so that number of independent degrees of freedom in the pure spinor is 22, and the pure spinor cancels the central charge of the GS fields X^m and \theta^a.
Here a miracle occurs...
Let's now jump ahead. Imagine that you suddenly and seriously open Nathan's paper about the loop amplitudes in this formalism:
On its 49 pages, it contains a lot of operators, operator product expansions, and so forth. On page 27, equation (5.1), you will finally find the prescription for the complete loop amplitude. I was always feeling a little bit uneasy that the "b" ghost was not an elementary field, so you don't get the measure for the moduli spaces of Riemann surfaces in the same way as in the RNS formalism.
But after Hiroši Ooguri convinced me that Nathan's proof of the finiteness of the amplitude is a complete proof, not just an "almost complete proof", I decided to be more critical about my skepticism. Indeed, there does not seem to be anything wrong if you define the amplitudes in this direct way (5.1) - a formula that includes composite expressions for the "b" ghost as well as the picture changing operators (PCOs).
The punch line is that one can show that the correlators are equal to the correlators in the RNS formalism summed over the complex structures (different periodic or antiperiodic boundary conditions for the worldsheet fermions around various cycles).
Incidentally, this summation over the complex structures yields a much more convergent integrand which simplifies Nathan's proof of the finiteness - one does not obtain any divergences from the boundaries of the moduli space (the boundaries describe degenerating Riemann surfaces). Nathan uses the usual arguments that all divergences can be reinterpreted as the infrared divergences, and with the help of a few vanishing theorems that are not hard to prove in his formalism, he can complete the proof of the perturbative finiteness of the superstring S-matrix, falsifying Lee Smolin's scary scenarios about the possible mysterious problems of superstring theory at higher loops.
Hiroši Ooguri has verified a significant fraction of the details of Nathan's calculations - so Nathan is not the only one who now claims that the proof of finiteness has been completed. Let me emphasize that if I ever suggested that this work of Nathan is probably a proof - or almost a proof - it does not mean that I am aware of any problems with it! It just means that I have not checked all the details.
Let me hope that Hiroši will eventually understand that we can't just claim that "we're sure that the proof is correct" because it was written by an ingenious friend of ours. If we did so, we would be like our friends in loop quantum gravity who also claim that they have proved everything - except that the proofs usually don't mean much. Yes, now I have good reasons to be convinced that Nathan's proof of the perturbative finiteness of superstring amplitudes is a complete proof - and the more I look at the details of the paper, the more my doubts evaporate. But once again, I have not checked everything. It's a difficult stuff.
I agree with Hiroši that Nathan's paper is one of the most fascinating papers of 2004. Nathan reviewed it at Strings 2004, and the talk may be a better starting point to learn it:
And the strength of the formalism does not stop here:
- The BRST quantization in the pure spinor formalism can also be showed to be equivalent to the "semi light cone" quantization of the Green-Schwarz string - see a paper by Berkovits and Marchioro
- The pure spinor GS formalism can deal with the Ramond-Ramond backgrounds, unlike the RNS formalism, which can potentially become a very useful language to study AdS5 x S5 - the most popular example of the AdS/CFT correspondence, for example. Nathan has shown that the theory on this background is conformal in hep-th/0411170 which may be a good starting point for further investigations, e.g. an attempt to study the highly curved AdS space that should be dual to a weakly coupled gauge theory.
I would like to know whether I am wrong about the following statement: if the pure spinor formalism - that only deals with the Riemann surfaces and not the supersurfaces - is equivalent to the RNS calculations, does not it prove that the supermoduli space of the supersurfaces must be a split supermanifold, because it can be "projected" to its bosonic subspace?
Hiroši disagrees, and he's probably right. There's a possibility that the RNS loop calculation does not really exist, and the pure spinor language is the "smarter route to take" here. But in that case, I would still like to see more clearly why the n-loop amplitudes in Nathan's formalism are unitary order by order in the perturbation theory. It seems that this fact is not being proved by the equivalence to the RNS loop amplitudes - that don't have to have a simple prescription.
Don't get me wrong, I definitely feel that it's correct that the amplitudes constructed as these integrals over the moduli space should give a unitary S-matrix. Is there a proof? Of course, the equivalence to the Hamiltonian light cone gauge GS computation is also enough - the unitarity then follows from the hermiticity of the Hamiltonian.
Hiroši wrote me another interesting comment:
- I would also like to point out that the fact that the superpotential terms of the CY compactification of the type II superstring can be computable using the topological string theory is a special case of the Berkovits formalism. There you agree that the amplitudes are expressed as integrals over the bosonic moduli space of Riemann surfaces and the results are manifestly finite.
That's really impressive, but frankly speaking, someone will have to help me how the pure spinors relate to topological string theory. Yes, I got a help - and the answer may be explained in the paper by Berkovits, Ooguri, and Vafa.
