# Is there a T-dual of Witten's twistor topological string theory?

In late 2003, Edward Witten released a paper that revived the interest in Roger Penrose's twistors among particle physicists. The scattering amplitudes of gluons in $N=4$ gauge theory in four dimensions were expressed in a simple way using the twistor variables. Witten also proposed a particular model, the topological B-model on the $CP^{3|4}$ twistor space, to generate all these amplitudes.

These methods began their own life but the topological B-model became largely silent, perhaps partly because the phenomenologists who fell in love with these things haven't been trained in string theory, especially not in the topological one. However, many twistor-related discoveries in the recent 3 years - which were made without Witten's constructive picture - lead me to ask whether Witten's theory actually knows about these matters.

In particular, the "dual superconformal symmetry" was first noticed by Drummond et al. in 2006 and derived by stringy methods by Alday & Maldacena in 2008 or so. The 3+1 dimensions on the CFT boundary may be T-dualized to produce another copy of the Yang-Mills theory that is superconformally invariant once again. Scattering amplitudes have been converted to the expectation values of piecewise linear Wilson loops in the dual theory - the segments have the directions and length of the light-like momenta of the scattering particles. My question is

Can you also "T-dualize" Witten's topological B-model to obtain another one in which the scattering amplitudes are computed in a different way?

If you think that the answer is Yes, I would also like to know what is the "dual prescription" for the supersymmetric Yang-Mills amplitudes and whether the D1- and D5-branes in Witten's original models are replaced by other D1- and D5-branes or, for example, by D3-branes.

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I think this is a very nice question. It might take some time until it may be answered since it is one of few research questions on the site so far. I greatly acknowledge your work done for the community @Luboš Motl. You may also want to consider area51.stackexchange.com/proposals/23848/theoretical-physics as a place to post research question as soon as the site starts :) Greets – Robert Filter Jan 17 '11 at 13:33
Sir Roger Penrose! ;) In any case, good question, but I think as this is in advanced string theory question, you may be lucky to get answers here. – Noldorin Jan 17 '11 at 14:36

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Anyone who can answer this question correctly is likely to write up a journal article and publish it first, before answering it here. :) – QGR Jan 17 '11 at 15:20

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Hi! I don't expect to get a satisfactory answer, with all my respect to the big shot physicists here. After all, Nima Arkani-Hamed who has discussed it with Edward Witten hasn't answered it, either. But who knows? :-) If someone writes an article about it after reading this, good for science and for this server, too. (I have only partial clues how the T-dual theory could work.) Noldorin: sorry, we abandoned the kingdom in 1918 and kings were replaced by presidents. These days, we don't use Sir for the president to be respectful. Instead, he is the Professor. ;-) – Luboš Motl Jan 17 '11 at 17:58

## 2 Answers

Luboš would know this already (he's acknowledged in this paper), but Neitzke and Vafa conjectured in 2004 that the mirror manifold of $CP^{3|4}$ is a quadric surface $Q$ in $CP^{3|3}$ x $CP^{3|3}$, and mirror symmetry is a type of T-duality. There were a few follow-ups, including a paper by Sinkovics and Verlinde which studies classical $N=4$ super-Yang-Mills on the quadric, which in the very last paragraph asks whether the quantum scattering amplitudes can also be recovered from $Q$. After that, I can find nothing. But at least it's a place to begin!

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This is a very good reminder, @Mitchell! I would forget about this mirror manifold, especially because I never found it too natural... +1 but let's keep the question open. The newest citation of Verlinde-Anamaria is from 2006 which is long before the dual superconformal symmetry etc. – Luboš Motl Feb 8 '11 at 19:08

The only thing which might be done is to cast the question in different forms. The CY supermanifold $CP^{3|4}$ for the "4" corresponding to a spinor field and "3" coordinates might be cast into $J^5(C) = R\oplus J^4\oplus C^4$, so the twistor components are contained in a $5\times5$ self adjoint matrix. By extension or analogue the question is whether this has some higher Jordan algebraic or a $J^3(O)$ realization. The cubic form gives $OP^2 \sim OP^1$, which might (I stress might without any strong evidence) mean the $D1$ is dual to a $D2$ or $M2$. The scalar part of this cubic form is the Chern-Simons form. As for any duality with the $D5$ (or $NS5$ “black brane”) that would have to be determined. The CS Lagrangian has a winding number transformation $L \rightarrow L + 2πNk$, which might then have a coordinate dual $x \rightarrow x + 2πiR$ winding or compactification.

A chance to ponder per chance to solve. This might be one way of trying to think about it.

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Thank you for your answer and welcome to the site! I don't have the background to get your answer but for those who do: Could you please format your formula using the MathJax notation employed here? Just set everything into two \$'s and it will be displayed like via LaTeX. Greets – Robert Filter Jan 17 '11 at 21:14
If there is something wrong with my suggestion, please let me and everyone know --- just be kind about. If it is wrong, then we need to think about something else. Cheers – Lawrence B. Crowell Jan 17 '11 at 22:18
I fixed some latex. @Lawrence you just need to put your math markup in  – user346 Jan 17 '11 at 22:27

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Thanks, Lawrence, but if your advanced text using as unexpected yet mutually unrelated objects as the Jordan algebra over octonions, M2-branes, NS5-branes, and the CS Lagrangian with winding numbers (?) has anything to do with my question, it may take centuries for the likes of me to understand the relationship. ;-) – Luboš Motl Jan 20 '11 at 8:48
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