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.


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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    $\begingroup$ 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. $\endgroup$ – 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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