Let me start with a sentence that occurs quite often on this blog, and it is no coincidence: The most interesting hep-th paper today is arguably the first one. Nima Arkani-Hamed, Freddy Cachazo, Clifford Cheung, and Jared Kaplan wrote about
a duality for the S-matrix (click).They present a concise formula for scattering amplitudes in N=4 supersymmetric Yang-Mills up to all orders in perturbation theory and they market it as a new "weak-weak dual description" for this S-matrix. The most informative formulae are equations (9) and (10), analyzed in the rest of the paper:
This is a formula for the scattering amplitude of "n" gluons with "k" of them having the opposite chirality than others. The amplitude is written as an integral over a "kn"-dimensional space of rectangular matrices "C" (with no twistorial indices!). These matrices may be interpreted as parameters of "k"-dimensional planes embedded in an "n"-dimensional space: the space of matrices is therefore an "(n choose k)-1"-dimensional projective space (plus an irrelevant scaling variable).
The integral includes a measure which is a "cyclic" product of "n" inverse determinants of "k x k" matrices. They're multiplied by the integrand, which is a product of "k" delta-functions in a twistor "four-bosonic plus four-fermionic" space. I guess that there should exist a generalized superspace formula that unifies all the n-gluon formulae for a fixed "n" but all values of "k" into one formula that allows the helicities to be arbitrary and that includes the other superpartners as well. It may even be written somewhere in the paper already. (Well, they surprisingly find that different helicities are obtained from the same integral over different charts...)
Even now, their formula has the clear ambition to extend the BCFW twistor rules for the scattering amplitudes to all orders of perturbation theory. This obviously brings two possible question marks:
- whether some quantities describing the loop scattering amplitudes of N=4 SYM may be well-defined, despite the infrared divergences
- whether we should be surprised that only the perturbative portion of such amplitudes is being calculated, and whether the lack of nonperturbative knowledge is a serious problem.
Another general question is whether such a new compact formula should be considered a "theory". Well, in some generalized sense, it is surely a "theory". As soon as the amplitudes are written as integrals over a space, we have a new "physical picture", even though it could look counterintuitive.
We may always say that the matrices "C" - the planes - describe some virtual degrees of freedom in some generalized Feynman diagrams.
Incidentally, it is impossible for me not to mention my own observation, namely a possible relationship with an unknown but interesting 1996 paper by Albert Schwarz, Grassmannian and String Theory that noticed that the moduli spaces of Riemann surfaces in perturbative string theory can be embedded into an infinite-dimensional Grassmannian manifold.
Schwarz has argued that such Grassmannians are natural objects as moduli spaces and made some guesses about the mechanisms of "localization" that produce the perturbative part from his Grassmannians which were supposed to contain the full nonperturbative answer, as well as a sketch of a possible proof of S-dualities in this framework.
It just happened that an integral over a Grassmannian, i.e. a space of subspaces of another linear space, occurred in this new formula, too, somewhat confirming Schwarz's general mathematical "prediction". You are invited to think about the integration variables as some new generalized moduli of a new generalized "worldsheet" or "worldvolume" optimized for the on-shell N=4 supersymmetric gauge theory.
Further research will say whether their line of reasoning leads somewhere: Witten's original twistor paper has already seen a lot of additional insights. The future results may uncover all mysteries hiding behind the N=4 SYM and its integrability and they may become even more far-reaching if they shed light on all possible integration variables that may emerge in formulae for amplitudes anywhere in physical theories dual to quantum gravity.