We have explored much of the remarkable physics and mathematics of scattering amplitudes in planar N = 4, as seen through the lens of on-shell diagrams as the primary objects of study. Let us conclude by making some brief comments on further avenues of research.

One immediate extension of our work is the continued study of theories with N < 4 supersymmetry, whose most basic features we sketched out in Chapter 15. For N ≥ 1, we expect that all-loop BCFW recursion holds just as for N = 4, together with its realization in terms of on-shell diagrams. For non-supersymmetric theories, the forward-limit of tree amplitudes are singular, and thus don't directly give us the single-cuts of the loop-integrand [81]. More thought is needed to establish a connection between on-shell diagrams and the full amplitude, though it is likely that fully understanding the on-shell diagrams will continue to play an important role in determining N = 0 amplitudes as well.

The general connection between on-shell diagrams and the Grassmannian has nothing to do with any particular theory, only with the general picture of amalgamating basic three-particle amplitudes, and the connection to the positive Grassmannian in particular holds for any planar theory of massless particles in 4 dimensions. Only the form on the Grassmannian changes from theory to theory. As briefly discussed in Chapter 15, the essential physical novelty of gauge theories with N ≤2 supersymmetry is the presence of UV divergences. The most physical, Wilsonian way to think about UV divergences makes critical use of off-shell ideas, and so a major challenge is finding the correct way of thinking about such physics in a directly on-shell language. It is fascinating to see that the UV and IR singularities, together with UV/IR decoupling, are reflected directly in on-shell diagrams through simple structures in the Grassmannian. A clear goal would be to understand the physics of the renormalization group along these lines.

Another obvious extension is to push beyond the planar limit, starting already with N = 4; in this case, there is no longer an obvious notion of “the loop integrand,” and thus we must learn how to establish a connection between on-shell diagrams and the full scattering amplitude along the lines of the BCFW construction in the planar limit.