The traditional formulation of quantum field theory—encoded in its very name—is built on the two pillars of locality and unitarity [2]. The standard apparatus of Lagrangians and path integrals allows us to make these two fundamental principles manifest. This approach, however, requires the introduction of a large amount of unphysical redundancy in our description of physical processes. Even for the simplest case of scalar field theories, there is the freedom to perform field redefinitions. Starting with massless particles of spin 1 or higher, we are forced to introduce even larger, gauge redundancies [2].

Over the past few decades, there has been a growing realization that these redundancies hide amazing physical and mathematical structures lurking within the heart of quantum field theory. This has been seen dramatically at strong coupling in gauge/gauge (see, e.g. [3–5]) and gauge/gravity dualities [6]. The past decade has uncovered further remarkable new structures in field theory even at weak coupling, seen in the properties of scattering amplitudes in gauge theories and gravity (for reviews, see [7–12]). The study of scattering amplitudes is fundamental to our understanding of field theory, and fueled its early development in the hands of Feynman, Dyson, and Schwinger among others. It is therefore surprising to see that even here, by committing so strongly to particular, gauge-redundant descriptions of the physics, the usual formalism is completely blind to astonishingly simple and beautiful properties of the gauge-invariant physical observables of the theory.

Many of the recent developments have been driven by an intensive exploration of N = 4 supersymmetric Yang–Mills (SYM) in the planar limit [12, 13]. The all-loop integrand for scattering amplitudes in this theory can be determined by a generalization of the BCFW recursion relations [14], in a way that is closely tied to remarkable new structures in algebraic geometry, associated with contour integrals over the Grassmannian G(k, n) [15–18]. This makes both the conformal and long-hidden dual conformal invariance of the theory (which together close into the infinite-dimensional Yangian symmetry) completely manifest [19]. It is remarkable that a single function of external kinematical variables can be interpreted as a scattering amplitude in one space-time, and as a Wilson loop in another (for a review, see [12]). Each of these descriptions makes a commitment to locality in its own space-time, making it impossible to see the dual picture.