The Amplituhedron

In 2013, building on previous Grassmannian formulations and on Hodges’ ideas in scattering amplitudes, Arkani-Hamed and Trnka defined the amplituhedron $$\mathcal {A}_{n,k,m}$$ as the image of the positive Grassmannian $$\textrm{Gr}^{\ge 0}_{k,n}$$ under a totally positive linear map. Regarded as a non-linear generalization of (cyclic) polytopes inside the Grassmannian, it is a semialgebraic set with beautiful and rich combinatorics. The $$m=4$$ amplituhedron $$\mathcal {A}_{n,k,4}$$ encodes tree-level amplitudes in $$\mathcal {N}=4$$ SYM. The $$m=2$$ amplituhedron $$\mathcal {A}_{n,k,2}$$ , toy-model for the $$m=4$$ case, it also enters the geometry of some loop amplitudes, tree-level form factors of half-BPS operators and correlators in planar $$\mathcal {N}=4$$ SYM. In this chapter, after a review on the positive Grassmannian, we introduce and prove new properties of the amplituhedron. We define the amplituhedron-analogue of the matroid stratification of the Grassmannian, which we call sign stratification. We classify all positroid tiles of the amplituhedron $$\mathcal {A}_{n,k,2}$$ – full-dimensional images of positroid cells of $${{\,\textrm{Gr}\,}}^{\ge 0}_{k,n}$$ on which the amplituhedron map is injective—and we give them an intrinsic sign-characterization in the amplituhedron. We use this result to prove Arkani-Hamed–Thomas–Trnka’s sign-flip conjecture on $$\mathcal {A}_{n,k,2}$$ . Finally, we review how to extract tree-level amplitudes of $$\mathcal {N}=4$$ SYM from the $$m=4$$ amplituhedron.