Brave new categorical spectral positive Schubert geometry and the categorical Dual Amplituhedron

Abstract

This ArXiv preprint of my doctoral dissertation, which, at this stage, has not yet been accepted by the doctoral thesis committee, is intended both to lay the groundwork for a series of papers, and to confirm the existence of a first proposed solution to the "Dual Amplituhedron" problem posed in 2014 by Princeton physicists N. Arkani-Hamed and J. Trnka, a mathematical object which encapsulate the calculation of scattering amplitudes in high-energy particle colliders. The first part of this thesis deals with a rewriting of the positive real Grassmannian and performing the singular gluing of its positroid varieties in a new way via Spectral Algebraic Geometry of J. Lurie on "structured" spaces and a categorification of his "Tannaka Duality for Quasi-coherent Stacks", finding in this formal moduli problem, a compact yet holistic formulation via perverse intersection complexes of P. Deligne. This algebraico-geometric perspective paired with a synthetic differential-geometric perspective, namely the Differential Cohesion of B. Lawvere and U. Schreiber, subsumes infinitesimal thickenings, crystalline cohomology of De Rham stacks, the "Modalities of Structured Geometries" and the unification of their cohomologies of the underlying concrete topological etale algebraic space. The second part uses this rewriting of positive Schubert geometry and "The Cohomology of Brauer-Grothendieck Spaces" of B. To\"en and B. Antieau, to show that, instead of the Grassmannian spectral Deligne-Mumford stack which is autodual in the infinity-category of prestable infinity-categories of modules on E\infty-ring spectra, the Amplituhedron, categorified in a functor between infinity topo\"i, possesses a concrete non-trivial dual, and has a De Rham volume, facts of interest for the expected Duality between the Standard Model of particles and String Theory. This new construction yields the Dual Amplituhedron.