We compute the modification of the w 1+∞ algebra of soft graviton, gluon and scalar currents in the Celestial CFT due to non-minimal couplings. We find that the Jacobi identity is satisfied only when the spectrum and couplings of the theory obey certain constraints. We comment on the similarities and essential differences of this algebra to W 1+∞ .

We continue the study of positive geometries underlying the Grassmannian string integrals, which are a class of "stringy canonical forms", or stringy integrals, over the positive Grassmannian mod torus action, G + (k, n)/T . The leading order of any such stringy integral is given by the canonical function of a polytope, which can be obtained using the Minkowski sum of the Newton polytopes for the regulators of the integral, or equivalently given by the so-called scattering-equation map. The canonical function of the polytopes for Grassmannian string integrals, or the volume of their dual polytopes, is also known as the generalized bi-adjoint φ 3 amplitudes. We compute all the linear functions for the facets which cut out the polytope for all cases up to n = 9, with up to k=4 and their parity conjugate cases. The main novelty of our computation is that we present these facets in a manifestly gauge-invariant and cyclic way, and identify the boundary configurations of G + (k, n)/T corresponding to these facets, which have nice geometric interpretations in terms of n points in (k−1)-dimensional space. All the facets and configurations we discovered up to n = 9 directly generalize to all n, although new types are still needed for higher n.

We propose to use tensor diagrams and the Fomin-Pylyavskyy conjectures to explore the connection between symbol alphabets of n-particle amplitudes in planar $$ \mathcal{N} $$ N = 4 Yang-Mills theory and certain polytopes associated to the Grassmannian Gr(4, n). We show how to assign a web (a planar tensor diagram) to each facet of these polytopes. Webs with no inner loops are associated to cluster variables (rational symbol letters). For webs with a single inner loop we propose and explicitly evaluate an associated web series that contains information about algebraic symbol letters. In this manner we reproduce the results of previous analyses of n ≤ 8, and find that the polytope $$ {\mathcal{C}}^{\dagger}\left(4,9\right) $$ C † 4 9 encodes all rational letters, and all square roots of the algebraic letters, of known nine-particle amplitudes.