Braid arrangement bimonoids and the toric variety of the permutohedron

TL;DR

This paper reveals that the permutohedral space has a bimonoid structure that unifies various combinatorial Hopf algebras and relates to Feynman amplitudes, offering a geometric perspective on these algebraic objects.

Contribution

It establishes the permutohedral space as a fundamental bimonoid object that geometrically interprets many combinatorial Hopf algebras and connects to Feynman graph structures.

Findings

Permutohedral space has a cocommutative bimonoid structure.

This structure aligns with combinatorial Hopf algebras.

Feynman amplitudes relate to integrals over permutohedral space.

Abstract

We show that the toric variety of the permutohedron (=permutohedral space) has the structure of a cocommutative bimonoid in species, with multiplication/comultiplication given by embedding/projecting-onto boundary divisors. In terms of Losev-Manin's description of permutohedral space as a moduli space, multiplication is concatenation of strings of Riemann spheres and comultiplication is forgetting marked points. In this way, the bimonoid structure is an analog of the cyclic operad structure on the moduli space of genus zero marked curves. Covariant/contravariant data on permutohedral space is endowed with the structure of cocommutative/commutative bimonoids by pushing-forward/pulling-back data along the (co)multiplication. Many well-known combinatorial objects index data on permutohedral space. Moreover, combinatorial objects often have the structure of bimonoids, with multiplication/comultiplication given by merging/restricting objects in some way. We prove that the bimonoid structure enjoyed by these indexing combinatorial objects coincides with that induced by the bimonoid structure of permutohedral space. Thus, permutohedral space may be viewed as a fundamental underlying object which geometrically interprets many combinatorial Hopf algebras. Aguiar-Mahajan have shown that classical combinatorial Hopf theory is based on the braid hyperplane arrangement in a crucial way. This paper aims to similarly establish permutohedral space as a central object, providing an even more unified perspective. The main motivation for this work concerns Feynman amplitudes in the Schwinger parametrization, which become integrals over permutohedral space if one blows-up everything in the resolution of singularities. Then the Hopf algebra structure of Feynman graphs, first appearing in the work of Connes-Kreimer, coincides with that induced by the bimonoid structure of permutohedral space.