Hopf monoids, permutohedral cones, and generalized retarded functions

TL;DR

This paper links the combinatorial structure of Hopf monoids with geometric objects like permutohedral cones, revealing new insights into quantum field theory's causal perturbation framework.

Contribution

It constructs a geometric realization of the Hopf monoid of set compositions using permutohedral cones, connecting algebraic, geometric, and quantum field theory concepts.

Findings

Identifies monomial basis with characteristic functions of permutohedral tangent cones.

Shows Steinmann relations correspond to functions restricted to chambers.

Provides a new geometric interpretation of Epstein-Glaser renormalization.

Abstract

The commutative Hopf monoid of set compositions is a fundamental Hopf monoid internal to vector species, having undecorated bosonic Fock space the combinatorial Hopf algebra of quasisymmetric functions. We construct a geometric realization of this Hopf monoid over the adjoint of the (essentialized) braid hyperplane arrangement, which identifies the monomial basis with signed characteristic functions of the interiors of permutohedral tangent cones. We show that the indecomposable quotient Lie coalgebra is obtained by restricting functions to chambers of the adjoint arrangement, i.e. by quotienting out the higher codimensions. The resulting functions are characterized by the Steinmann relations of axiomatic quantum field theory, demonstrating an equivalence between the Steinmann relations, tangent cones to (generalized) permutohedra, and having algebraic structure internal to species. Our results give a new interpretation of a construction appearing in the mathematically rigorous formulation of renormalization by Epstein-Glaser, called causal perturbation theory. In particular, we show that operator products of time-ordered products correspond to the H-basis of the cocommutative Hopf monoid of set compositions, and generalized retarded products correspond to a spanning set of its primitive part Lie algebra.