TY - JOUR

T1 - Hopf monoids, permutohedral cones, and generalized retarded functions

AU - Norledge, William

AU - Ocneanu, Adrian

N1 - Funding Information:
Funding. The second author was partly supported by NSF grants DMS-9970677, DMS-0200809, DMS-0701589. Both authors were partly supported by Templeton Religion Trust grant TRT-0159.
Publisher Copyright:
© 2023 Association Publications de l’Institut Henri Poincaré Published by EMS Press This work is licensed under a CC BY 4.0 license.

PY - 2023/12/1

Y1 - 2023/12/1

N2 - 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.

AB - 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.

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U2 - 10.4171/AIHPD/159

DO - 10.4171/AIHPD/159

M3 - Article

AN - SCOPUS:85166175714

SN - 2308-5827

VL - 10

SP - 555

EP - 609

JO - Annales de l'Institut Henri Poincare (D) Combinatorics, Physics and their Interactions

JF - Annales de l'Institut Henri Poincare (D) Combinatorics, Physics and their Interactions

IS - 3

ER -