Renormalization groupoids in algebraic topology

TL;DR

This paper explores the algebraic structures underlying noncommutative cobordism spectra, linking renormalization Hopf algebras to characteristic numbers and automorphisms in algebraic topology.

Contribution

It introduces a renormalization Hopf algebra framework for automorphisms of noncommutative cobordism spectra, connecting algebraic topology with quantum electrodynamics concepts.

Findings

Characterization of automorphisms via a Hopf algebra of formal diffeomorphisms

Representation of the structure as a groupoid scheme over Z

Applications to symplectic toric manifolds and combinatorics

Abstract

Continuing work begin in arXiv:1910.12609, we interpret the Hurewicz homomorphism for Baker and Richter's noncommutative complex cobordism spectrum $M\xi$ in terms of characteristic numbers (indexed by quasi-symmetric functions) for complex-oriented quasitoric manifolds, and show that automorphisms or cohomology operations on this representation are defined by a `renormalization' Hopf algebra of formal diffeomorphisms at the origin of the noncommutative line, previously considered (over $Q$) in quantum electrodynamics. The resulting structure can be presented in purely algebraic terms, as a groupoid scheme over $Z$ defined by a coaction of this Hopf algebra on the ring of noncommutative symmetric functions. We sketch some applications to symplectic toric manifolds, combinatorics of simplicial spheres, and statistical mechanics.