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What is Renormalization Groupoids in Algebraic Topology about?

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.

Author

Morava, Jack

Published

2020

Language

EN