Combinatorial Dyson–Schwinger equations and inductive data types

Kock, Joachim

doi:10.1007/s11467-015-0544-3

Cited by 6 publications

(2 citation statements)

References 32 publications

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“…The close connections with Type Theory is an important byproduct of the present contribution, expanded upon in a companion paper [31]. In short, Green functions are revealed to be inductive types, in a precise technical sense, thus formalising the classical wisdom that Dyson-Schwinger equations express self-similarity properties of their solutions.…”

Section: Introductionmentioning

confidence: 67%

Polynomial functors and combinatorial Dyson–Schwinger equations

Kock

2017

Journal of Mathematical Physics

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We present a general abstract framework for combinatorial Dyson-Schwinger equations, in which combinatorial identities are lifted to explicit bijections of sets, and more generally equivalences of groupoids. Key features of combinatorial Dyson-Schwinger equations are revealed to follow from general categorical constructions and universal properties. Rather than beginning with an equation inside a given Hopf algebra and referring to given Hochschild 1-cocycles, our starting point is an abstract fixpoint equation in groupoids, shown canonically to generate all the algebraic structure. Precisely, for any finitary polynomial endofunctor P defined over groupoids, the system of combinatorial Dyson-Schwinger equations X = 1 + P(X) has a universal solution, namely the groupoid of P-trees. The isoclasses of P-trees generate naturally a Connes-Kreimer-like bialgebra, in which the abstract Dyson-Schwinger equation can be internalised in terms of canonical B + -operators. The solution to this equation is a series (the Green function) which always enjoys a Faà di Bruno formula, and hence generates a sub-bialgebra isomorphic to the Faà di Bruno bialgebra. Varying P yields different bialgebras, and cartesian natural transformations between various P yield bialgebra homomorphisms and sub-bialgebras, corresponding for example to truncation of Dyson-Schwinger equations. Finally, all constructions can be pushed inside the classical Connes-Kreimer Hopf algebra of trees by the operation of taking core of P-trees. A byproduct of the theory is an interpretation of combinatorial Green functions as inductive data types in the sense of Martin-Löf Type Theory (expounded elsewhere). Contents 0 Introduction 2 1 Combinatorial Dyson-Schwinger equations 6 2 Polynomial functors in one variable; basic aspects of the theory 8 3 Polynomial functors in many variables 16 4 Groupoids 18 5 Polynomial functors over groupoids, and P-trees 22 6 Abstract combinatorial Dyson-Schwinger equations 26 7 Bialgebras and B + -operators 27 8 Natural transformations 34 9 Foissy equations 36 10 Trees versus graphs 41

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Section: Introductionmentioning

confidence: 67%

Polynomial functors and combinatorial Dyson–Schwinger equations

Kock

2017

Journal of Mathematical Physics

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“…The Green function is the sum of all operadic trees, weighted by symmetry factors (and it is crucial for these symmetry factors to come out right to use operadic trees rather than combinatorial trees); it appears as solution to a certain abstract combinatorial Dyson-Schwinger equation in the category of groupoids. Further relationships with Category Theory and Logic are explored in [10] and [11].…”

Section: Trees -Combinatorial Versus Operadic In the Usual Connes-krmentioning

confidence: 99%

Perturbative Renormalisation for Not-Quite-Connected Bialgebras

Kock

2015

Lett Math Phys

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We observe that the Connes-Kreimer Hopf-algebraic approach to perturbative renormalisation works not just for Hopf algebras but more generally for filtered bialgebras B with the property that B0 is spanned by grouplike elements (e.g. pointed bialgebras with the coradical filtration). Such bialgebras occur naturally both in Quantum Field Theory, where they have some attractive features, and elsewhere in Combinatorics, where they cover a comprehensive class of incidence bialgebras. In particular, the setting allows us to interpret Möbius inversion as an instance of renormalisation.

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Feynman diagrams and their algebraic lattices

Borinsky¹,

Kreimer²

2017

Resurgence, Physics and Numbers

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We present the lattice structure of Feynman diagram renormalization in physical QFTs from the viewpoint of Dyson-Schwinger-Equations and the core Hopf algebra of Feynman diagrams. The lattice structure encapsules the nestedness of diagrams. This structure can be used to give explicit expressions for the counterterms in zero-dimensional QFTs using the lattice-Moebius function. Different applications for the tadpole-free quotient, in which all appearing elements correspond to semimodular lattices, are discussed. * DK thanks the Alexander von Humboldt Foundation and the BMBF for support by an Alexander von Humboldt Professorship. It is a pleasure for both authors to thank David Broadhurst, Spencer Bloch, Dominique Manchon and Karen Yeats for helpful discussions. We also thank Fréderic Fauvet for organizing the workshop Resurgence, Physics and Numbers, Centro de Giorgi, Pisa, May 2015, and hospitality there.

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Combinatorial Dyson–Schwinger equations and inductive data types

Cited by 6 publications

References 32 publications

Polynomial functors and combinatorial Dyson–Schwinger equations

Polynomial functors and combinatorial Dyson–Schwinger equations

Perturbative Renormalisation for Not-Quite-Connected Bialgebras

Feynman diagrams and their algebraic lattices

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