Conjectured Enumeration of irreducible Multiple Zeta Values, from Knots and Feynman Diagrams

TL;DR

This paper tests a conjecture relating the enumeration of irreducible multiple zeta values (MZVs) to knot theory and Feynman diagrams, using extensive computational methods up to high weights and depths, supporting the conjecture's validity.

Contribution

The authors perform large-scale analytical and numerical computations that verify the BK conjecture on the enumeration of irreducible MZVs, extending previous work with high-weight and depth data.

Findings

Conjecture on the generating function for irreducible MZVs is consistent with computed data.

Computations up to weights 44, 37, 42, 27 at depths 2-5 support the conjecture.

Results reinforce the connection between knot theory, Feynman diagrams, and MZV enumeration.

Abstract

Multiple zeta values (MZVs) are under intense investigation in three arenas -- knot theory, number theory, and quantum field theory -- which unite in Kreimer's proposal that field theory assigns MZVs to positive knots, via Feynman diagrams whose momentum flow is encoded by link diagrams. Two challenging problems are posed by this nexus of knot/number/field theory: enumeration of positive knots, and enumeration of irreducible MZVs. Both were recently tackled by Broadhurst and Kreimer (BK). Here we report large-scale analytical and numerical computations that test, with considerable severity, the BK conjecture that the number, $D_{n,k}$, of irreducible MZVs of weight $n$ and depth $k$, is generated by $\prod_{n\ge3}\prod_{k\ge1}(1-x^n y^k) ^{D_{n,k}}=1-\frac{x^3y}{1-x^2}+\frac{x^{12}y^2(1-y^2)}{(1-x^4)(1-x^6)}$, which is here shown to be consistent with all shuffle identities for the corresponding iterated integrals, up to weights $n=44, 37, 42, 27$, at depths $k=2, 3, 4, 5$, respectively, entailing computation at the petashuffle level. We recount the field-theoretic discoveries of MZVs, in counterterms, and of Euler sums, from more general Feynman diagrams, that led to this success.