Relations between multiple zeta values and delta values from Drinfeld's associator series
Cameron James Deverall Kemp
TL;DR
The paper develops two parallel constructions of Drinfeld’s associator $\Phi_{\mathrm{KZ}}(A,B)$—one with coefficients in multiple zeta values (MZVs) and another with delta values $\delta_{s_1,...,s_k}$—and shows their equality by expanding to fifth order in the deformation parameter $\\hbar$. By matching the order-by-order terms, it derives explicit relations between MZVs and delta values, such as $\\zeta_2=2\\delta_2+(\\ln2)^2$, and more intricate identities involving higher-weight zetas and delta-values. The MZV-series produces a relatively streamlined fifth-order formula in compact Lie-theoretic form, while the delta-series yields a rich set of linear relations governed by integrals $\\mathcal{I}_{\\ell_1...\\ell_r}$ and the constant $\\ln2$. Overall, the work uncovers both known and novel connections between polylogarithmic values, MZVs, and delta-values, enriching the algebraic framework of associators and their polylogarithmic evaluations.
Abstract
It is shown that novel relations between multiple zeta values and single-variable multiple polylogarithms at 1/2 (delta values) can be derived by comparing two distinct, yet a priori equal, series formulae for the Drinfeld associator (from the Knizhnik-Zamolodchikov connection). In particular, we demonstrate that two new relations are found by comparing the fifth order terms of each series formula.