M Theory Lesson 84

I was hoping somebody could point me to papers on 2-categories and double shuffle relations for MZV algebras. These arise from the two shuffle products, on series or on integral forms, of zeta values. Both of these products may be thought to arise from associahedra combinatorics, as discussed in the work of Brown. A recent paper, by Zagier et al, extends these relations to a set that can characterise the full algebra of MZVs. It does so by introducing an infinite series

$A(x) = exp(\sum_{2}^{\infty} \frac{-1}{n} \zeta (n) x^n)$

with zeta value coefficients. New results include a proof that the weight 3 and 4 zeta values form one dimensional algebras. For example,

$4 \zeta (3,1) = \zeta (4) = \zeta (3,1) + \zeta (2,2) = \zeta (2,1,1)$

There is a conjecture due to Zagier which states that the dimensions of the $\mathbb{Q}$ vector spaces for weight $n$ obey the recurrence relation

$d_n = d_{n – 2} + d_{n – 3}$

with $d_{0} = 1$ and $d_{1} = 0$. The appendix discusses the weight $n$ and depth $d$ generalisation due to Broadhurst and Kreimer, which I believe appeared in the Feynman diagram paper listed as a reference.

3 Responses so far »

  1. 1

    Matti Pitkanen said,

    Dear Kea,

    the link to the Zagier’s paper does not work for some reason.

  2. 2

    Kea said,

    Sorry! Should be fixed now.

  3. 3

    Doug said,

    Hi Kea, Matti,

    I need to use an university computer to read the Zagier paper in PS.

    The Brown paper in PDF appears to relate to blog posts of Terence Tao and Lieven le Bruyn?

    I am curious why PF was dropped from your blog links?

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