Magic Motives

Speaking of orbifold Euler characteristics, let's put the magic formula

$f\left(n\right)=\prod _{m=0}^{n-1}\frac{m!}{(m+n)!}$

in terms of Euler characteristics. First, let $m=2g-2$ be the Euler characteristic of a closed surface of genus $g$. This already suggests allowing non-orientable surfaces to account for odd values of $m$. Then consider moduli spaces $M_{m,n}$ for $(n+1)$ punctured surfaces. The orbifold Euler characteristic of such a space will be denoted by $E_{m,n}$. Using Mulase's expression for $E_{m,n}$ and assuming it may be extended to the non-orientable case, one finds a natural definition for the moment coefficients of the form

$f\left(n\right)=\frac{1}{(n+1)!}\prod _{m=0}^{n-1}{b}_{m+2}{E}_{m,n}^{-1}$

which is a product over surfaces of genus $g$ limited by $n$, and where $b_i$ is a Bernoulli number (for even $m$ these are defined in terms of zeta values for odd negative reals). One should take more care with the non-orientable factors, but this simple exercise shows that the zeta moments are naturally dependent on categorical invariants associated to complex moduli.