Magic Motives

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

$f(n) = \prod_{m=0}^{n – 1} \frac{m!}{(m + n)!}$

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

$f(n) = \frac{1}{(n + 1)!} \prod_{m=0}^{n – 1} b_{m + 2} E_{m,n}^{-1}$

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.


3 Responses so far »

  1. 1

    L. Riofrio said,

    Orbits are definitely on our minds. Perhaps there is some relation including planetary and atomic orbits. Thanks for the May 28 reference, I just linked badk to it.

  2. 2

    Doug said,

    Is the lower half plane simply a mirror reflexion on the boundary line -1_0_+1

    Upper HP

    Lower HP

    So figure 1.3, page 11, will a mirror image be the lower half plane?

  3. 3

    Doug said,

    Speculation on orbits, planetary and atomic.?.?

    Suppose planetary orbits or multiple electron shells are Parallel One Loops functioning as a type of Solenoid?

    Biot-Savart Law and Applications
    Some examples of geometries where the Biot-Savart law can be used to advantage in calculating the magnetic field resulting from an electric current distribution.

    Iron Core Solenoid
    An iron core has the effect of multiplying greatly the magnetic field of a solenoid compared to the air core solenoid on the left.

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