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
    http://mathworld.wolfram.com/UpperHalf-Plane.html

    Lower HP
    http://mathworld.wolfram.com/LowerHalf-Plane.html

    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.

    http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/biosav.html

    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.

    http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/elemag.html#c4


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