on June 19, 2008 by lieven in geometry, groups, Comments (1)
Monstrous frustrations
Thanks for clicking through… I guess.
If nothing else, it shows that just as much as the stock market is fueled by greed, mathematical reasearch is driven by frustration (or the pleasure gained from knowing others to be frustrated).
I did spend the better part of the day doing a lengthy, if not laborious, calculation, I’ve been postponing for several years now. Partly, because I didn’t know how to start performing it (though the basic strategy was clear), partly, because I knew beforehand the final answer would probably offer me no further insight.
Still, it gives the final answer to a problem that may be of interest to anyone vaguely interested in Moonshine :
What does the Monster see of the modular group?
I know at least two of you, occasionally reading this blog, understand what I was trying to do and may now wonder how to repeat the straightforward calculation. Well the simple answer is : Google for the number 97239461142009186000 and, no doubt, you will be able to do the computation overnight.
One word of advice : don’t! Get some sleep instead, or make love to your partner, because all you’ll get is a quiver on nine vertices (which is pretty good for the Monster) but having an horrible amount of loops and arrows…
If someone wants the details on all of this, just ask. But, if you really want to get me exited : find a moonshine reason for one of the following two numbers :
(the dimension of the monster-singularity upto smooth equivalence), or,
(the dimension of the moduli space).
Hi, I'm a professor of mathematics at the University of Antwerp, in Belgium. My research areas are non-commutative algebra and non-commutative geometry. Sometimes, you can also find me here :
mark a. thomas
August 21, 2008 @ 2:42 am
I am not sure about finding a moonshine connection with the numbers 7.916138…10^38 and 1.5759188…10^39 but both of these numbers are close to the important physics large numbers in the range ~10^40 (dimensionless) as expounded upon by Dirac and Harrison. Basically it is a physics form hbarc/Gm1m2 ~ 10^40 where m1 and m2 could be nucleon masses say a proton to proton or neutron to proton or electron masses. If you invert the relation you get a form not too unrelated to Newton’s gravity form Gm1m2/hbarc ~ 10^-39. A very very small and weak form of gravity indeed (and dimensionless). (Currently, string physicists use the form 1/Mpl^2 ~ 10^-39 (Mpl = planck mass) to show that gravity is weak at the electroweak end of the scale but notice it is still dimensionful.) What makes these physics forms interesting is that they contain constants involving both quantum and relativity. And besides the 10^40 representing possibly the stronger gauge couplings (invert to gravity weakness) 10^40 also represents the classically large distances of the Hubble length compared to a proton size. Strange stuff. Myself, I believe that the correct form uses the neutron mass squared as Mpl^2/mn^2 = 1.6889…10^38 (dimensionless)which is also hbarc/Gmnmn. I have good reasons for this as it is involved in a physics calculation of the monster symmetry.
(4/a^4)(Mpl^2/me^2)[((Mpl^2/mn^2)^1/2^16 -1.00)^-1]^1/2^11 = 8.0801742…*10^53