In addition, the mass ratios of the fundamental fermions are determined by the group relationships to the j-invariant of the Monster Group.
These mass predictions were based upon the mass ratios being determined by the j-invariant function of elliptic modular functions and of fractional linear transformations, i.
This connection to the Monster Group is present already in determining the lepton and quark mass ratios, which are proportional to the j-invariant of elliptic modular functions, the same j-invariant that is the partition function for the Monster Group in a quantum field theory .
In particular, we prove that the almost complex structure J of an almost Kahler manifold is soldered to a submanifold iff the latter is a J-invariant, totally geodesic submanifold.
The tensor field J is algebraically adapted to the submanifold N iff N is J-invariant, which we shall assume hereafter.
The almost complex structure J is soldered to the J-invariant submanifold N iff the second fundamental form of N is given by the formula
If (M, J, g) is an almost Kahler manifold, the almost complex structure J is soldered to the submanifold N iff N is a J-invariant, totally geodesic submanifold of M.
My principal goal is to show that if a 4th quark family exists, the physical rules of the Universe follow directly from mathematical properties dictated by the Fischer-Greiss Monster Group via the Monster's j-invariant function and the Mobius transformation in discrete spacetime, with everything related to the Golay-24 information code for the Leech lattice.
Lepton and quark approximate mass values are determined by the j-invariant function of elliptic modular functions, being related to the above subgroups and Mobius transformations in both discrete lattice spaces and continuous spaces.
What the others have not realized is the direct connection in a discrete internal symmetry space from the Monster to the lepton and quark states via the j-invariant of elliptic modular functions.
As I explain in the next section, the most direct connection of M with the SM of leptons and quarks is via the j-invariant of elliptic modular functions
Here I review the connection between the j-invariant and the discrete symmetry groups for the leptons and quarks.