26th January, 2011.
E-infinity communication No. 77
A moonshine conjecture from E-infinity (number theoretical motivation)
One of Alexander Grothendieck’s greatest insights was to follow Andre Weil’s hint at the deep connection between topological characteristic of a variety and its number theoretical aspect, i.e. its diplomatic aspects. Topologizing physics within a number theoretical framework seems to be an obvious characteristic of El Naschie’s E-infinity theory.
In the present communication we discuss a surprising relation between the totality of all Stein spaces, the compact and non-compact Lie symmetry groups on the one side and super string theory, path integral and the summing over dimensions procedure of E-infinity theory as well as the inverse fine structure constant = 137. The relation seems at first sight so bizarre and unreal that it is justifiably called the moonshine conjecture. In fact it has some similarity with the original moonshine conjecture and it is best to start by introducing the relation between the monster symmetry group and the coefficient of the j-function. The story starts when it was noticed that the minimal dimension for the monster is only one less than the first coefficient in the j-function. Thus we have D(min monistor) = b ̶ 1 = 196884 = 196883. The relation was clarified and the conjecture proven by Borcherds, a student of Conway (see El Naschie’s paper on the subject, CS&F, 32, (2007), pp. 383-387 as well as his paper “Symmetry groups prerequisite for E-infinity”, CS&F, 35, (2008), pp. 202-211 as well as “On the sporadic 196884-dimensional group, strings and E-infinity spacetime”, CS&F, 10(6), (1999), pp. 1103-1109.
We start by observing that the sum of the dimensions of the 17 two and three Stein spaces is exactly 686. This is equal 5 times 137 plus one. On the other hand the sum of the dimensions of the 12 compact and non-compact Lie symmetry groups is 1151. This is one short of 1152 which is 9 times 128, the electroweak inverse coupling of electromagnetics. This value (9)(128) = 1152 plays an important role in calculating the quantum states spectrum of the Heterotic string theory as can be seen in the excellent book of M. Kaku. Adding 686 to 1151 one finds 1837. Next we consider the total number of dimensions of the 12 non-compact Lie groups which comes to 1325. On the other hand the total number of the 8 non-compact 2 and 3 Stein spaces is given by 527. This is one short of Witten’s 528 states of a 5-Bran theory in 11 dimensions. Adding 527 and 1325 one finds 1852. The grand total is thus 1837 + 1852 = 3689. Now we embed 3689 in the ten dimensions of super strings and find that 3689 + 10 = 3699. Here comes the first incredible surprise because 3699 = (27)(137) = 3699 where = 137.
The second surprise in when we consider the “energy” stored in the “isometries” of the symmetry groups. Starting with the curvature of E-infinity spacetime = 26 + k we see that ( )( ) = (26 + k)(26 + k) which comes to 685.5. This is almost equal to 686 of the sum over all two and three Stein spaces. This is one of the best and simplest justifications ever for the theory of summing over symmetry group dimensions. Next we consider the intrinsic dimension of E7. This is dim E8(intrinsic) = 57. The transfinitely corrected compactified value is 57 + 1 + 3k 58.5. The energy is thus given by (58.54101966)2. This gives us 3427.050983. Here comes our next and final surprise for this communication. Dividing the energy by 25 one finds = 137.082039. The numerics indicate that there is indeed a deep connection between energy, symmetry and the electromagnetic fine structure constant. Members of the E-infinity group may like to think about a water tight proof for the above as well as pointing to more intricate relations.