Monstrous Moonshine - Borcherds' Proof

Borcherds' Proof

Richard Borcherds' proof of the conjecture of Conway and Norton can be broken into five major steps as follows:

1. A vertex algebra V is constructed that is a graded algebra affording the moonshine representations on M, and it is verified that the monster module has a vertex algebra structure invariant under the action of M. V is thus called the monster vertex algebra.
2. A Lie algebra is constructed from V using the Goddard–Thorn "no-ghost" theorem from string theory; this is a generalized Kac-Moody Lie algebra.
3. A denominator identity for is constructed that is related to the coefficients of .
4. A number of twisted denominator identities are constructed that are similarly related to the series .
5. The denominator identities are used to determine the numbers c_m, using Hecke operators, Lie algebra homology and Adams operations.

Thus, the proof is completed. Borcherds was later quoted as saying "I was over the moon when I proved the moonshine conjecture", and "I sometimes wonder if this is the feeling you get when you take certain drugs. I don't actually know, as I have not tested this theory of mine."