  A systematic analysis of the genus two vacuum amplitudes of chiral self-dual
conformal field theories is performed. It is explained that the existence of a
modular invariant genus two partition function implies infinitely many
relations among the structure constants of the theory. All of these relations
are shown to be a consequence of the associativity of the OPE, as well as the
modular covariance properties of the torus one-point functions. Using these
techniques we prove that for the proposed extremal conformal field theories at
c=24k a consistent genus two vacuum amplitude exists for all k, but that this
does not actually check the consistency of these theories beyond what is
already testable at genus one.
