Higher genus partition functions of meromorphic conformal field theories

It is shown that the higher genus vacuum amplitudes of a meromorphic conformal field theory determine the affine symmetry of the theory uniquely, and we give arguments that suggest that also the representation content with respect to this affine symmetry is specified, up to automorphisms of the finite Lie algebra. We illustrate our findings with the self-dual theories at c=16 and c=24; in particular, we give an elementary argument that shows that the vacuum amplitudes of the E_8\times E_8 theory and the Spin(32)/Z_2 theory differ at genus g=5. The fact that the discrepancy only arises at rather high genus is a consequence of the modular properties of higher genus amplitudes at small central charges. In fact, we show that for c\leq 24 the genus one partition function specifies already the partition functions up to g\leq 4 uniquely. Finally we explain how our results generalise to non-meromorphic conformal field theories.