Higher genus partition functions of meromorphic conformal field theories

TL;DR

Higher genus vacuum amplitudes in meromorphic conformal field theories uniquely determine their affine symmetry and likely their representation content, with distinctions between theories becoming apparent at higher genus levels.

Contribution

This work demonstrates that higher genus vacuum amplitudes uniquely identify the affine symmetry and potentially the representation content of meromorphic conformal field theories.

Findings

Genus g=5 amplitudes distinguish E8×E8 and Spin(32)/Z2 theories.

For c≤24, genus one partition function determines amplitudes up to g≤4.

Higher genus amplitudes reveal differences only at high genus due to modular properties.

Abstract

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.