Genus two partition functions of chiral conformal field theories

TL;DR

This paper systematically analyzes genus two vacuum amplitudes in chiral self-dual conformal field theories, revealing that their consistency conditions are derived from OPE associativity and modular covariance, and confirming the existence of consistent amplitudes for extremal theories at all levels.

Contribution

It demonstrates that the relations among structure constants from genus two amplitudes follow from fundamental CFT principles and confirms the existence of consistent genus two vacuum amplitudes for extremal theories at all levels.

Findings

Genus two partition functions imply relations from OPE associativity and modular covariance.

Consistent genus two vacuum amplitudes exist for extremal theories at all levels.

Genus two analysis does not provide new constraints beyond genus one.

Abstract

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.