Communications in Number Theory and Physics

Volume 4 (2010)

Number 2

Genus two partition functions of chiral conformal field theories

Pages: 295 – 363



Matthias R. Gaberdiel (Institut für Theoretische Physik, ETH Zürich, Switzerland)

Christoph A. Keller (California Institute of Technology, Pasadena, Calif., U.S.A.)

Roberto Volpato (Institut für Theoretische Physik, ETH Zürich, Switzerland)


A systematic analysis of the genus two vacuum amplitudes of chiral self-dual conformal field theories is performed. Itis explained that the existence of a modular invariant genus two partition function implies infinitely many relationsamong the structure constants of the theory. All of these relations are shown to be a consequence of the associativityof the operator product expansion, 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 genustwo vacuum amplitude exists for all $k$, but that this does not actually check the consistency of these theoriesbeyond what is already testable at genus one.

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