Communications in Number Theory and Physics

Volume 4 (2010)

Number 2

Genus two partition functions of chiral conformal field theories

Pages: 295 – 363

DOI: http://dx.doi.org/10.4310/CNTP.2010.v4.n2.a2

Authors

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)

Abstract

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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