Communications in Number Theory and Physics

Volume 6 (2012)

Number 1

Symmetries of K3 sigma models

Pages: 1 – 50

DOI: http://dx.doi.org/10.4310/CNTP.2012.v6.n1.a1

Authors

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

Stefan Hohenegger (Institut für Theoretische Physik, ETH Zurich, Switzerland)

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

Abstract

It is shown that the supersymmetry-preserving automorphismsof any non-linear $\sigma$-model on K3 generate a subgroupof the Conway group Co$_1$. This is the stringygeneralization of the classical theorem, due to Mukai andKondo, showing that the symplectic automorphisms of any K3manifold form a subgroup of the Mathieu group $\MM_{23}$.The Conway group Co$_1$ contains the Mathieu group$\MM_{24}$ (and therefore in particular $\MM_{23}$) as asubgroup. We confirm the predictions of the Theorem withthree explicit conformal field theory (CFT) realizations of K3: the ${\mathbbT}^4/\ZZ_2$ orbifold at a self-dual point, and the twoGepner models $(2)^4$ and $(1)^6$. In each case wedemonstrate that their symmetries do not form a subgroup of$\MM_{24}$, but lie inside Co$_1$ as predicted by ourTheorem.

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