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# 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

#### 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.