We study the moduli space M of N = (4, 4) superconformal field theories with central charge c = 6. After a slight emendation of its global description we find the locations of various known models in the component of M associated to K3 surfaces. Among them are the Z 2 and Z 4 orbifold theories obtained from the torus component of M. Here, SO(4, 4) triality is found to play a dominant role. We obtain the B-field values in direction of the exceptional divisors which arise from orbifolding. We prove T-duality for the Z 2 orbifolds and use it to derive the form of M purely within conformal field theory. For the Gepner model (2) 4 and some of its orbifolds we find the locations in M and prove isomorphisms to nonlinear σ models. In particular we prove that the Gepner model (2) 4 has a geometric interpretation with Fermat quartic target space.

In view of a potential interpretation of the role of the Mathieu group M 24 in the context of strings compactified on K3 surfaces, we develop techniques to combine groups of symmetries from different K3 surfaces to larger 'overarching' symmetry groups. We construct a bijection between the full integral homology lattice of K3 and the Niemeier lattice of type A 24 1 , which is simultaneously compatible with the finite symplectic automorphism groups of all Kummer surfaces lying on an appropriate path in moduli space connecting the square and the tetrahedral Kummer surfaces. The Niemeier lattice serves to express all these symplectic automorphisms as elements of the Mathieu group M 24 , generating the 'overarching finite symmetry group' (Z 2 ) 4 ⋊ A 7 of Kummer surfaces. This group has order 40320, thus surpassing the size of the largest finite symplectic automorphism group of a K3 surface by orders of magnitude. For every Kummer surface this group contains the group of symplectic automorphisms leaving the Kähler class invariant which is induced from the underlying torus. Our results are in line with the existence proofs of Mukai and Kondo, that finite groups of symplectic automorphisms of K3 are subgroups of one of eleven subgroups of M 23 , and we extend their techniques of lattice embeddings for all Kummer surfaces with Kähler class induced from the underlying torus.

This work develops the correspondence between orbifolds and free fermion models. A complete classification is obtained for orbifolds X/G with X the product of three elliptic curves and G an abelian extension of a group (Z_2)^2 of twists acting on X. Each such quotient X/G is shown to give a geometric interpretation to an appropriate free fermion model, including the geometric NAHE+ model. However, the semi-realistic NAHE free fermion model is proved to be non-geometric: its Hodge numbers are not reproduced by any orbifold X/G. In particular cases it is shown that X/G can agree with some Borcea-Voisin threefolds, an orbifold limit of the Schoen threefold, and several further orbifolds thereof. This yields free fermion models with geometric interpretations on such special threefolds.Comment: 46 pages; typos corrected and references adde

Prompted by the Mathieu Moonshine observation, we identify a pair of 45-dimensional vector spaces of states that account for the first order term in the massive sector of the elliptic genus of K3 in every Z 2 -orbifold CFT on K3. These generic states are uniquely characterized by the fact that the action of every geometric symmetry group of a Z 2orbifold CFT yields a well-defined faithful representation on them. Moreover, each such representation is obtained by restriction of the 45-dimensional irreducible representation of the Mathieu group M 24 constructed by Margolin. Thus we provide a piece of evidence for Mathieu Moonshine explicitly from SCFTs on K3.The 45-dimensional irreducible representation of M 24 exhibits a twist, which we prove can be undone in the case of Z 2 -orbifold CFTs on K3 for all geometric symmetry groups. This twist however cannot be undone for the combined symmetry group (Z 2 ) 4 A 8 that emerges from surfing the moduli space of Kummer K3s. We conjecture that in general, the untwisted representations are exclusively those of geometric symmetry groups in some geometric interpretation of a CFT on K3. In that light, the twist appears as a representation theoretic manifestation of the maximality constraints in Mukai's classification of geometric symmetry groups of K3.

in admiration and gratitude: To an extraordinary scientist, an unforgettable teacher, and a model of altruism.Abstract. A maximal subgroup of the Mathieu group M 24 arises as the combined holomorphic symplectic automorphism group of all Kummer surfaces whose Kähler class is induced from the underlying complex torus. As a subgroup of M 24 , this group is the stabilizer group of an octad in the Golay code. To meaningfully combine the symmetry groups of distinct Kummer surfaces, we introduce the concepts of Niemeier markings and overarching maps between pairs of Kummer surfaces. The latter induce a prescription for symmetry-surfing the moduli space, while the former can be seen as a first step towards constructing a vertex algebra that governs the elliptic genus of K3 in an M 24 -compatible fashion. We thus argue that a geometric approach from K3 to Mathieu Moonshine may bear fruit.