K3 string theory, lattices and moonshine

Abstract

In this paper, we address the following two closely related questions. First, we complete the classification of finite symmetry groups of type IIA string theory on K 3 × R 6 , where Niemeier lattices play an important role. This extends earlier results by including points in the moduli space with enhanced gauge symmetries in spacetime, or, equivalently, where the world-sheet CFT becomes singular. After classifying the symmetries as abstract groups, we study how they act on the BPS states of the theory. In particular, we classify the conjugacy classes in the T-duality group O + ( 4,20 ) which represent physically distinct symmetries. Subsequently, we make two conjectures regarding the connection between the corresponding twining genera of K 3 CFTs and Conway and umbral moonshine, building upon earlier work on the relation between moonshine and the K 3 elliptic genus.

Some comments on symmetric orbifolds of K3

“…In some cases, the possible twining genera φ g for a given Frame shape are Jacobi forms with a non-trivial complex-valued multiplier. This happens if and only if the length of the shortest cycle in the Frame shape is > 2, see [10]. In this case, there are always at least two different twining genera at n = 1, with complex conjugate multipliers [10].…”

“…This happens if and only if the length of the shortest cycle in the Frame shape is > 2, see [10]. In this case, there are always at least two different twining genera at n = 1, with complex conjugate multipliers [10]. For the Frame shapes 6 4 and 4 2 8 2 , where there are more than two candidate twining genera, we separate by a horizontal line the twining genera with different multipliers.…”

“…Both Ψ g and Φ g can be defined explicitly in terms of infinite products, whose exponents depend on the Fourier coefficients of φ C K3 g ′r for all powers g ′r . The set of possible twining genera of NLSM on K3 are known [10,11] (although, for some symmetries g, there are more than one 'candidate' twining genus, and it is not known which one is actually realized in a given NLSM, see [11]), so the computation can be effectively performed. Standard arguments (reviewed in section 3.3) show that a twining genus φ K3 [n] g is invariant under deformations of the moduli that preserve the symmetry g, i.e.…”

Calabi-Yau manifolds and sporadic groups

“…Since the elliptic genus only counts BPS states of the theory, one might have hoped that the BPS states have M 24 as symmetry group, but the full spectrum has a different symmetry group. By calculating explicit twined elliptic genera [18,19] one finds that there are more twined elliptic genera for K3 spaces than conjugacy classes of M 24 . The symmetry groups of the K3 manifold at different points in moduli space have found a beautiful string theory explanation in [20], where the authors study type IIA string theory on K3×T 3 .…”

“…So one should ask which group is relevant for the elliptic genus which counts a subset of states independently of where we are in moduli space. These questions are not easily answered but it has been shown that twined elliptic genera for K3 manifolds give all the McKay-Thompson series of Mathieu moonshine plus several extra twined elliptic genera that one would not have expected based on Mathieu moonshine alone [18]. At the same time the idea of symmetry surfing has been pursued, in which one tries to find an explicit M 24 symmetry by combining the symmetry groups at different points in K3 moduli space [13][14][15][16].…”