Landau-Ginzburg orbifolds and symmetries of K3 CFTs

Abstract

Recent developments in the study of the moonshine phenomenon, including umbral and Conway moonshine, suggest that it may play an important role in encoding the action of finite symmetry groups on the BPS spectrum of K3 string theory. To test and clarify these proposed K3-moonshine connections, we study Landau-Ginzburg orbifolds that flow to conformal field theories in the moduli space of K3 sigma models. We compute K3 elliptic genera twined by discrete symmetries that are manifest in the UV description, though often inaccessible in the IR. We obtain various twining functions coinciding with moonshine predictions that have not been observed in physical theories before. These include twining functions arising from Mathieu moonshine, other cases of umbral moonshine, and Conway moonshine. For instance, all functions arising from M 11 ⊂ 2.M 12 moonshine appear as explicit twining genera in the LG models, which moreover admit a uniform description in terms of its natural 12-dimensional representation. Our results provide strong evidence for the relevance of umbral moonshine for K3 symmetries, as well as new hints for its eventual explanation.

K3 string theory, lattices and moonshine

“…Motivated by the connection between the stringy K3 symmetries and moonshine, our analysis of world-sheet parity symmetries of NLSMs (see Sect. 3.2) and results regarding LandauGinzburg orbifolds [17], in this paper we conjecture (Conjecture 5) that the proposed twining genera arising from umbral and Conway moonshine as defined in [22] and [42] capture all of the possible discrete stringy symmetries of any NLSM in the K 3 CFT moduli space. Moreover, we conjecture (Conjecture 6) that each of the umbral and Conway moonshine functions satisfying certain basic assumptions (that the symmetry preserves at least four planes in the defining 24-dimensional representation) is realized as the physical twining genus of a certain K 3 NLSM.…”

“…In Sect. 4.3, we will discuss results of the aforementioned investigation of LG orbifolds [17], while in Sect. 4.4 we will analyze the constraints on such genera coming from modularity.…”

“…singularities locally of the form C 2 /G, where G is a finite subgroup of SU (2) C [37]. Note that, upon including this kind of singular surfaces, the moduli space of hyperkähler structures (with fixed volume) on K3 admits a simple description as (a discrete quotient of) the It was conjectured in [22] that φ N g are candidates for twining genera arising from K 3 NLSMs when g preserves a four-plane; this conjecture has passed a few consistency tests and was further tested in [17]. For a given N with a non-trivial root system, we will denote the set of Jacobi forms arising in this way as (N ) := {φ N g |g is a four-plane preserving element of G N }.…”

“…As a consequence, one might expect both Conway and umbral moonshine to play a role in describing the twining genera. Secondly, UV descriptions of K3 NLSMs given by LandauGinzburg (LG) orbifolds furnish evidence that suggests that the Conway functions alone are not sufficient to capture all the twining genera [17] (see also Sect. 4.3).…”

Calabi-Yau manifolds and sporadic groups

“…we consider the orbifold by Z k+2 of the tensor product of 15 or 10 minimal modes of A k+1 type for k = 1 and k = 2, respectively. It was explained how to calculate the elliptic genus for the orbifolded tensor product of minimal models in [23,50] (this was applied to moonshine in [51]). Each A k+1 minimal model is obtained from a chiral multiplet Φ with superpotential W = Φ k+2 k+2 and gives a multiplicative contribution of…”