Generalised Mathieu Moonshine

Abstract

The Mathieu twisted twining genera, i.e. the analogues of Norton's generalised Moonshine functions, are constructed for the elliptic genus of K3. It is shown that they satisfy the expected consistency conditions, and that their behaviour under modular transformations is controlled by a 3-cocycle in H 3 (M24, U (1)), just as for the case of holomorphic orbifolds. This suggests that a holomorphic VOA may be underlying Mathieu Moonshine. Anexample 30 4.3. Computation of a twisted twining genus 32 5. Conclusions 34 5.1. Summary 34 5.2. Open problems and future work 35 Acknowledgements 37 Appendix A. Details on the unobstructed twisted twining genera 37 A.1. The characters φ 2B,4A 2 (group 13) φ 4B,4A 3 (group 23) and φ 4B,4A 4 (group 24) 38 A.2. The cases φ 3A,3A 3 (group 33) and φ 3A,3B 1 (group 34) 40 Appendix B. Some group cohomology 42 Appendix C. Projective representations of finite groups 43 C.1. Central extension and orbifolds 45 Appendix D. Centralisers C M 24 (g) and (projective) character tables 46 D.1. Character Tables 47

Generalised Moonshine and Holomorphic Orbifolds

“…Indeed, since ℓ(12B) = 12, it follows that α must correspond to a generator of H 3 (M 24 , U(1)). With the help of the software GAP [33], we have verified that a generator reproducing the multiplier phases (3.22) exists and is unique [19]. ‡ Note that for a finite group G one has the isomorphisms…”

“…Once the 3-cocycle α is known, one can use (3.16) and (3.17) to deduce the precise modular properties of each twisted twining genus φ g,h . It turns out that, in many cases, these properties can only be satisfied if φ g,h vanishes identically [19]. In particular, there are two kinds of potential obstructions that can force a certain twisted twining genus to vanish (i) Consider three pairwise commuting elements g, h, k ∈ M 24 .…”

“…Although this establishes Mathieu moonshine, there is a major outstanding question: what is the M 24 -analogue of the Monster module V ♮ ? In [19] we gave evidence that some kind of holomorphic vertex operator algebra (VOA) should be underlying Mathieu moonshine. The main point was to extend the previous results on twining genera to the complete set of twisted twining genera φ g,h (τ, z), corresponding to the M 24 -analogues of Norton's generalised moonshine functions f (g, h; τ ) for the Monster.…”

“…Our aim in this note is to give a short review of the generalised Mathieu moonshine phenomenon uncovered in [19], focussing on the main ideas rather than technical details. For completeness we include a discussion of holomorphic orbifolds and group cohomology which are the key ingredients in our work, as well as some background on Norton's generalised moonshine conjecture, which served as strong motivation for [19].…”

Anomalies, Extensions and Orbifolds

“…c 1,2 ψ 12 (k 3 )c 12,3 , g 1 g 2 g 3 ) (B.11) and (k 1 , g 1 )((k 2 , g 2 )(k 3 , g 3 )) = (k 1 , g 1 )(k 2 ψ 2 (k 3 )c 3,2 , g 2 g 3 ) = (k 1 ψ 1 (k 2 ψ 2 (k 3 )c 2,3 )c 1,23 , g 1 g 2 g 3 ). (B 12). …”