It is shown that the supersymmetry-preserving automorphisms of any non-linear σ-model on K3 generate a subgroup of the Conway group Co 1 . This is the stringy generalization of the classical theorem, due to Mukai and Kondo, showing that the symplectic automorphisms of any K3 manifold form a subgroup of the Mathieu group M 23 . The Conway group Co 1 contains the Mathieu group M 24 (and therefore in particular M 23 ) as a subgroup. We confirm the predictions of the Theorem with three explicit conformal field theory (CFT) realizations of K3: the T 4 /Z 2 orbifold at a self-dual point, and the two Gepner models (2) 4 and (1) 6 . In each case we demonstrate that their symmetries do not form a subgroup of M 24 , but lie inside Co 1 as predicted by our Theorem.

The analogue of the McKay-Thompson series for the proposed Mathieu group action on the elliptic genus of K3 is analysed. The corresponding NS-sector twining characters have good modular properties and satisfy remarkable replication identities. These observations provide strong support for the conjecture that the elliptic genus of K3 carries indeed an action of the Mathieu group M 24 .

It has recently been conjectured that the elliptic genus of K3 can be written in terms of dimensions of Mathieu group M24 representations. Some further evidence for this idea was subsequently found by studying the twining genera that are obtained from the elliptic genus upon replacing dimensions of Mathieu group representations by their characters. In this paper we find explicit formulae for all (remaining) twining genera by making an educated guess for their general modular properties. This allows us to identify the decomposition of all expansion coefficients in terms of dimensions of M24-representations. For the first 500 coefficients we verify that the multiplicities with which these representations appear are indeed all non-negative integers. This represents very compelling evidence in favour of the conjecture.Comment: 23 pages; an error corrected, references update

We evaluate the partition function of three dimensional theories of gravity in the quantum regime, where the AdS radius is Planck scale and the central charge is of order one. The contribution from the AdS vacuum sector can -with certain assumptionsbe computed and equals the vacuum character of a minimal model CFT. The torus partition function is given by a sum over geometries which is finite and computable. For generic values of Newton's constant G and the AdS radius ℓ the result has no Hilbert space interpretation, but in certain cases it agrees with the partition function of a known CFT. For example, the partition function of pure Einstein gravity with G = 3ℓ equals that of the Ising model, providing evidence that these theories are dual. We also present somewhat weaker evidence that the 3-state and tricritical Potts models are dual to pure higher spin theories of gravity based on SL(3) and E 6 , respectively.

The Mathieu twisted twining genera, i.e., the analogues of Norton's generalized 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 (M 24 , U(1)), just as for the case of holomorphic orbifolds. This suggests that a holomorphic VOA may be underlying Mathieu Moonshine.