Mathieu Moonshine

String theory provides a fantastic framework to experiment with novel ideas rooted in the physics of elementary particles. Recently, considerations related to a specific type of superstrings propagating on a K3 surface (a 4-dimensional hyperkahler manifold) have led to the discovery of a phenomenon called Mathieu Moonshine, following an observation by Eguchi, Ooguri and Tachikawa in 2010. It manifests itself in a particular rewriting of the elliptic genus \(\mathbf{E_{K3}(\tau,z)}\) of K3, which is a weak Jacobi form of weight 0 and index 1 encoding topological invariants of K3 surfaces.

This rewriting of the elliptic genus of K3 exploits the well-known N=4 world sheet superconformal symmetry of such superstring theories. More precisely, the elliptic genus may be written as a linear combination of two massless characters \(Ch(0;\tau,z)\) and \(Ch(\frac{1}{2};\tau,z)\), as well as an infinite family of massive characters \(\{\,q^h\,Ch(\tau,z),\,h \in \mathbb{R}, h > \frac{1}{4},\,q=e^{2i\pi\tau}, \tau, z \in \mathbb{C}, {\cal Im}\,\tau >0 \}\) of unitary representations of the 2-dimensional N=4 superconformal algebra at central charge \( c=6\) in the following way, $$E_{K3}(\tau,z)=-2\,Ch(0; \tau,z)+20\,Ch(\frac{1}{2}; \tau,z)+e(q)\,Ch(\tau,z),$$ where all the coefficients of the \(q\)-series $$e(q)=90q+462q^2+1540q^3+4554q^4+11592q^5+\dots$$ coincide with twice the dimension of some representations of the sporadic group \( M_{24}\) (Mathieu 24). For instance, \(90=45+\overline{45} \), where \(45\) is a complex irreducible representation of \( M_{24}\), while \( 4554= 2\times 2277 \) where \( 2777 \) is a real irreducible representation of \( M_{24}\). The coefficients of higher powers of \(q \) correspond to dimensions of reducible representations of \( M_{24}\).

The role this sporadic group might play in superstring theory remains a mystery. Indeed, the elliptic genus \(E_{K3}(\tau, z)\) is a topological invariant that arises from the supertrace over the subsector of Ramond-Ramond states of every superconformal field theory on K3, and hence, it counts states with signs according to their bosonic or fermionic nature (unlike the partition function that counts states). That the net contribution should yield a well-defined representation of any group, let alone \(M_{24}\), is at the heart of our quest.

This problem is particularly exciting. Indeed, although rooted in string theory, it has ramifications in several distinguished areas of modern mathematics: finite group theory, algebraic geometry and number theory . Although reminiscent of the famous Monster Moonshine, the Mathieu Moonshine appears to be different in nature. Indeed, the former stems from the observation that the coefficients in the \(q\)-series expansion of the Hauptmodul \(J(\tau)\) of the group \(SL(2,\mathbb{Z})\) - later identified as the partition function of a conformal field theory at central charge \(c=24\) - are (sums of) dimensions of representations of the Griess-Fisher Monster group. On the other hand, the Mathieu Moonshine connects the (sums of) dimensions of representations of the sporadic group \(M_{24}\) with the elliptic genus of \(K3\), derived from the partition function of any conformal field theory at central charge \(c=6\), and closely related to a weakly holomorphic mock modular form of weight \(\mathbf{\frac{1}{2}}\) over \(\mathbf{SL(2,\mathbb{Z}})\). Although some techniques that proved successful in unravelling the mysteries of the Monster Moonshine have been successfully applied in the context of Mathieu Moonshine, mainly to provide circumstantial evidence of an \(M_{24}\) action on the states selected by the elliptic genus of \(K3\) and give a proof of principle of an action, we have so far failed to understand the exact mechanisms of such an action in the context of conformal field theory.

Together with Katrin Wendland, I have started a programme of research centred on the Mathieu Moonshine phenomenon in 2010.

Our initial motivation was to provide a concrete representation of the symplectic automorphisms of Kummer \(K3\) surfaces, which form an interesting class of \(K3\) surfaces, in terms of permutations of 24 elements that preserve the extended binary Golay code, i.e. in terms of elements of \(M_{24}\). Our work relies on lattice techniques developed by Nikulin and sharpens Kondo's proof of Mukai's seminal classification of finite symplectic automorphism groups of \(K3\) surfaces as subgroups of \(M_{23}\). Our first paper, available from arXiv:1107.3834, provides a detailed construction of a bijection between the lattice of integral homology of a Kummer surface and the Niemeier lattice \(N\) based on the Lie algebra \(A_1^{24}\). This bijection has very specific properties that enable to encode the action of symplectic automorphisms on Kummer surfaces as an action on the Niemeier lattice \(N\). Remarkably, we were able to show that the same bijection encodes the action on \(N\) of two different symplectic automorphism groups, corresponding to the Kummer surfaces built from a square torus and a tetrahedral torus. This prompted us to argue that the Niemeier lattice accommodates the combined action of these two symmetry groups, that we call the overarching symmetry group \(\mathbf{\mathbb{Z}_2^4\rtimes A_7}\), which does not correspond to any symplectic automorphism group of a K3 surface but nevertheless provides, through the Niemeier lattice, a `memory' of the symmetries of individual Kummer surfaces.

This overarching symmetry group is a subgroup of \(M_{23}\). However, we were able in our second paper arXiv:1303.2931 to extend our technique, not only to surf between two Kummer surfaces of maximal symmetry, but also between any two points in the moduli space of \(N=(4,4)\) superconformal field theories at central charge \(c=6\) describing strings propagating on Kummer surfaces. Again we can make sense of a `super' overarching symmetry encoded in the Niemeier lattice \(N\), which combines the actions of the symplectic automorphism groups of all Kummer surfaces. This `super' overarching symmetry group is \(\mathbf{\mathbb{Z}_2^4\rtimes A_8}\), a maximal subgroup of \(M_{24}\) which stabilizes an octad of the extended binary Golay code. Since our construction relies on an octad being preserved, \(\mathbb{Z}_2^4\rtimes A_8\) is the largest symmetry group we can access through surfing.

Having explored in some detail the geometrical symmetries of Kummer surfaces and their relation to \(M_{24}\), we turned our attention to \(\mathbf{\mathbb{Z}_2}\)-orbifolds of toroidal conformal field theories in 4 dimensions, which are known to describe strings propagating on (orbifold limits of) \(K3\) surfaces. We have obtained the result that the generic field content of \(\mathbb{Z}_2\)-orbifold CFTs on \(K3\) ensures the existence of a \(V_{45}^{CFT}\oplus\overline{V}^{CFT}_{45}\) space of states which naturally accounts for the massive net contribution to the elliptic genus at leading order. We then studied how closely these two vector spaces are related to the vector spaces on which the complex \( 45 (\overline{45})\)-dimensional irreducible representations of \(M_{24}\) are constructed by Margolin. This is the first ever attempt at producing physical states that account for the number 90 appearing as the first coefficient in the \(q\)-series \(e(q)\) entering in the \(K3\) elliptic genus expression quoted above, and therefore at approaching the \(M_{24}\) mystery from a concrete conformal field theory standpoint. Our investigations show that the representations on \(V_{45}^{CFT}\) and \(\overline{V}_{45}^{CFT}\) can be induced from the two irreducible 45-dimensional complex conjugate representations of \(M_{24}\), providing evidence for Mathieu Moonshine from the point of view of superconformal field theories on \(K3\). Our results are presented in arXiv:1303.3221.