Beyond Mathieu Moonshine: a look at large N = 4 Algebras.

\small{The conformal field theory approach to calculate the elliptic genus of $K3$ surfaces has revealed the Mathieu moonshine phenomenon, which highlights relations between the `small' $\mathcal N=4$ superconformal algebra at central charge $c=6$, the sporadic group Mathieu 24 and mock modular forms. Here we take a look at a family of 'large' $\mathcal N=4$ superconformal algebras, labelled $\mathcal A_\gamma, \gamma \in [\hf, \infty [$ (from which one can recover the small $\mathcal N=4$ algebras in some limit), in the hope that a moonshine-like phenomenon might be observed. We consider realizations of $\mathcal A_\gamma$ and its closely related family of non-linear algebras $\Atg$ on $SU(3)=WS(3)\times SU(2)\times U(1)$, where $WS(3)$ is a 4-dimensional Wolf space, i.e. a quaternionic symmetric space. The underlying physical models are supersymmetric Wess-Zumino-Novikov-Witten models describing superstring propagation on the $SU(3)$ group manifold, for which explicit partition functions can be constructed. In order to exhibit the $\Atg$ (and $\mathcal A_\gamma$) symmetries of these models at the level of partition functions, we construct character sum rules which encode how products of affine $\widehat{su(3)}$ characters with a character for four `Wolf space' fermions decompose as sums of $\Atg$ characters. We find close analytic forms for the corresponding branching functions in a theory with $\Atg$ symmetry where the levels of the two affine $\widehat{su(2)}$ subalgebras of $\Atg$ are $\ktp=2$ and $\ktm=1$, and we discover that they form a vector-valued mock modular form of weight $1/2$. To arrive at this result, we used the transformation laws of the $\Atg$ characters under the modular group $SL(2, \mathbb{Z})$, which we derive in the twisted Ramond sector.