Duality in $N=4$ Liouville theory and moonshine phenomena

We consider the $N=4$ Liouville theory by varying the linear dilaton coupling constant $Q$. It is known that, at two different values of coupling constant $Q=\sqrt{\frac{2}{N}},-(N-1)\sqrt{\frac{2}{N}}$, the system exhibits two different small $N=4$ superconformal symmetries with central charge $c=6$ and $c=6(N-1)$, respectively. In the context of string theory these two theories are considered to describe the Coulomb and Higgs branches of the theory and are expected to be dual to each other. We study the Mathieu and umbral moonshine phenomena in these two theories and discuss their dual descriptions. We mainly consider the case of $A_N$-type modular invariants.