Duality in N=4 Liouville Theory and Moonshine Phenomena

Understand

We consider the ${\cal N}=4$ Liouville theory by varying the linear dilaton coupling constant $\cal{Q}$.

• It is known that at two different values of coupling constant ${\cal Q}=\sqrt{{2\over N}},-(N-1)\sqrt{{2\over N}}$ system exhibits two different small ${\cal N}=4$ 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 Coulumb and Higgs branch of the theory and expected to be dual to each other.
• We study the Mathieu and umbral moonshine phenomena in these two theories and discuss their dual description.