Affine Lie Algebras and S-Duality of N=4 Super Yang-Mills Theory for ADE Gauge Groups on K3

We attempt to determine the partition function of $\mathcal{N}=4$ super Yang-Mills theory for $ADE$ gauge groups on $K3$ and investigate the relation with affine Lie algebras. In particular we describe eta functions, which compose SU(N) partition function, by level $N$ $A_{N-1}$ theta functions. Moreover we find $D,E$ theta functions, which satisfy the Montonen-Olive duality for $D,E$ partition functions.