Extremal correlators and Hurwitz numbers in symmetric product orbifolds

We study correlation functions of single-cycle chiral operators in ${\mathrm{Sym}}^N{T}^4$, the symmetric product orbifold of $N$ supersymmetric four-tori. Correlators of twist operators are evaluated on covering surfaces, generally of different genera, where fields are single valued. We compute some simple four-point functions and study how the sum over inequivalent branched covering maps splits under operator product expansions. We then discuss extremal $n$-point correlators, i.e. correlators of $n-1$ chiral and one antichiral operators. They obey simple recursion relations involving numbers obtained from counting branched covering maps with particular properties. In most cases we are able to solve explicitly the recursion relations. Remarkably, extremal correlators turn out to be equal to Hurwitz numbers.