Extremal Correlators and Hurwitz Numbers in Symmetric Product Orbifolds

Abstract

We study correlation functions of single-cycle chiral operators in 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 OPEs. We then discuss extremal n-point correlators, i.e. correlators of n-1 chiral and one anti-chiral 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.

Figures (3)

• Figure 1: The eight classes of different diagrams contributing to a generic polynomial five-point function. The number over each line is the number of propagators joined in that line. The four vertices at finite positions are $b,c,d,e$, and $a$ is the vertex at infinity. Below each diagram we indicate the ordering of the vertices. The commutator denotes that vertices commute, and parenthesis indicate the possible position of an operator.
• Figure 2: An example of a generic graph of a polynomial map. The blue lines connect the first $p-1$ ramification points and the dashed black lines go to ramification point $n_p$. In this case we have a particular example of a diagram contributing to $\langle(\sigma_{[2]})^{17}\,\sigma_{[18]}\rangle$.
• Figure 3: The different classes of diagrams contributing to the 'near polynomial' four-point function. Thin blue lines represent a single propagator of the graph.