Extremal Correlators and Hurwitz Numbers in Symmetric Product Orbifolds

TL;DR

This paper explores the relationship between extremal correlators in symmetric product orbifolds and Hurwitz numbers, providing explicit calculations and recursion relations that reveal a deep mathematical connection.

Contribution

It establishes that extremal correlators in symmetric product orbifolds are equal to Hurwitz numbers, linking physical correlators to enumerative geometry.

Findings

Extremal correlators obey simple recursion relations.

Explicit solutions to recursion relations are obtained.

Extremal correlators are equal to Hurwitz numbers.

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.