Origami: real structure, enumeration and quantum modularity

TL;DR

This paper introduces real origami structures, counts them using zonal polynomials, and links their generating functions to quantum modular forms, while also connecting classical origami to Hurwitz numbers and exploring related conjectures.

Contribution

It defines real origami with real structures, provides explicit enumeration formulas, and reveals their generating functions as quantum modular forms, extending the understanding of origami in algebraic and quantum contexts.

Findings

Explicit formulas for counting genus 2 and 3 real origami

Generating functions are quantum modular forms

Connection between classical origami and double Hurwitz numbers

Abstract

We define real origami (that is, origami equipped with a real structure) and enumerate them using the combinatorics of zonal polynomials. We explicitly express in terms of sums of divisors the numbers of genus 2 real origami with 2 simple zeros and the numbers of genus 3 real origami with 2 double zeros showing that their generating functions are quantum modular forms. Furthermore, we show that by replacing zonal polynomials with Schur polynomials we can effectively count the classical (complex) origami. As a byproduct, we establish a connection between classical origami and a specific class of double Hurwitz numbers. Finally, we discuss some conjectures and open questions involving Jack functions, quantum modular forms, and integrable hierarchies.