Origami: real structure, enumeration and quantum modularity

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.