Origami: real structure, enumeration and quantum modularity

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 the numbers of genus 2 real origami with 2 simple zeros in terms of the sums of divisors, showing that their generating function is a quantum modular form. Furthermore, we show that replacing zonal polynomials with Schur polynomials we can effectively count the classical (complex) origami. As a by-product, 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.