The spectral curve and the Schrödinger equation of double Hurwitz numbers and higher spin structures (Q382482)

The authors aim to provide a solution to the conjecture of existence of quantum curves for ``three series'' of infinitely many cases of generalized Hurwitz numbers. More explicitly, the authors consider two generalization of Hurwitz numbers. One is the double Hurwitz numbers that counts the ramified coverings of \(\mathbb{P}^{1}\) with two special fibers. The other generalization they use is called \(\mathit{r-spin}\) Hurwitz numbers. Lastly, the authors consider the mixed case of aforementioned generalizations. For each of these generalizations, the work contains the proof of the existence of quantum curve or stationary Schrödinger equation. For each case, the formula for principal specialization of partition function is derived and the formula for spectral curves is produced.