Topological recursion associates to a spectral curve, a sequence of differential forms, and provides a candidate to be a Tau-function and Baker--Akhiezer function of an integrable system. The Baker--Akhiezer function would then obey an ODE, which can be called a "quantization" of the spectral curve. We will review some aspects of these topics.
Links:
[1] https://www.mpim-bonn.mpg.de/de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/de/node/3444
[3] https://www.mpim-bonn.mpg.de/de/node/8493