Topological recursion (TR) is a universal recursive formalism that associates to a spectral curve an infinite family of multidifferentials on that curve. Its applications span a wide range of fields, including enumerative geometry, random matrix theory, topological string theory, quantum spectral curves, and conjecturally knot theory..
Recently, a new fundamental duality within TR has been understood: the so-called $x$-$y$ duality. This duality admits several incarnations across different applications of TR. In this talk, I will present this duality and explain how it extends the framework of TR for certain curves in $mathbb{C}^*$. Furthermore, I will show how the $x$-$y$ duality can be used to effectively compute string amplitudes (i.e., Gromov–Witten invariants) and quantum curves for specific mirror curves of toric Calabi-Yau threefolds..

https://indico.math.cnrs.fr/event/16035/