Highest weight vectors, shifted topological recursion and quantum curves

We extend the theory of topological recursion by considering Airy structures whose partition functions are highest weight vectors of particular $\mathcal{W}$-algebra representations. Such highest weight vectors arise as partition functions of Airy structures only under certain conditions on the representations. In the spectral curve formulation of topological recursion, we show that this generalization amounts to adding specific terms to the correlators $\omega_{g,1}$, which leads to a ``shifted topological recursion'' formula. We then prove that the wave-functions constructed from this shifted version of topological recursion are WKB solutions of families of quantizations of the spectral curve with $\hbar$-dependent terms. In the reverse direction, starting from an $\hbar$-connection, we find that it is of topological type if the exact same conditions that we found for the Airy structures are satisfied. When this happens, the resulting shifted loop equations can be solved by the shifted topological recursion obtained earlier.