Whittaker vectors at finite energy scale, topological recursion and Hurwitz numbers

We upgrade the results of Borot--Bouchard--Chidambaram--Creutzig to show that the Gaiotto vector in $4d$ $\mathcal{N} = 2$ pure supersymmetric gauge theory admits an analytic continuation with respect to the energy scale (which can therefore be taken to be finite, instead of infinitesimal), and is computed by topological recursion on the (ramified) half Seiberg--Witten spectral curve. This has a number of interesting consequences for the Gaiotto vector: relations to intersection theory on $\overline{\mathcal{M}}_{g,n}$ in at least two different ways, Hurwitz numbers, quantum curves, and (almost complete) description of the correlators as analytic functions of $\hslash$ (instead of formal series). The same method is used to establish analogous results for the more general Whittaker vector constructed in the recent work of Chidambaram--Do{\l}{\k{e}}ga--Osuga.