Whittaker vectors for $$\mathcal {W}$$ -algebras from topological recursion

We identify Whittaker vectors for \(\mathcal {W}^{\textsf{k}}(\mathfrak {g})\)-modules with partition functions of higher Airy structures. This implies that Gaiotto vectors, describing the fundamental class in the equivariant cohomology of a suitable compactification of the moduli space of G-bundles over \(\mathbb {P}^2\) for G a complex simple Lie group, can be computed by a non-commutative version of the Chekhov–Eynard–Orantin topological recursion. We formulate the connection to higher Airy structures for Gaiotto vectors of type A, B, C, and D, and explicitly construct the topological recursion for type A (at arbitrary level) and type B (at self-dual level). On the physics side, it means that the Nekrasov partition function for pure \(\mathcal {N} = 2\) four-dimensional supersymmetric gauge theories can be accessed by topological recursion methods.