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

Abstract

We identify Whittaker vectors for ${\mathcal{W}}_k(\mathfrak{g})$-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.

arXiv comment79 pages, 1 figure; v2: Proposition 4.12 corrected, intro references updated; v3: tiny revisions

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• Whittaker vectors at finite energy scale, topological recursion and Hurwitz numbers
• $\mathfrak{b}$-Hurwitz numbers from refined topological recursion