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.

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

References in corpus (13)

• Remodeling the B-model
• The Volume Conjecture, Perturbative Knot Invariants, and Recursion Relations for Topological Strings
• Weil-Petersson volume of moduli spaces, Mirzakhani's recursion and matrix models
• Topological expansion of beta-ensemble model and quantum algebraic geometry in the sectorwise approach
• Think globally, compute locally
• CFT and topological recursion
• Topological expansion of the Bethe ansatz, and non-commutative algebraic geometry
• Topological expansion of mixed correlations in the hermitian 2 Matrix Model and x-y symmetry of the F_g invariants
• Topological expansion of the Bethe ansatz, and quantum algebraic geometry
• Topological recursion for Kadomtsev-Petviashvili tau functions of hypergeometric type
• Higher Airy structures and topological recursion for singular spectral curves
• Topological recursion on the Bessel curve
• Super Topological Recursion and Gaiotto Vectors For Superconformal Blocks

Cited by in corpus (6)

• Super Topological Recursion and Gaiotto Vectors For Superconformal Blocks
• Symplectic (Non-)Invariance of the Free Energy in Topological Recursion
• Taking limits in topological recursion
• Whittaker vectors at finite energy scale, topological recursion and Hurwitz numbers
• Topological recursion on transalgebraic spectral curves and Atlantes Hurwitz numbers
• $\mathfrak{b}$-Hurwitz numbers from refined topological recursion