The ABCDEFG of Instantons and W-algebras

TL;DR

This work extends the AGT-type correspondence to pure N=2 gauge theories with arbitrary gauge groups by equating Nekrasov's one-instanton partition function with the norm of a Gaiotto–Whittaker coherent state in the corresponding W-algebra. It leverages free-field realizations of W-algebras and a 6d (2,0) framework with outer-automorphism foldings to treat twisted sectors for non-simply-laced groups. The authors perform explicit 1-instanton checks across A_n, D_n, B_n, C_n, G_2, and F_4, showing exact agreement (up to normalization) between Z_{G,1} and ⟨𝒢|𝒢⟩, and provide detailed constructions of the coherent states via Kac–Shapovalov matrices. This work broadens the scope of 4d/2d dualities and clarifies the roles of Langlands duals and twisted sectors in relating gauge theory data to W-algebra representations.

Abstract

For arbitrary gauge groups, we check at the one-instanton level that the Nekrasov partition function of pure N=2 super Yang-Mills is equal to the norm of a certain coherent state of the corresponding W-algebra. For non-simply-laced gauge groups, we confirm in particular that the coherent state is in the twisted sector of a simply-laced W-algebra.

Figures (3)

• Figure 1: The relation between a non-simply-laced Lie algebra $G$, its associated simply-laced algebra $\Gamma$, and the outer automorphism used to fold $\Gamma$ to obtain $G$.
• Figure 2: Top: the Seiberg-Witten solution of pure $\mathcal{N}=2$ super Yang-Mills theory with gauge group $G$ in terms of 6d $\mathcal{N}=(2,0)$ theory of type $\Gamma$ on $C=\mathbb{CP}^1$ with the $\mathbb{Z}_r$ twist line from $z=0$ to $z=\infty$. Middle: the $S^1$ reduction to the 5d maximally supersymmetric Yang-Mills theory with gauge group $G$ on a segment, with a suitable half-BPS boundary condition on both ends. Bottom: In the 2d description, the coherent state $\langle\underline{\mathcal{G}}|$ is produced by the BPS boundary condition. It is then propagated along the horizontal direction and annihilated by $|\underline{\mathcal{G}}\rangle$.
• Figure 3: Dynkin diagrams of simple Lie algebras, our labeling of the simple roots, and the comarks. The extended node is shown by a black blob.