Counting Exceptional Instantons

TL;DR

This work extends Nakajima–Yoshioka blow-up recursion to compute the instanton partition function for $\,N=2$ SYM with arbitrary gauge groups, including exceptional groups $E_6$ and $E_7$, and validates the results by matching with the Hall-Littlewood limit of the superconformal index for corresponding non-Lagrangian theories. The authors derive a general recursion for the instanton series, reproduce the known 1-instanton formula, and explore higher instantons, uncovering a universal structure for the 2-instanton term expressed through group characters. Comparison with the exceptional index confirms the proposed lifting prescription and supports the interpretation of the instanton moduli space as the Higgs branch of the full theory. The results pave the way toward all-orders universal expressions and connections to $q$-deformed $\mathcal{W}$-algebras, with potential insights into the geometry of exceptional instanton moduli spaces.

Abstract

We show how to obtain the instanton partition function of N=2 SYM with exceptional gauge group EFG using blow-up recursion relations derived by Nakajima and Yoshioka. We compute the two instanton contribution and match it with the recent proposal for the superconformal index of rank 2 SCFTs with E6, E7 global symmetry.