5-dim Superconformal Index with Enhanced En Global Symmetry

TL;DR

This paper computes the 5d ${\cal N}=1$ superconformal index for Sp$(N)$ gauge theories with $N_f$ fundamental flavors on $S^1\times S^4$ using localization, including both perturbative and Nekrasov-type instanton contributions. The authors demonstrate global symmetry enhancement from $SO(2N_f)\times U(1)_I$ to $E_{N_f+1}$ at the conformal point, by reorganizing index terms into $E_{N_f+1}$ adjoint and higher adjoint representations for $N_f\le5$, and discuss obstacles for $N_f=6,7$ due to pole ambiguities in multi-instanton sectors. They present explicit results for ${\rm Sp}(1)$ up to four instantons and for ${\rm Sp}(2)$ up to one instanton, highlighting a universal pattern where instanton sectors populate higher adjoint powers in the enhanced symmetry. The findings connect 5d SCFT dynamics to brane constructions (D4/D8/O8), Nekrasov instanton physics, and potential reductions to 4d ellipsoids or holographic duals, offering a concrete nonperturbative probe of symmetry enhancement in higher-dimensional QFTs.

Abstract

The five-dimensional $\mathcal{N}=1$ supersymmetric gauge theory with Sp(N) gauge group and SO(2N_f) flavor symmetry describes the physics on N D4-branes with $N_f$ D8-branes on top of a single O8 orientifold plane in Type I' theory. This theory is known to be superconformal at the strong coupling limit with the enhanced global symmetry $E_{N_f+1}$ for $N_f\le 7$. In this work we calculate the superconformal index on $S^1\times S^4$ for the Sp(1) gauge theory by the localization method and confirm such enhancement of the global symmetry at the superconformal limit for $N_f\le 5$ to a few leading orders in the chemical potential. Both perturbative and (anti)instanton contributions are present in this calculation. For $N_f=6,7$ cases some issues related the pole structure of the instanton calculation could not be resolved and here we could provide only some suggestive answer for the leading contributions to the index. For the Sp(N) case, similar issues related to the pole structure appear.