Equivalence of several descriptions for 6d SCFT

TL;DR

<$The paper demonstrates an exact equivalence between the elliptic genus of the 6d $\mathcal{N}=(1,0)$ $Sp(1)$ theory with $10$ flavors and a tensor multiplet, the Nekrasov partition function of the 5d $\mathcal{N}=1$ $Sp(2)$ theory with $10$ flavors, and the Nekrasov partition function of the 5d $\mathcal{N}=1$ $SU(3)$ theory with $10$ flavors, under explicit parameter maps derived from Type IIB 5-brane webs (Tao diagrams) and Higgsing arguments. The authors compute the 5d $SU(3)$ partition function via the Tao diagram and verify the equivalence order by order in the instanton fugacity $q$ and a Coulomb modulus (two-string sector), and also analyze the 5d $Sp(2)$ partition function up to one-instanton exactly, proposing a corrected two-instanton structure guided by Weyl-invariant completions and consistency with the elliptic genus. The results provide strong evidence for a common UV fixed point for the three theories and expand the toolkit for high-flavor BPS counting using Tao diagrams, with discussed avenues to generalize to higher rank and other duality webs.

Abstract

We show that the three different looking BPS partition functions, namely the elliptic genus of the 6d $\mathcal{N}=(1,0)$ $Sp(1)$ gauge theory with $10$ flavors and a tensor multiplet, the Nekrasov partition function of the 5d $\mathcal{N}=1$ $Sp(2)$ gauge theory with $10$ flavors, and the Nekrasov partition function of the 5d $\mathcal{N}=1$ $SU(3)$ gauge theory with $10$ flavors, are all equal to each other under specific maps among gauge theory parameters. This result strongly suggests that the three gauge theories have an identical UV fixed point. Type IIB 5-brane web diagrams play an essential role to compute the $SU(3)$ Nekrasov partition function as well as establishing the maps.

Figures (13)

• Figure 1: The type IIA brane configuration which gives rise to the 6d $Sp(1)$ gauge theory with $10$ flavors and a tensor multiplet.
• Figure 2: The 5-brane web which gives rise to the 5d $Sp(2)$ gauge theory with $10$ flavors.
• Figure 3: The 5-brane web which gives rise to the 5d $SU(3)$ gauge theory with $10$ flavors.
• Figure 4: The extrapolation of the upper and lower external 5-branes to the origin.
• Figure 5: Shrinking the Coulomb branch modulus and an effect of the resolution of an $O7^-$-plane.