The Superconformal Index of the E_6 SCFT

TL;DR

This work derives an explicit integral representation for the superconformal index of the rank-one ${ m E}_6$ SCFT by leveraging Argyres–Seiberg duality to invert gauging in a weakly-coupled $SU(3)$ theory with ${N_f=6}$.The index of the ${ m E}_6$ theory is obtained as a closed form through a nontrivial inversion formula for elliptic hypergeometric integrals, and its expansion reveals the expected Higgs, Coulomb, and stress-tensor sectors along with Joseph relations for the Higgs branch.The authors organize the generalized quivers into a 2d TQFT with associativity checks that reproduce S-duality frames of the ${ m A}_2$ theories, performing perturbative verifications that also yield new elliptic Gamma identities.The results demonstrate how duality-based methods can yield exact indices for strongly coupled SCFTs and point toward extensions to higher-rank theories (e.g., ${ m E}_7$) and deeper mathematical structures underlying dualities.

Abstract

We derive an integral representation for the superconformal index of the strongly-coupled N=2 superconformal field theory with E_6 flavor symmetry. The explicit expression of the index allows highly non-trivial checks of Argyres-Seiberg duality and of a class of S-dualities conjectured by Gaiotto.

Figures (7)

• Figure 1: Moduli spaces for $\mathcal{N}=2$$SU(n)$ gauge theory with $2n$ flavors, (a) for $n=2$ and (b) for $n=3$ (in fact, for any $n >2$). The shaded region in (a) is $H/SL(2,\mathbb Z)$ while in (b) it is $H/\Gamma^0(2)$, where $H$ is the upper half plane.
• Figure 2: $SU(3)$ SYM with $N_f = 6$. The $U(6)$ flavor symmetry is decomposed as $SU(3)_{\mathbf z} \otimes U(1)_a \oplus SU(3)_{\mathbf y} \otimes U(1)_b$. S-duality $\tau \to -1/\tau$ interchanges the two $U(1)$ charges.
• Figure 3: Argyres-Seiberg duality for $SU(3)$ SYM with $N_f = 6$.
• Figure 4: The three structure constants of the TQFT. The dots represent $U(1)$ punctures and the circled dots $SU(3)$ punctures.
• Figure 5: The relevant four-punctured spheres for $A_2$ theories. The three different degeneration limits of a four-punctured sphere correspond to different S-duality frames. For example, in $(a)$ two of the degeneration limits (when a $U(1)$ puncture collides with an $SU(3)$ puncture) correspond to the weakly-coupled $N_f=6$$SU(3)$ theory, the third limit (when two like punctures collide) corresponds to the Argyres-Seiberg theory. In $(d)$ the degeneration limits correspond to the different duality frames of $SU(2)$ SYM with $N_f=4$ theory plus a decoupled hypermultiplet.