Exceptional Indices

TL;DR

The paper analyzes the Hall-Littlewood limit of the class S superconformal index, introducing a practical diagnostic for when the index sum diverges (bad), contains decoupled free sectors (ugly), or converges (good). It reframes index computation as a residue calculus that reduces flavor symmetry step-by-step and explores the nuances of pole structures, including disappearing and higher-order poles, to extend the method beyond S-duality-connected theories. It provides explicit constructions and examples, including rank-two E6, and connects indices to Higgs-branch Hilbert series and multi-instanton moduli spaces, offering new expressions for higher-rank E_n theories. The results illuminate how decoupled sectors reshape the index and enable predictions of symmetry enhancements and Higgs-branch structure in otherwise challenging SCFTs.

Abstract

Recently a prescription to compute the superconformal index for all theories of class S was proposed. In this paper we discuss some of the physical information which can be extracted from this index. We derive a simple criterion for the given theory of class S to have a decoupled free component and for it to have enhanced flavor symmetry. Furthermore, we establish a criterion for the "good", the "bad", and the "ugly" trichotomy of the theories. After interpreting the prescription to compute the index with non-maximal flavor symmetry as a residue calculus we address the computation of the index of the bad theories. In particular we suggest explicit expressions for the superconformal index of higher rank theories with E_n flavor symmetry, i.e. for the Hilbert series of the multi-instanton moduli space of E_n.

Figures (11)

• Figure 1: Association of the flavor fugacities for a generic puncture. Punctures are classified by embeddings of $SU(2)$ in $SU(k)$, so they are specified by the decomposition of the fundamental representation of $SU(k)$ into irreps of $SU(2)$, that is, by a partition of $k$. Graphically we represent the partition by an auxiliary Young diagram $\Lambda$ with $k$ boxes, read from left to right. In the figure we have the fundamental of $SU(26)$ decomposed as $\mathbf{5} + \mathbf{5} + \mathbf{4} + \mathbf{4} + \mathbf{4} + \mathbf{2} + \mathbf{1} + \mathbf{1}$. The commutant of the embedding gives the residual flavor symmetry, in this case $S(U(3)\times U(2)\times U(2)\times U(1))$, where the $S(\dots)$ constraint amounts to removing the overall $U(1)$. The $\tau$ variable is viewed here as an $SU(2)$ fugacity, while the Latin variables are fugacities of the residual flavor symmetry. The $S(\dots)$ constraint implies that the flavor fugacities satisfy $(ab)^{5}(cde)^4f^2gh=1$.
• Figure 2: The factors ${\frak a}^i_k$ associated to a generic Young diagram. The upper index is the row index and the lower is the column index. In $\bar{\frak a}^i_k$ one takes the inverse of flavor fugacities while $\tau$ is treated as real number. As before, the flavor fugacities in this example satisfy $(ab)^{5}(cde)^4f^2gh=1$.
• Figure 3: The $SU(2k+1)$ Riemann surface corresponding to the rank $k$ interacting SCFT with flavor symmetry $SO(4k+6)\times U(1)$. The height of the special punctures is $k$.
• Figure 4: Getting L-shaped puncture from a maximal one.
• Figure 5: Auxiliary Young diagram. The flavor symmetry is decreased by raising the right-most box to the position indicated by the arrow.