Notes on Enhancement of Flavor Symmetry and 5d Superconformal Index

TL;DR

The work investigates the enhancement of flavor symmetry in 5d SU(2) UV fixed-point theories using the 5d superconformal index computed via localization and refined topological string techniques. By decomposing the index into perturbative and instanton pieces and expressing results in SU(2) characters, it provides evidence that the UV theories realize E_{N_f+1} flavor symmetries for N_f = 0,1,2, with higher cases explored through geometric engineering of F_m surfaces. A key methodological advance is the identification and removal of non-full spin content, which isolates the genuine UV fixed-point contribution and yields positive, representation-theoretic index expansions. The results unify CY/brane-web constructions with algebraic structures (AGT-related arguments) to support the proposed symmetry enhancements and set the stage for extending the analysis to all N_f up to 7.

Abstract

The UV fixed point theory of SU(2) gauge theory with N_f = 0,1,...,7 flavors is believed to have the enlarged E_{N_f +1} flavor symmetry. Actually it is not easy to check this conjecture because the UV theory is strongly-coupled, however, computation of certain SUSY protected quantities provides strong evidence for the enhancement of flavor symmetry. We study the superconformal index for SU(2) gauge theory with N_f = 0, 1 flavors in details, and we give a support for the enhancement by studying combinatorial structure of the superconformal indexes of these theories. We also give a nontrivial evidence that the local F_2 geometry leads to the E_1 superconformal field theory.

Figures (4)

• Figure 1: The first deformation which moves the external lines is the mass deformation of the theory $E_1$. The second arrow denotes the breathing mode parametrized by $Q_F$.
• Figure 2: In the limit $\vert x\vert \ll 1$ the pole region of the integrant of the formula (\ref{['indexE0']}) splits into near the origin and faraway region.
• Figure 3: The local $\mathbb{F}_2$ geometry and its six-dimensional contribution. (a) 5-brane web for the local $\mathbb{F}_2$ geometry. (b) on the coincided two 5-branes, additional six-dimensional degrees of freedom appear.
• Figure 4: We can regularize the 5-brane web by terminating the external branes on 7-branes. The 7-branes are illustrated by the blue circles. We conjecture that the regularized theory can be computed by eliminating the contribution from the non-full-spin content, and the resulting 5d theory is equal to the local $\mathbb{F}_0$ theory, i.e. the $E_1$ SCFT.