Infinitely many 4D <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi mathvariant="script">N</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:math> SCFTs with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>a</mml:mi><mml:mo>=</mml:mo><mml:mi>c</mml:mi></mml:math> and beyond
Monica Jinwoo Kang, Craig Lawrie, Jaewon Song
Abstract
We study a set of four-dimensional $\mathcal{N}=2$ superconformal field theories (SCFTs) $\stackrel{^}{\mathrm{\ensuremath{\Gamma}}}(G)$ labeled by a pair of simply laced Lie groups $\mathrm{\ensuremath{\Gamma}}$ and $G$. They are constructed out of gauging a number of ${\mathcal{D}}_{p}(G)$ and $(G,G)$ conformal matter SCFTs; therefore, they do not have Lagrangian descriptions in general. For $\mathrm{\ensuremath{\Gamma}}={D}_{4},{E}_{6},{E}_{7},{E}_{8}$, and some special choices of $G$, the resulting theories have identical central charges ($a=c$) without taking any large $N$ limit. Moreover, we find that the Schur indices for such theories can be written in terms of that of $\mathcal{N}=4$ super-Yang-Mills theory upon rescaling fugacities. Especially, we find that the Schur index of ${\stackrel{^}{D}}_{4}(SU(N))$ theory for $N$ odd is written in terms of MacMahon's generalized sum-of-divisor function, which is quasimodular. For generic choices of $\mathrm{\ensuremath{\Gamma}}$ and $G$, it can be regarded as a generalization of the affine quiver gauge theory obtained from $\text{D}3$-branes probing singularity of type $\mathrm{\ensuremath{\Gamma}}$. We also comment on a tantalizing connection regarding the theories labeled by $\mathrm{\ensuremath{\Gamma}}$ in the Deligne-Cvitanovi\ifmmode \acute{c}\else \'{c}\fi{} exceptional series.