Spectral curves of $\mathcal{N}=1$ theories of class $\mathcal{S}_k$

TL;DR

The work develops a Coulomb-branch spectral-curve framework for four-dimensional $\mathcal{N}=1$ theories in class $\mathcal{S}_k$, obtained by orbifolding class $\mathcal{S}$ theories. It leverages M-theory constructions and a Gaiotto-like punctured-curve dictionary to derive explicit IR curves (e.g., for SCQCD$_k$) and reexpress them as UV four-punctured (and trinioned) Riemann surfaces with a novel $k$-sheeted structure. A key finding is the emergence of minimal punctures as branch points with fractional power, along with two distinct maximal punctures (one ordinary and one fractional) and a rich network of both inherited and orbifold-induced branch cuts. The paper also analyzes weak- and strong-coupling limits, including the free trinion and non-Lagrangian $T_N^k$ theories, and discusses the implications for future 4D/2D dualities, potential AGT$_k$ extensions, and higher-dimensional uplifts.

Abstract

We study the Coulomb branch of class $\mathcal{S}_k$ $\mathcal{N} = 1$ SCFTs by constructing and analyzing their spectral curves.

Figures (13)

• Figure 1: The IIA brane set-up from which we calculate the IR curve $\Sigma$ for the $SU(2)$ case. The thick dashed line depicts the $\mathbb{Z}_k$ orbifold point, for $k=2$. For each D4 brane the mirror images are also depicted using grey dotted lines.
• Figure 2: The quiver diagram for the orbifolded linear quiver of $\mathcal{N}=2$ with $M=4$. The color groups are labelled by $(i,c)$ where ${i=1,\ldots,k}$ is the $\mathbb{Z}_k$ orbifold index that labels the mirror images and $c=0,\ldots,M$ is the label from the original $\mathcal{N}=2$ theory.
• Figure 3: The spectral curve for $\mathcal{N}=2$ SCQCD with $SU(2)$ gauge group and $N_f=4$ flavors. On the left we depict the SW curve $\Sigma$. On the right we depict the four-punctured sphere $\mathcal{C}_{0,4}$, the Gaiotto curve, whose double-cover is $\Sigma$.
• Figure 4: The IR spectral curve of $\mathcal{N}=1$$SU(2)$ SCQCD$_k$ obtained as a $2k^{th}$ cover of the four-punctured sphere in class $\mathcal{S}_k$. On the left we draw the IR curve $\Sigma$ and explicitly denote the branch cut structure. On the right side we depict only the four-punctured sphere $\mathcal{C}_{0,4}^{(k)}$.
• Figure 5: The IR spectral curve of $\mathcal{N}=1$$SU(3)$ SCQCD$_k$ obtained as a $3k^{th}$ cover of the four-punctured sphere in class $\mathcal{S}_k$. On the left we draw the IR curve $\Sigma$ and the branch cuts and on the right the four-punctured sphere $\mathcal{C}_{0,4}^{(k)}$.