$\mathcal{N} = 1$ superconformal theories with $D_N$ blocks

TL;DR

This work provides a complete four-dimensional field-theoretic realization of the BBBW class of ${\mathcal N}=1$ SCFTs obtained from M5-branes on curves in Calabi–Yau three-folds, by introducing the ${D_N}$ family of building blocks and their cousins ${\widetilde{D}}_N$ and ${\widetilde{}}_N$. By performing exact ${a}$-maximization and matching to the M5 anomaly polynomial, the authors reproduce the central charges for all BBBW models, including inaccessible cases with negative line-bundle degrees, and they classify heavy operators associated with M2-wrapping, with their ${R}$-charges matching holographic expectations. They derive a rich set of chiral ring relations across ${T_N}$, ${D_N}$, and flipped puncture constructions, establishing a web of ${\mathcal N}=1$ dualities and providing precise operator-counting rules for relevant operators in varied topologies (high genus, torus, sphere). The results offer a rigorous four-dimensional underpinning for these holographic theories, illuminate the role of nilpotent Higgsing in building blocks, and set the stage for further generalizations to other ${\mathcal S}$-type and higher-rank constructions with exact central-charge and operator data.

Abstract

We study the chiral ring of four-dimensional superconformal field theories obtained by wrapping M5-branes on a complex curve inside a Calabi-Yau three-fold. We propose a field theoretic construction of all the theories found by Bah, Beem, Bobev and Wecht by introducing new building blocks, and prove several $\mathcal{N} = 1$ dualities featuring the latter. We match the central charges with those computed from the M5-brane anomaly polynomial, perform the counting of relevant operators and analyze unitarity bound violations. As a byproduct, we compute the exact dimension of "heavy operators" obtained by wrapping an M2-brane on the complex curve.

Figures (17)

• Figure 1: The building blocks for $\mathcal{N}=1$ generalized quiver theories. From left to right: $T_N$, $D_N$, $\widetilde{D}_N$, and ${ \widetilde{}}_N$. The $\mu_X$'s refer to the moment maps associated with the available $\mathop{\mathrm{SU}}\nolimits(N)_X$ flavor symmetries (represented by legs) in a block. Each leg can be used as a "tube" connecting two blocks in a generalized quiver, whereby the flavor symmetries connected by the tube (say $X$ and $Y$) are gauged together in an $\mathcal{N}=2$ or $\mathcal{N}=1$ way, i.e. via the superpotential term $\mathop{\mathrm{Tr}}\nolimits \Phi (\mu_X -\mu_Y)$ and $\mathop{\mathrm{Tr}}\nolimits \mu_X \mu_Y$ respectively.
• Figure 2: Examples of $\mathcal{N}=1$ theories engineered by $N$ M5-branes wrapping a holomorphic curve $\mathcal{C}_3$ of genus $g=3$. (There are four other quiver topologies with same genus bbbw.) The various possibilities depicted in figures \ref{['fig:g3p4']}, \ref{['fig:g3p1q3']}, \ref{['fig:g36Dn']} correspond to different choices of degrees $p$, $q$ of the two line bundles over $\mathcal{C}_3$, such that $2g-2=4=p+q$. To have $p>0, q<0$ we need to introduce some $D_N$ blocks in the generalized quiver.
• Figure 3: Examples of $\mathcal{N}=1$ theories engineered by $N$ M5-branes wrapping a torus. Figure \ref{['fig:torus-1']} shows the "minimal" theory consisting of a single $D_N$ block (i.e. $p=1$), its two flavor symmetries being gauged together. Figure \ref{['fig:torus-6']} shows the $p=6$ case. All gaugings are $\mathcal{N}=2$.
• Figure 4: Examples of $\mathcal{N}=1$ theories engineered by $N$ M5-branes wrapping the sphere. There is no available flavor symmetry (hence no punctures on $\mathcal{C}_0$), as is clear from the absence of external legs in the quivers. The value of $z=\frac{p-q}{p+q}$ is always negative and integer in this case, $z=-2,-3,-4,\ldots$, given that $p>q$ and $p+q =-2<0$.
• Figure 5: Coupling a $T_N$ block to a linear quiver with $N-2$ gauge groups of decreasing rank. A subgroup $\mathop{\mathrm{SU}}\nolimits(N-1)$ of $\mathop{\mathrm{SU}}\nolimits(N)_C$ has been gauged by the first $\mathop{\mathrm{SU}}\nolimits(N-1)$ gauge group on the right.