4d SCFTs from negative-degree line bundles

TL;DR

This work extends BBBW-type 4d $ abla \,N=1$ SCFTs to cases with negative line-bundle degrees by leveraging generalized $T_N^{(m)}$ building blocks. Through meticulous $a$-maximization and anomaly matching, the authors derive explicit central charges $a$ and $c$ and operator dimensions for genus $g$ surfaces with $n$ maximal punctures, showing results depend only on topological data via the twist $z=(p-q)/(p+q)$ and $ ext{χ}=-2g+2-n$. The constructions reproduce BBBW gravity results for both $g>1$ and torus cases, including the large-$N$ scaling $a,c ightarrow ext{const} imes N^3$, while highlighting possible field-theoretic corrections from decoupled operators in certain regimes. The paper also analyzes the Higgsed ($g=0$) sector, demonstrates agreement with BBBW in many cases, and outlines a comprehensive framework for general $C_{g,n}$ via chaining $T_N^{(m_i)}$ blocks, paving the way for further explorations of dualities, moduli spaces, and special-point phenomena in these extended class S theories.

Abstract

We construct 4d $\mathcal{N}=1$ quantum field theories by compactifying the (2,0) theories on a Riemann surface with genus $g$ and $n$ punctures, where the normal bundle decomposes into a sum of two line bundles with possibly negative degrees $p$ and $q$. Until recently, the only available field-theoretic constructions required the line bundle degrees to be nonnegative, although supergravity solutions were constructed in the literature for the zero-puncture case for all $p$ and $q$. Here, we provide field-theoretic constructions and computations of the central charges of 4d $\mathcal{N}=1$ SCFTs that are the IR limit of M5-branes wrapping a surface with general $p$ or $q$ negative, for general genus $g$ and number of maximal punctures $n$.

Figures (6)

• Figure 1: A UV generalized quiver description for the $T_N^{(m)}$ theories. The blue color of the $SU(N)_{A,B,C}$ flavor groups corresponds to punctures with sign $\sigma_{A,B,C}=+1$. The red boxes correspond to closed $\sigma=-1$ punctures, i.e. red-colored $SU(N)$ flavor groups that were Higgsed by giving vevs to adjoint operators $M^{(i)}$. The singlets $M_j^{(i)}$ are the leftover components of the fluctuations of the $M^{(i)}$ about the vevs. The diagram to the right of the quiver is used in later figures as a shorthand.
• Figure 2: Options for gluing 2 $T_N^{(m_i)}$ blocks to form a genus $2$ surface. Shaded triangles correspond to $T_N^{(m_i)}$ theories with $\sigma_i=-1$, while unshaded triangles have $\sigma_i=+1$. Shaded circular nodes correspond to $\mathcal{ N} =1$ vector multiplets, while unshaded circular nodes correspond to $\mathcal{ N} =2$ vector multiplets. Blocks of differing colors should be glued by an $\mathcal{ N} =1$ vector.
• Figure 3: $a$ as a function of the twist $z$ for quivers constructed from $2g-2$ glued $T_N^{(m_i)}$ blocks, plotted for $g=11$ and various $N$.
• Figure 4: UV quiver for the $T_N^{(m)}$ theory with $\sigma_i=+1$, Higgsing the $SU(N)$ flavor nodes. The rightmost figure depicts a shorthand we use throughout, for reference.
• Figure 5: A genus 2, single-puncture example of a possible generalization of the diagrams in Figure \ref{['fig:genus2ex']}. In our notation, these all have $n_{tot}=1$, $n_{dif}=-1$, and $\ell_1+\ell_2=2,n_1+n_2= \ell_1'+\ell_2'= n_1'+n_2'=3,h_2=2$. All three diagrams have the same IR central charges.