Compactification of 6d minimal SCFTs on Riemann surfaces

TL;DR

The paper investigates how 6d $N=(1,0)$ rank-1 SCFTs without continuous flavor symmetries compactify on punctured Riemann surfaces to yield 4d $\mathcal N=1$ theories, deriving 4d anomalies by integrating the 6d anomaly polynomial and proposing explicit 4d quivers for the $G=SU(3)$ and $G=SO(8)$ cases. It connects these 4d constructions to 5d circle reductions and twisted punctures, with central charges $a$ and $c$ governed by the 6d group data and the surface genus, and shows that conformal manifolds match the complex-structure moduli, while dualities arise from different pair-of-pants decompositions. The work provides concrete Lagrangian-like quivers built from trifundamental and bifundamental chiral fields, validates anomaly matching and index computations, and analyzes puncture closing and twists to reveal a rich web of dualities and RG flows. Overall, it advances a systematic framework linking 6d rank-1 SCFT data to 4d $N=1$ theories via Riemann-surface compactifications, with predictions that tie mapping-class-group-like dualities to the geometry of the compactification surface.

Abstract

We study compactifications on Riemann surfaces with punctures of N=(1,0) 6d SCFTs with a one dimensional tensor branch and no continuous global symmetries. The effective description of such models on the tensor branch is in terms of pure gauge theories with decoupled tensor. For generic Riemann surfaces, the resulting theories in four dimensions are expected to have N=1 supersymmetry. We compute the anomalies expected from the resulting 4d theories by integrating the anomaly polynomial of the 6d theory on the Riemann surface. For the cases with 6d gauge models with gauge groups SU(3) and SO(8) we further propose a field theory construction for the resulting 4d theories. For the 6d SU(3) theory, we argue that the theories in four dimensions are quivers with SU(3) gauge nodes and free chiral fields. The theories one obtains from the 6d SO(8) gauge theory are quivers with SU(4) gauge groups and chiral fields with R charge a half. In the last case the theories constructed for general Riemann surfaces involve gauging of symmetries appearing at strong coupling. The conformal manifolds of the models are constructed from gauge couplings and baryonic superpotentials. We support our conjectures by matching the dimensions of the conformal manifolds with complex structure moduli of the Riemann surfaces, matching anomalies between six and four dimensions, and checking the dualities related to different pair of pants decompositions of the surfaces. As a simple application of the results we conjecture that SU(3) gauge theory with nine flavors in four dimensions has a duality group acting on the seven dimensional conformal manifold which is the mapping class group of sphere with ten marked points.

Figures (8)

• Figure 1: The sphere with four punctures. The squares are $SU(3)$ flavor and circles $SU(3)$ gauge. We have additional $SU(3)$ global symmetries rotating the three bifundamentals. The bifundamentals in the middle link have a baryonic superpotential turned on breaking the $SU(3)$ flavor symmetry. The wavy lines denote the fact that the fields have the baryonic superpotential. All in all we are left with four $SU(3)$ symmetries.
• Figure 2: Building blocks of the theories. First we have the trifundamental, and then trifundamental with one of the three $SU(3)$ symmetries broken by baryonic superpotential which preserves other two $SU(3)$ flavor symmetries.
• Figure 3: Four puncture sphere
• Figure 4: Example of genus three model.
• Figure 5: Closing a puncture of the four punctured sphere. The circles are $SU(3)$ punctures and crosses empty ones.