Surveying 4d SCFTs twisted on Riemann surfaces

TL;DR

This work develops a unified framework to obtain 2d $\mathcal{N}=(0,2)$ SCFTs by twisted compactification of 4d $\mathcal{N}=1,2,3,4$ SCFTs on constant-curvature Riemann surfaces, using 4d conformal supergravity to solve Killing-spinor equations under abelian R-symmetry and flavor fluxes. It provides a comprehensive classification of preserved supersymmetry for each amount of 4d supersymmetry, detailing the allowed flux configurations on $\Sigma$ and the resulting 2d algebras, including two-step reduction mechanisms where applicable. A central contribution is the c-extremization construction that yields the 2d central charge and the exact R-current in terms of 4d anomaly data and fluxes, explicitly relating 4d 't Hooft anomalies to 2d anomalies. The results offer a robust toolkit for generating families of 2d SCFTs with controlled supersymmetry and anomaly structure, with potential holographic duals and extensions to punctured surfaces or higher-dimensional compactifications. Key mathematical tools include Killing-spinor analysis in conformal supergravity, flux-induced topological twists, anomaly polynomials, and the c-extremization procedure that captures current mixing across dimensions.

Abstract

Within the framework of four dimensional conformal supergravity we consider $\mathcal{N}=1,2,3,4$ supersymmetric theories generally twisted along the abelian subgroups of the R-symmetry and possibly other global symmetry groups. Upon compactification on constant curvature Riemann surfaces with arbitrary genus we provide an extensive classification of the resulting two dimensional theories according to the amount of supersymmetry that is preserved. Exploiting the c-extremization prescription introduced in arXiv:1211.4030 we develop a general procedure to obtain the central charge for 2d $\mathcal{N}=(0,2)$ theories and the expression of the corresponding R-current in terms of the original 4d one and its mixing with the other abelian global currents.