SCFTs

Superconformal field theories (SCFTs) are quantum field theories that are invariant under the superconformal group, combining conformal symmetry with supersymmetry. They possess scale invariance and special conformal transformations, and their full symmetry algebra imposes strong constraints on the spectrum and dynamics. The local operators of an SCFT organize into superconformal multiplets, including short (BPS) multiplets whose scaling dimensions are protected by supersymmetry. Because of these protections, many correlation functions and spectra can be computed exactly across coupling, making SCFTs a fertile ground for nonperturbative insight.

In four dimensions, SCFTs are classified by the amount of supersymmetry: N=1, N=2, N=3, or N=4, with respective R-symmetry groups U(1), SU(2), SU(3), and SU(4). The best-known example is N=4 super Yang–Mills theory, which is believed to remain conformal at all values of the gauge coupling and is central to the AdS/CFT correspondence, dual to type IIB string theory on AdS5 × S5. Other 4d SCFTs arise at conformal fixed points of renormalization group flows and in various quiver constructions.

In three dimensions, SCFTs include theories with N=2, N=3, N=4, N=6, and N=8 supersymmetry. A notable example is ABJM theory, an N=6 Chern–Simons–matter theory with a holographic dual in AdS4. In two dimensions, SCFTs with extended supersymmetry (for instance N=(2,2) and N=(4,4)) play a central role in string theory, mirror symmetry, and the study of exactly solvable models. In six dimensions, the (2,0) SCFTs are highly symmetric and strongly interacting, lacking a conventional Lagrangian description but influential in the broader landscape of quantum field theory.

Applications of SCFTs include deriving exact results via localization, constraining operator dimensions through unitarity and representation theory, and exploring holographic dualities, dualities between different field theories, and properties of moduli spaces and anomalies.