N=4 SYM, S-Duality, and Higher-Dimensional Interfaces
Four-dimensional N=4 Yang–Mills theory combines strong local constraints with nontrivial global choices. This chapter separates its action, moduli, and finiteness properties from the proposed strong–weak dualities and from the higher-dimensional constructions that organize them.
Enter this chapter
Section titled “Enter this chapter”Begin with Extended Supersymmetry Gauge Dynamics: Scope and Structure to separate the dynamical consequences of eight and sixteen real supercharges.
Helpful background. Electric and Magnetic One-Form Symmetries; Genuine Line Spectra, Discrete Theta Data, and Theory Specification; Primaries, Descendants, and Conformal Multiplets provide the one-form symmetries, genuine lines, and conformal multiplets used here.
Topics
Section titled “Topics”- Extended Supersymmetry Gauge Dynamics: Scope and Structure — Sets the scope of extended-supersymmetry dynamics.
- N=4 SYM Field Content, Action, and Superconformal Data — Defines N=4 fields and local superconformal data.
- N=4 Moduli, BPS States, and Protected Sectors — Distinguishes vacuum geometry from protected spectra.
- Finiteness, Conformality, and Nonperturbative Limits — Separates perturbative finiteness from wider existence claims.
- Line Operators, Global Forms, and Discrete Theta Data — Specifies lines and discrete data beyond the gauge algebra.
- Montonen–Olive and S-Duality — States the electric–magnetic duality map and its tests.
- Duality Groupoids, Walls, and Generalized-Symmetry Refinements — Tracks transformations between globally distinct theories.
- Five- and Six-Dimensional Supersymmetric Fixed Points, Branches, and Status — Distinguishes higher-dimensional fixed points from their effective branches.
- Six-Dimensional Origins, Compactification, and Duality Frames — Transports accepted higher-dimensional data through compactification.
How the ideas fit together
Section titled “How the ideas fit together”The familiar transformation of the complex coupling is only part of S-duality. Electric and magnetic line charges transform too, and a transformation can take one global gauge form or discrete-theta choice to another. The full family of theories therefore carries more information than a coupling acted on by a group. Compactification can explain this organization once the higher-dimensional starting theory is specified, but cannot serve as an independent proof of every property assumed for that starting point. Aharony, Seiberg, and Tachikawa 2013, §§ 1–2.
Where to go next
Section titled “Where to go next”Continue to Two-Dimensional Supersymmetric QFT, GLSMs, and Mirror Symmetry to organize supercharges and multiplets by two-dimensional chirality.