N=2 Gauge Dynamics and Seiberg–Witten Geometry

Seiberg–Witten theory encodes two-derivative N=2 gauge dynamics in periods of a geometric family. This chapter connects local effective couplings to global charge transport and singularities, using pure SU(2) as the main example while distinguishing central-charge formulas from the actual BPS spectrum.

Enter this chapter

Begin with N=2 Multiplets, Lagrangians, and Vacuum Branches to identify the multiplets and vacuum branches of an N=2 theory.

Helpful background. Hamiltonian Group Actions and Moment Maps; Branches, Sheets, Analytic Continuation, and Monodromy; Monopoles and Dyons provide the symplectic geometry, analytic continuation, and monopole charges used here.

Topics

1. N=2 Multiplets, Lagrangians, and Vacuum Branches — Builds vector and hypermultiplet theories.
2. Coulomb Branches and the Abelian Low-Energy Theory — Identifies the generic abelian Coulomb-branch description.
3. Prepotentials and Special Kähler Geometry — Packages local couplings in a holomorphic prepotential.
4. Charge Lattices, Duality Frames, and Local Systems — Transports integral charges between duality frames.
5. Singularities, Vanishing Cycles, and Monodromies — Interprets vanishing charges and monodromy.
6. Seiberg–Witten Curves, Differentials, and Periods — Extracts physical periods from a curve and differential.
7. The Pure SU(2) Seiberg–Witten Solution — Reconstructs the pure-SU(2) solution.
8. Matter, Higher Rank, and Integrable-System Structures — Extends the construction to matter and higher rank.
9. BPS Spectra, Chambers, and Wall Crossing — Adds state existence and stability chambers to period data.
10. Argyres–Douglas Theories, Class S, and Non-Lagrangian Interfaces — Identifies interacting singular sectors and class-S constructions.
11. N=1 Deformations, Monopole Condensation, and Controlled Confinement — Uses a small N=1 deformation to study monopole condensation.

How the ideas fit together

Periods define central charges locally, but transport around a singularity mixes their electric and magnetic components. Charges must transform compatibly so that the physical pairing is unchanged. A vanishing period signals failure of the photon-only effective description and the need to consider a light charged sector. It does not by itself enumerate all BPS states or their stability chambers. The global solution therefore combines analytic geometry, integral charge data, and separate spectral input. Seiberg and Witten 1994, §§ 2–6.

Where to go next

Continue to N=4 SYM, S-Duality, and Higher-Dimensional Interfaces to separate the dynamical consequences of eight and sixteen real supercharges.

References

• Seiberg, Nathan, and Edward Witten. “Electric–Magnetic Duality, Monopole Condensation, and Confinement in $\mathcal N=2$ Supersymmetric Yang–Mills Theory.” Nuclear Physics B 426 (1994): 19–52; erratum 430 (1994): 485–486. arXiv:hep-th/9407087.