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Supersymmetric Vacua, Moduli Geometry, and BPS Sectors

Supersymmetric vacua are zero-energy configurations modulo gauge equivalence; BPS states saturate charge-dependent energy bounds. This chapter connects their algebraic and geometric descriptions while keeping metric corrections, existence, and stability separate from protected relations.

Enter this chapter

Section titled “Enter this chapter”

Begin with F- and D-Flatness and Gauge Quotients to solve flatness equations before dividing by genuine gauge redundancy.

Helpful background. Hamiltonian Group Actions and Moment Maps; Gauge Orbits, Gauss Constraints, and Stabilizers; Bogomolny Bounds and First-Order Equations provide the moment maps, gauge quotients, and energy bounds for solitons used here.

Topics

Section titled “Topics”
  1. F- and D-Flatness and Gauge Quotients — Constructs the physical vacuum quotient.
  2. Gauge-Invariant Coordinates and Classical Moduli Varieties — Uses invariant coordinates with the correct quotient conditions.
  3. Kähler and Hyperkähler Quotients in Supersymmetric QFT — Adds symplectic and metric structure to vacuum geometry.
  4. Moduli-Space Metrics and Quantum Corrections — Distinguishes protected complex data from corrected metrics.
  5. Singular Loci and Emergent Low-Energy Degrees of Freedom — Identifies light sectors near singular loci.
  6. BPS Particles and Central Charges — Relates particle masses to central charges.
  7. BPS Solitons, Walls, Strings, Vortices, and Junctions — Derives soliton bounds and preserved supercharges.
  8. Framed BPS States, Line Defects, and Defect Moduli — Defines the Hilbert space in the presence of a BPS defect.
  9. Marginal Stability, Chambers, and Wall Crossing — Tracks allowed decays as central-charge phases align.

How the ideas fit together

Section titled “How the ideas fit together”

A BPS bound says what the mass would be if an appropriate state exists and saturates it. It does not guarantee existence in every chamber of moduli space. When central charges of possible constituents align, a bound state can reach a decay threshold and disappear while the algebraic bound remains valid. Framed states similarly depend on the chosen defect. This distinction connects moduli geometry to wall crossing without identifying a protected charge formula with a complete spectrum. Gaiotto, Moore, and Neitzke 2013, §§ 2–3.

Where to go next

Section titled “Where to go next”

Continue to Holomorphy, Anomalies, and Exact Quantum Constraints to promote couplings to background chiral fields before applying holomorphy.

References

Section titled “References”
  • Gaiotto, Davide, Gregory W. Moore, and Andrew Neitzke. “Framed BPS States.” Advances in Theoretical and Mathematical Physics 17, no. 2 (2013): 241–397. DOI. Open PDF, arXiv v2.