Supersymmetric Vacua, Moduli Geometry, and BPS Sectors

Supersymmetric vacuum problems split into two linked spines. The moduli spine starts from $F=0$ and $D=0$, divides by genuine gauge redundancy, reexpresses the result in invariant coordinates, and then asks which complex, symplectic, metric, and singularity data survive quantum corrections. The BPS spine starts from a central or tensorial charge, derives a positivity bound and preserved-supercharge projector, and only afterward asks whether a state or soliton exists, is stable, and contributes to a protected index in a specified chamber.

This chapter keeps those layers separate. A singular quotient does not by itself prove that a new particle is massless, and saturation of a formal BPS bound does not prove that the corresponding state exists.

Helpful background. Hamiltonian Group Actions and Moment Maps supplies reduction, Gauge Orbits, Gauss Constraints, and Stabilizers supplies the quotient cautions, and Bogomolny Bounds and First-Order Equations supplies the general energy-completion method.

Parent volume. Supersymmetry and Duality

Jump to: choose a route · compare the two spines · use the chapter guide · review the chapter

Diagnose your preparation

Auxiliary equations. Ready: you can derive $V=g^{i\bar j}W_i\bar W_{\bar j}+\mathcal P^{\mathsf T}h^{-1}\mathcal P/2$ and state every positivity assumption. Enter F- and D-Flatness. Unsure: repair with Gauge–Matter Systems, F- and D-Term Potentials, and FI Data.

Quotients and stabilizers. Ready: you distinguish an orbit from an orbit closure, know why a nonfree action produces strata, and can count a generic quotient dimension. Enter Gauge-Invariant Coordinates. Unsure: use the gauge orbit background page above and Constraints, Reduction, and Dirac Brackets.

Kähler geometry. Ready: you can distinguish a complex variety from a metric and can state a moment-map level. Enter Kähler and Hyperkähler Quotients. Unsure: use Kähler Sigma Models.

Effective theories. Ready: you can identify a mass matrix, a Wilsonian scale, and the point where an omitted field becomes light. Enter Singular Loci and Emergent Degrees. Unsure: first read Moduli-Space Metrics and Quantum Corrections.

Short multiplets. Ready: you can derive a positive norm from the supersymmetry algebra and distinguish a shortening condition from state existence. Enter BPS Particles and Central Charges. Unsure: use BPS Bounds, Shortening, and Multiplet Recombination.

Topological sectors. Ready: you can complete an energy functional into squares plus a boundary charge and state the boundary conditions. Enter BPS Solitons. Unsure: use the Bogomolny background page above.

Charge lattices and defects. Ready: you can distinguish an ordinary bulk Hilbert space from a Hilbert space defined with a line defect and can evaluate a symplectic charge pairing. Enter Framed BPS States or Wall Crossing. Unsure: use Disorder Operators and Singular Boundary Conditions.

Choose a route

• Construct a classical moduli space. Read flatness and gauge quotients → invariant coordinates and relations → Kähler or hyperkähler reduction. This route ends with a quotient whose stabilizers, patches, and metric status are explicit.

• Decide what a singularity means. Read metrics and quantum corrections before singular-locus diagnostics. The result distinguishes a bad coordinate, a nonfree quotient, a branch intersection, and a genuine massless threshold.

• Derive a BPS particle bound. Start with central charges and particle shortening. Continue to marginal stability only after the charge lattice and protected index have been named.

• Construct an extended BPS object. Read BPS solitons, walls, strings, vortices, and junctions. This route derives the first-order equations and projector but keeps existence, normalizable zero modes, and quantum stability as separate checks; the classical construction is treated systematically in Weinberg 2012, ch. 8.

• Work in the presence of a line defect. Read framed BPS states and defect moduli before framed wall crossing. The defect’s singular boundary condition and core charge are part of the Hilbert-space definition Gaiotto, Moore, and Neitzke 2013, pp. 241–397, arXiv:1006.0146.

Two spines with one discipline

The moduli spine is

$$(F,D)=0 \longrightarrow \{F=D=0\}/G \longleftrightarrow \mathcal Z/\!/G_{\mathbb C} \longrightarrow \text{invariants and strata} \longrightarrow \text{low-energy geometry}.$$

The middle equivalence needs a reductive complexified group and the correct stability condition; unstable orbits cannot simply be divided out. The final arrow needs a scale separation: integrating out a field whose mass vanishes at the locus under study produces a singular and incomplete effective description.

The BPS spine is

$$\text{charge and algebra} \longrightarrow M\geq|Z| \longrightarrow \text{projector and shortening} \longrightarrow \text{existence and stability} \longrightarrow \Omega(\gamma;u,\mathfrak c).$$

$\Omega$ denotes a specified protected index for charge $\gamma$, moduli $u$, and chamber $\mathfrak c$. The algebra fixes the inequality; dynamics decides whether a saturating object exists; a protected index retains only signed or graded information; and wall crossing controls its chamber dependence, not the complete unprotected spectrum.

Both spines use the same discipline: state the space, group, charges, boundary conditions, stability notion, effective scale, and protected datum before applying a powerful formula.

Exact chapter guide

1. F- and D-Flatness and Gauge Quotients constructs the vacuum set and compares compact and complexified quotient descriptions.
2. Gauge-Invariant Coordinates and Classical Moduli Varieties finds invariant generators, relations, dimensions, branches, tangent spaces, and singular strata.
3. Kähler and Hyperkähler Quotients in Supersymmetric QFT distinguishes one moment map from a triplet and separates complex/algebraic data from the quotient metric Hitchin et al. 1987, pp. 535–589.
4. Moduli-Space Metrics and Quantum Corrections identifies which structures are generally corrected and which extended-supersymmetry statements are genuinely protected.
5. Singular Loci and Emergent Low-Energy Degrees of Freedom combines Jacobian, stabilizer, mass-matrix, and EFT diagnostics.
6. BPS Particles and Central Charges derives the mass inequality and saturation projector while separating existence and stability.
7. BPS Solitons, Walls, Strings, Vortices, and Junctions derives representative wall and vortex equations from energy and supersymmetry.
8. Framed BPS States, Line Defects, and Defect Moduli defines the defect Hilbert space, core/halo split, framed charge sectors, and framed index.
9. Marginal Stability, Chambers, and Wall Crossing derives the primitive jump and states exactly which protected quantity it constrains.

Recurrent convention checks

For quotients, the moment-map sign and FI level are local choices; check them by differentiating the final potential. For BPS particles, the phase of $Z$ and the antisymmetric pairing $\langle\gamma_1,\gamma_2\rangle$ fix the projector and wall-crossing sign. For solitons, orientation fixes the sign of the topological term. Reversing a convention must reverse all three linked objects, while invariant masses, tensions, quotient dimensions, and index jumps remain unchanged.

Review the chapter

Quotient reconstruction. For $n$ charge-one fields with positive FI level, derive $\sum_i|z_i|^2=r$ modulo $U(1)$ and its complexified description. A complete answer identifies $\mathbb{CP}^{n-1}$, the excluded origin, the residual stabilizer, and complex dimension $n-1$.

Singularity diagnosis. For a rank-one determinantal variety, compare the Jacobian rank at a generic point and at the origin. Then identify which gauge bosons become massless at the corresponding field configurations. Algebra and the mass matrix must tell the same story before “emergent field” is claimed.

BPS-bound derivation. Diagonalize the rest-frame $\mathcal N=2$ algebra after writing $Z=|Z|e^{i\alpha}$. A successful derivation obtains $M\geq|Z|$ from two positive norms and identifies the supercharges that annihilate a saturating state.

Wall-crossing diagnosis. For $\gamma=\gamma_1+\gamma_2$, locate the wall $\arg Z_{\gamma_1}=\arg Z_{\gamma_2}$ and explain why the triangle inequality is saturated there. State the protected index and the pairing convention before writing a jump formula.

If these tasks can be completed only by suppressing a stabilizer, boundary condition, charge phase, or chamber label, repair that missing record before continuing to Holomorphy, Anomalies, and Exact Quantum Constraints or the dimension-specific strong-dynamics chapters.

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.
• Hitchin, Nigel J., Anders Karlhede, Ulf Lindström, and Martin Roček. “Hyper-Kähler Metrics and Supersymmetry.” Communications in Mathematical Physics 108, no. 4 (1987): 535–589. DOI.
• Weinberg, Erick J. Classical Solutions in Quantum Field Theory: Solitons and Instantons in High Energy Physics. Cambridge: Cambridge University Press, 2012, ch. 8. DOI.