BPS Spectra, Chambers, and Wall Crossing

Seiberg–Witten periods determine the central charge of any allowed electromagnetic vector, but a BPS spectrum is additional chamber-dependent information. For pure $SU(2)$, a strong-coupling chamber contains two hypermultiplets, while a weak-coupling chamber contains a vector multiplet and two infinite dyon towers. A phase-ordered wall-crossing identity relates their protected indices.

Required background. Curves and periods supplies $Z_\gamma(u)$, while marginal stability and wall crossing fixes the index and ordering conventions. Helpful background. Framed BPS states gives the halo interpretation.

Spectrum data at a point

To state a BPS spectrum, specify

$$(u,\Gamma,\langle\ ,\ \rangle,Z,mathcal H,Omega),$$

where $u$ is the vacuum, $\Gamma$ the transported charge lattice, $\mathcal H$ a chosen half-plane of central-charge phases distinguishing particles from antiparticles, and $\Omega(\gamma;u)$ the protected second helicity supertrace.

The mass formula

$$M_\gamma=|Z_\gamma|$$

applies if a BPS state exists. It does not decide $\Omega(\gamma;u)$.

A chamber is a connected region in which no relevant central-charge rays align and no BPS state crosses the boundary of $\mathcal H$. Within it, protected indices are locally constant after charges are parallel transported.

Walls of marginal stability

For a possible decay $\gamma=\gamma_1+\gamma_2$, the wall is

$$\operatorname{Im}\!\left( Z_{\gamma_1}\overline{Z_{\gamma_2}} \right)=0, \qquad \operatorname{Re}\!\left( Z_{\gamma_1}\overline{Z_{\gamma_2}} \right)>0.$$

At the wall,

$$|Z_\gamma|=|Z_{\gamma_1}|+|Z_{\gamma_2}|,$$

so a bound state can reach infinite separation. Which side is stable depends on the Dirac pairing and the oriented phase order.

For pure $SU(2)$ with constituent charges associated with the two finite singularities, the curve of marginal stability can be written as the locus where the relevant period ratio is real, for example

$$\operatorname{Im}\frac{a_D}{a}=0$$

away from zeros of $a$, with a further sign condition selecting alignment. Its precise drawing depends on branch cuts; it passes through the singular points and separates strong- and weak-coupling chambers.

A pure-SU(2) charge convention

Use period charges $\gamma=(p,q)$ with

$$Z_\gamma=pa_D+qa.$$

For the particle lattice convention used in the monodromy formulas, normalize the physical pairing as

$$\langle(p,q),(p',q')\rangle =2(pq'-qp').$$

The factor two reflects the $SU(2)$ root/weight normalization and is the same factor appearing in the local monodromies. Other conventions absorb it into electric charges.

Choose the two strong-chamber particle charges

$$\gamma_1=(1,0), \qquad \gamma_2=(-1,1).$$

$\gamma_1$ is the monopole. $\gamma_2$ is the antiparticle orientation of the dyon charge $(1,-1)$ used to label the second singularity. They satisfy

$$\langle\gamma_1,\gamma_2\rangle=2.$$

Choose the phase half-plane so both are particles. In the strong chamber,

$$\Omega(\gamma_1)=\Omega(\gamma_2)=+1,$$

and, for the standard pure theory, these are the only stable particle hypermultiplets in that half-plane.

The weak-coupling spectrum

Crossing to weak coupling produces:

• hypermultiplets of charges

  $$\gamma_n^+=(n+1)\gamma_1+n\gamma_2=(1,n), \qquad n\ge0,$$

• hypermultiplets of charges

  $$\gamma_n^-=n\gamma_1+(n+1)\gamma_2=(-1,n+1), \qquad n\ge0,$$

• a vector multiplet of charge

  $$\gamma_W=\gamma_1+\gamma_2=(0,1).$$

In the second-helicity convention,

$$\Omega(\gamma_n^\pm)=+1, \qquad \Omega(\gamma_W)=-2.$$

Antiparticles carry the opposite charges and lie outside the chosen half-plane. Rescaling the electric unit can make the $W$ charge appear as two and all dyon labels change accordingly; the pairing and central charges must be rescaled with it.

The tower is a statement about the protected one-particle spectrum in the weak chamber. Multi-particle states are not additional primitive $Omega$ entries. The strong- and weak-coupling spectra and their change across the marginal-stability curve are derived in Bilal and Ferrari 1996, §§3–6.

The ordered wall-crossing factorization

Let $\mathcal K_\gamma$ be the classical Kontsevich–Soibelman transformation with exponent $\Omega(\gamma)$. Order products by increasing central-charge phase in the chosen half-plane. For the 2-Kronecker pairing above, the strong-chamber product

$$\mathcal K_{\gamma_1}\mathcal K_{\gamma_2}$$

is equal to the weak-chamber phase-ordered product containing

$$\left( \prod_{n=0}^{\infty} \mathcal K_{(n+1)\gamma_1+n\gamma_2} \right) \mathcal K_{\gamma_1+\gamma_2}^{-2} \left( \prod_{n=\infty}^{0} \mathcal K_{n\gamma_1+(n+1)\gamma_2} \right).$$

The arrows on the product limits indicate opposite phase order on the two towers. If a convention orders rays clockwise rather than counterclockwise, both sides reverse. The vector exponent $-2$ is essential: omitting it spoils the symplectomorphism identity. The underlying stability-data wall-crossing transformation is formulated in Kontsevich and Soibelman 2008, §2.3; its four-dimensional BPS application and the treatment of mutually nonlocal corrections are developed in Gaiotto, Moore, and Neitzke 2010, §2.2 and §5.

The identity can be checked on formal variables to any bounded charge order. Such a truncation verifies the declared finite sector, not the entire infinite product unless a formal convergence or algebraic argument is supplied.

Determining a chamber numerically

Given numerical periods at $u$:

1. transport the charge basis from a fixed base point;

2. compute $Z_{\gamma_1}$ and $Z_{\gamma_2}$ with error bounds;

3. evaluate

   $$\operatorname{Im}(Z_{\gamma_1}\overline{Z_{\gamma_2}});$$

4. verify that its magnitude exceeds numerical uncertainty away from a wall;

5. order all included rays by a continuously unwrapped phase;

6. apply the appropriate chamber spectrum and test the ordered product.

Using a principal-value phase independently at each point can create artificial order jumps at $\arg Z=\pi$.

Higher-rank methods

In general $\mathcal N=2$ theories, BPS quivers, spectral networks, semiclassical quantization, string webs, and protected indices can determine spectra in selected chambers. Each method has a domain:

• a BPS quiver requires a finite positive basis in a chosen half-plane;
• a spectral network requires a suitable curve/differential presentation;
• semiclassics applies where solitons are large and weakly coupled;
• wall crossing propagates seed data but does not create those seeds.

Completeness must be argued separately. A finite chamber does not imply every chamber is finite.

Protected versus unprotected information

$\Omega$ is stable under deformations away from walls because long multiplets cancel. It can jump when a bound state meets the continuum. It does not determine:

• the full spin-refined spectrum unless a refined index is used;
• long-multiplet degeneracies;
• BPS wavefunctions or sizes;
• scattering amplitudes;
• whether an allowed charge has states with net zero index.

State these omissions whenever exporting a spectrum.

Common pitfalls

Mixing charge normalizations. The factor two can appear in the pairing, in electric charge labels, or in the central-charge convention; it cannot be changed in only one place.

Calling the weak tower chamber-independent. Most of its states decay on the marginal-stability curve.

Using wall crossing without seed states. The identity constrains changes given a spectrum in one chamber; it does not determine that input from periods alone.

Exercises

Using $\gamma_1=(1,0)$ and $\gamma_2=(-1,1)$:

1. verify their pairing is two;
2. compute $\gamma_0^\pm$, $\gamma_1^\pm$, and $\gamma_W$;
3. identify which strong-chamber singular charge is represented by $-\gamma_2$.

Solution

$$\langle\gamma_1,\gamma_2\rangle =2(1\cdot1-0\cdot(-1))=2.$$

The first tower entries are

$$\gamma_0^+=(1,0), \quad \gamma_1^+=(1,1), \quad \gamma_0^-=(-1,1), \quad \gamma_1^-=(-1,2),$$

and $\gamma_W=(0,1)$. Finally, $-\gamma_2=(1,-1)$ is the dyon charge whose central charge vanishes at the second finite singularity in the pure-solution convention.

References

• Bilal, Adel, and Frank Ferrari. “The Strong-Coupling Spectrum of the Seiberg–Witten Theory.” Nuclear Physics B 469 (1996): 387–402. arXiv:hep-th/9602082.
• Gaiotto, Davide, Gregory W. Moore, and Andrew Neitzke. “Four-Dimensional Wall-Crossing via Three-Dimensional Field Theory.” Communications in Mathematical Physics 299 (2010): 163–224. arXiv:0807.4723.
• Kontsevich, Maxim, and Yan Soibelman. “Stability Structures, Motivic Donaldson–Thomas Invariants and Cluster Transformations.” 2008. arXiv:0811.2435.