BPS spectra and wall crossing

When a particle reaches a possible decay threshold, what tells us whether it actually disappears? The period calculation locates the threshold. A protected spectral index supplies different information: how the stable one-particle content changes. Here we reproduce a small but substantive part of the pure $SU(2)$ answer, including the negative vector-multiplet index and the first new dyon charges.

Begin with BPS masses and stability and turning a reproduction into a research question. The new tool is a product of explicit rational maps. No knowledge of spectral networks or quiver representations is required.

We study pure four-dimensional N=2 $SU(2)$ Yang–Mills in a generic flat-space Coulomb vacuum. The theory has eight real supercharges, four manifest in N=1 notation, and gauge rank one. The vacuum preserves all eight; a massive BPS particle preserves four combinations. We keep the global group and particle charge unit fixed throughout.

Fix the physical input before the algebra

Use the period column $(a_D,a)^T$ and charge row $\gamma=(n_m,n_e)$:

$$Z_\gamma=n_ma_D+n_ea,\qquad M_{\rm BPS}=\sqrt2\,|Z_\gamma|. \tag{D15.1}$$

The periods have mass dimension one; the Coulomb coordinate $u$ has dimension two. A mass formula applies to an existing BPS state. It does not manufacture a state for every row.

Choose the same lower-half-plane continuation as the prerequisite. Define

$$\alpha=(-1,1),\qquad \beta=(1,0),\qquad \delta=\alpha+\beta=(0,1). \tag{D15.2}$$

These are the anti-dyon, monopole, and candidate W charge. The relevant electromagnetic pairing is $\langle\gamma,\gamma'\rangle=2(n_en_m'-n_mn_e')$, so $\langle\alpha,\beta\rangle=2$. It is the physical particle pairing, not the unit geometric intersection of the normalized curve cycles.

The input spectrum in the strong chamber contains one hypermultiplet in each of the $\alpha,\beta$ sectors. Their helicity indices are $\Omega_\alpha=\Omega_\beta=1$. We choose an angular sector of width less than $\pi$ containing these central charges and the positive cone between them; the opposite-charge CPT partners lie outside this sector. The protected index of a BPS vector multiplet is $-2$, rather than a negative count of states.

On the lower imaginary ray, the previously calculated wall is at $u/\Lambda^2=-0.86043668612567827i$. There $a_D/a=1/2$. The sample $u=-0.5i\Lambda^2$ lies on the strong side and $u=-i\Lambda^2$ on the weak side. With $w=a_D/a$, $Z_\beta=wa$ and $Z_\alpha=(1-w)a$; the same-phase condition is $w\in(0,1)$. The order of their phases reverses across the wall. Both constituents remain massive at the crossing.

The chamber input and wall-crossing identity used below are stated in GMN (2.26). Seiberg–Witten’s spectrum discussion explains why period monodromy alone cannot preserve a fixed list of particles. We use the later chamber formula, not the original paper’s schematic picture as an exact equation for the wall.

Encode an index by a coordinate transformation

Introduce two dimensionless formal variables $x=X_\alpha$ and $y=X_\beta$. They are bookkeeping coordinates, not the coordinates of the Seiberg–Witten curve and not the period values. A charge $\gamma=r\alpha+s\beta$ has monomial $X_\gamma=x^ry^s$.

In this particle sublattice the pairing is even. The source’s sign choice $\sigma(p,q)=(-1)^{pq}$ equals one because its electric-first charge is $(p,q)=(2n_e,n_m)$. Thus ordinary multiplication $X_\gamma X_{\gamma'}=X_{\gamma+\gamma'}$ suffices here. This simplification does not discard the sign prescription for arbitrary probe charges or change the global group.

Write $\mathcal K_{r,s}$ for the transformation associated with the charge $r\alpha+s\beta$. An index $\Omega$ gives

$$\mathcal K_{r,s}^{\Omega}: (x,y)\longmapsto \left( x(1-x^ry^s)^{2s\Omega}, y(1-x^ry^s)^{-2r\Omega} \right). \tag{D15.3}$$

The exponents follow directly from $\langle\alpha,r\alpha+s\beta\rangle=2s$ and $\langle\beta,r\alpha+s\beta\rangle=-2r$ in GMN (2.16). The monomial $x^ry^s$ is unchanged by its own transformation, so raising the map to an integer power really multiplies the exponents by that integer.

We compose these as maps on points: the rightmost factor acts first. For example, the strong-chamber product is

$$\mathcal S=\mathcal K_{1,0}\mathcal K_{0,1}: \quad x_{\mathcal S}=x(1-y)^2,\qquad y_{\mathcal S}=\frac{y}{[1-x(1-y)^2]^2}. \tag{D15.4}$$

If instead one implements their action on functions by substitution, the composition order reverses. Checking (D15.4) is a useful safeguard before a larger computation.

The wall-crossing statement is that the product ordered by the BPS phases remains the same as those phases exchange order, provided no ray crosses the chosen sector boundaries. Its factor exponents can change. We use this physical wall-crossing result as input; the following algebra reproduces its consequences rather than deriving it from the mass bound.

Recover the vector contribution

Start on the weak side with the reversed endpoint order and allow an unknown index $c$ at their sum:

$$\mathcal T_c=\mathcal K_{0,1}\mathcal K_{1,1}^{c} \mathcal K_{1,0}. \tag{D15.5}$$

Apply $\mathcal K_{1,0}$ first, and set $Y=y/(1-x)^2$ and $h=1-xY$. The other two steps give

$$x_{\mathcal T_c}=x h^{2c}(1-Yh^{-2c})^2,\qquad y_{\mathcal T_c}=Yh^{-2c}.$$

Count each $x$ and $y$ as degree one. Expansion through total degree three gives

$$\begin{aligned} x_{\mathcal T_c} &=x-2xy+xy^2-(4+2c)x^2y+O_{\geq4},\\ y_{\mathcal T_c} &=y+2xy+3x^2y+2cxy^2+O_{\geq4}. \end{aligned} \tag{D15.6}$$

Here $O_{\geq d}$ means only monomials of total degree at least $d$; it is a formal statement, not a numerical error estimate. The strong map has no $x^2y$ term, and its $xy^2$ coefficient in the second coordinate is $-4$. Both comparisons require

$$c=-2. \tag{D15.7}$$

This is the W-sector index on the weak side, with zero such index in the strong input. It has been recovered from the ordered transformation law and the two seed hypermultiplets. Merely imposing $Z_\alpha/Z_\beta>0$ would not yield this coefficient.

The interpretation still uses physics: a nonzero protected index implies a nonzero net BPS contribution, while identifying this charge sector as the W vector uses the known pure-theory multiplet content. An index by itself is not a complete list of spins or degeneracies.

Recover the next two dyons

The product with $c=-2$ is not yet equal to $\mathcal S$. At degree four its first-coordinate discrepancy is

$$x_{\mathcal T_{-2}}-x_{\mathcal S} =2x^3y+4x^2y^2+O_{\geq5}. \tag{D15.8}$$

Allow factors $\mathcal K_{2,1}^{d}$ and $\mathcal K_{1,2}^{e}$ in the phase order $\mathcal K_{0,1}\mathcal K_{1,2}^{e}\mathcal K_{1,1}^{-2} \mathcal K_{2,1}^{d}\mathcal K_{1,0}$. Their leading changes in the first coordinate are respectively $-2d\,x^3y$ and $-4e\,x^2y^2$. Other factors affect these new terms only at higher degree. Equation (D15.8) therefore requires $d=e=1$. The second coordinate gives the same result.

The recovered charges are $2\alpha+\beta=(-1,2)$ and $\alpha+2\beta=(1,1)$. They carry hypermultiplet index $+1$. Continuing the same process yields the sourced factorization

$$\mathcal K_{1,0}\mathcal K_{0,1} = \bigl(\mathcal K_{0,1}\mathcal K_{1,2}\mathcal K_{2,3}\cdots\bigr) \mathcal K_{1,1}^{-2} \bigl(\cdots\mathcal K_{3,2}\mathcal K_{2,1}\mathcal K_{1,0}\bigr). \tag{D15.9}$$

This is GMN (2.26) in the $\alpha,\beta$ coordinates. The two families have physical rows $(1,n)$ and $(-1,n+1)$ for $n\geq0$. Together with their CPT partners they produce the dyon tower of the weak chamber. The central factor is the W.

The full infinite identity is the source result. Our explicit coefficient calculation has so far established its first two nontrivial charge heights. We can extend that check systematically without pretending that a finite list proves an infinite spectrum.

A finite reproduction with an exact boundary

Define the height of $r\alpha+s\beta$ to be $r+s$. A factor of height $h$ first changes a coordinate at total degree $h+1$. Because every factor adds nonnegative powers, omitted factors of greater height cannot feed back into a lower degree.

Consequently, to test all indices of height at most $H$, retain coordinate polynomials through degree $H+1$. Insert all candidate charges in decreasing $s/r$ order, with the vertical endpoint first. Equal-slope factors commute. Fix the two endpoint factors from the strong input, then work through increasing heights.

Suppose lower heights have already been matched. Let $R_x$ and $R_y$ be the strong map minus the current weak map. For an interior charge $(r,s)$, its next index must obey

$$\Omega_{r,s} =-\frac{[x^{r+1}y^s]R_x}{2s} =\frac{[x^ry^{s+1}]R_y}{2r}. \tag{D15.10}$$

Square brackets extract the indicated coefficient. Equality of the two answers, integrality, and agreement after insertion are separate checks. This procedure also tests candidate charges whose answer is zero; it does not enter the dyon list as an assumption.

An exact rational-arithmetic reproduction through $H=10$ gives:

Charge coordinates	Physical row	Recovered index
$(1,1)$	$(0,1)$	$-2$
$(n,n+1)$, $n=1,2,3,4$	$(1,n)$	$+1$
$(n+1,n)$, $n=1,2,3,4$	$(-1,n+1)$	$+1$
All other interior pairs with $r+s\leq10$	$(s-r,r)$	$0$

Zero indices here are consistent with the known chamber spectrum. In an arbitrary theory, a zero index alone would not rule out cancelling BPS contributions.

There is a useful independent implementation. Put $q=\log x$, $p=\log y$ and give them Poisson bracket $\{q,p\}=2$. For $z=e^{rq+sp}$, the Hamiltonian

$$H_{r,s}=-\operatorname{Li}_2(z),\qquad \operatorname{Li}_2(z)=\sum_{n\geq1}\frac{z^n}{n^2} \tag{D15.11}$$

generates $\dot q=2s\log(1-z)$ and $\dot p=-2r\log(1-z)$. Since $\dot z=0$, its time-$\Omega$ flow is exactly (D15.3). On a monomial, the corresponding derivative is

$$\mathcal D_{r,s}(x^iy^j) =2(rj-si)\sum_{n\geq1}\frac{x^{i+nr}y^{j+ns}}{n}. \tag{D15.12}$$

At fixed total degree this raises degree and is nilpotent. Its exponential is therefore a finite sum. Exponentiating these derivative matrices on polynomials independently reproduces the rational-map calculation through degree eleven. This also checks the order reversal between point maps and their action on functions.

Change the cutoff, not the physical claim

A bounded extension is to increase the retained tower rather than change the gauge theory. Let $\mathcal T^{(L)}$ retain both factors of height $2n+1$ for $0\leq n\leq L$, together with the W factor. The first omitted dyons have height $2L+3$. Hence the first possible coordinate discrepancy is degree $2L+4$.

For a numerical diagnostic, evaluate the actual finite rational products at the auxiliary point $(x,y)=(0.06,0.09)$. The maximum absolute difference of the two coordinates from (D15.4) is

$L$	Coordinate discrepancy
$0$	$2.68243\times10^{-4}$
$1$	$2.84175\times10^{-6}$
$2$	$2.47058\times10^{-8}$
$4$	$1.47211\times10^{-12}$
$8$	$3.59297\times10^{-21}$

These are finite-product diagnostics at a nonsingular small point, not a uniform bound on the infinite product or a scan of Coulomb vacua. At $x^ry^s=1$ a rational factor can be singular; formal power-series validity does not authorize evaluation there.

Scale instead $(x,y)=t(0.3,0.4)$. The first omitted homogeneous term predicts that halving small $t$ reduces the leading discrepancy by $2^{2L+4}$. Separate calculations for $L=0,1,2,3$ reproduce the expected orders $4,6,8,10$. A wrong vector exponent fails already at degree three and cannot be repaired by adding high-charge dyons.

The deliverable is a table of recovered indices with an exact height boundary, a second implementation, and a negative control. Increasing arithmetic precision cannot repair an incorrect charge pairing or multiplication order. Increasing the height tests more of the known identity; neither operation creates a new BPS-spectrum result.

Exercises

1. A frame change that moves both seed rows

Replace $(a_D,a)$ by $(a_D+a,a)$. Find the new rows of $\alpha,\beta,\delta$ and verify their central charges and pairing. Must the indices or the formal calculation change? Explain why keeping the old rows would give a different problem.

Hint 1

The period column is multiplied by $C=\begin{pmatrix}1&1\\0&1\end{pmatrix}$. A charge row is multiplied by $C^{-1}$.

Hint 2

The formal variables are attached to physical seed charges, not to whichever rows happen to be coordinate basis vectors.

Solution

The rows become $\alpha'=(-1,2)$, $\beta'=(1,-1)$ and $\delta'=(0,1)$. Their central charges are $-(a_D+a)+2a=a-a_D$, $(a_D+a)-a=a_D$, and $a$. The pairing is $2[2\cdot1-(-1)(-1)]=2$.

Thus $x=X_\alpha$, $y=X_\beta$ still label the same physical charges, and every factor and index is unchanged. In general $(r\alpha+s\beta)'=(s-r,2r-s)$, with the same height in this seed cone. The period ratio changes, but neither phase alignment of the actual constituents nor their mass changes.

Keeping $\alpha=(-1,1)$ after changing the periods would instead assign central charge $-a_D$ to that row. It would no longer describe the original anti-dyon. This is a coordinate error, not wall crossing.

2. A multiple-cover term is not another W boson

The source writes $\mathcal K_\gamma=\exp(\sum_{n\geq1}e_{n\gamma}/n^2)$. Find the coefficient of $e_{2\delta}$ in $\log\mathcal K_\delta^{-2}$. Does it imply a particle index $\Omega(2\delta)=-1/2$? Suppose an extra factor $\mathcal K_{2,2}^{c}$ were inserted. At which degree and with what leading first-coordinate term would it become visible?

Hint 1

An exponent of a factor and a coefficient inside its logarithm are different objects.

Hint 2

Use (D15.3) with $(r,s)=(2,2)$ before composing with any other factors. Lower factors change its leading term only at higher degree.

Solution

The logarithm contains $-2(e_\delta+e_{2\delta}/4+e_{3\delta}/9+\cdots)$. The requested coefficient is $-1/2$. It is already generated by the single vector factor with index $-2$; it is not a fractional single-particle index.

An independent factor of charge $2\delta$ would multiply $x$ by $(1-x^2y^2)^{4c}$ and $y$ by its inverse. Its leading changes are $-4c\,x^3y^2$ and $+4c\,x^2y^3$, at degree five. Therefore a test through degree four cannot decide this index. The height-four extraction from (D15.10) gives $c=0$, once the lower factors and their multiple-cover contributions are included.

The full known chamber spectrum contains no separate $2\delta$ particle. The algebraic check establishes its zero protected index at the tested height; the distinction between index zero and an absence theorem remains necessary.

3. Diagnose a plausible but wrong spectrum

A calculation uses the correct two seed states but assigns the W exponent $+2$. It then tries to fix the disagreement by adding arbitrarily many dyons of height at least three. Can that work? After correcting the W exponent, determine how many dyons on each tower are needed to verify coordinate terms through degree eight. Interpret what such a successful test establishes.

Hint 1

Read the $x^2y$ coefficient in (D15.6). A factor of height $h$ first acts at degree $h+1$.

Hint 2

The dyons have heights $2n+1$, including the endpoint at $n=0$.

Solution

With $c=+2$, the unwanted coefficient is $-(4+4)=-8$. No factor of height at least three affects degree three, so no such tower can fix the error. The exponent must first be corrected to $-2$.

For degree eight, include all factors of height at most seven. On each tower this means $n=0,1,2,3$: the seed endpoint and three additional dyons. The next omitted pair has height nine and first acts at degree ten. Direct expansion agrees through degree nine as well; there is no height-eight factor in the known identity.

The result checks the ordered factorization to a stated finite order, under the supplied seed spectrum and wall-crossing law. It neither proves that the power series converges at every numerical point nor reconstructs all possible stable particles from central charges alone. A putative additional height-eight index would require a degree-nine extraction, rather than being ruled out by the degree-eight test itself.

Literature and a bounded next question

Literature checked on 20 October 2026. Read the physical stability discussion in SW1 first, then GMN section 2.2 for the precise maps and Appendix A for a recurrence-based route to the identities. Our truncation calculation is a reproduction of an established pure-$SU(2)$ spectrum, not a proposal that this spectrum is unknown.

A later direction is Bryan–Sułkowski’s reformulation using symmetric quivers. Its section 5 revisits the known pure theory, while the broader work addresses refined invariants and more difficult wild-spectrum examples. Refined spin data, quantum dilogarithms, and those different theories require additional preparation; their conventions should not be inferred by changing the integer two in (D15.3).

A useful next assignment remains within the present theory: increase the extraction to height twelve, keep every candidate interior charge, and locate the first discrepancy after deliberately omitting one known dyon. Predict its degree before running the calculation. Success means a reproducible factorization with a diagnosed error; failure first calls for checking signs, charge units, and ordering. A search for new spectra begins only after a separate theory, chamber, stability input, and literature question have been specified.