BPS Indices, Absolute Degeneracies, and Wall Crossing

A BPS index is stable under many continuous deformations because long multiplets cancel, whereas an absolute count is positive and can change when bound states appear or disappear. An index can determine the leading black-hole degeneracy only when cancellations, hair multiplets, charge conventions, and the chamber of moduli space are independently controlled.

Required background. The Witten Index, Vacuum Counting, and Its Failure Modes supplies the protected-trace logic; Microscopic Black-Hole Entropy: Claim and Ensemble Contract fixes the comparison.

Helpful background. Marginal Stability, Chambers, and Wall Crossing supplies the decay-wall geometry; Index Inversion, Recombination, and Protected-Spectrum Limits supplies protected-spectrum reconstruction limits.

Index, degeneracy, and helicity trace

In a charge sector $\Gamma$, let $d_j(\Gamma;t)$ count BPS states of spin $j$ at moduli $t$. Then

$$d(\Gamma;t)=\sum_j(2j+1)d_j(\Gamma;t)$$

is an absolute degeneracy, while a helicity supertrace is

$$B_{2k}(\Gamma;t) =\frac{1}{(2k)!} \operatorname{Tr}_{\Gamma,t} \left[(-1)^{2J_3}(2J_3)^{2k}\right].$$

The insertion soaks up fermion zero modes; its order must match the broken supersymmetries. Hair outside the horizon and center-of-mass multiplets contribute known factors that must be removed before comparing with a horizon index.

The elementary bound $\lvert B_{2k}\rvert\le C_k d$ can make a large index a lower bound on exponential growth, but a small index gives no upper bound on $d$: cancellations can be arbitrarily severe.

First application: a chamber with controlled signs

In four-dimensional $\mathcal N=4$ string compactifications, indexed dyon degeneracies are Fourier coefficients of a meromorphic Siegel modular form. A contour in chemical-potential space selects a chamber. After subtracting multicenter poles and using the single-center contour, the indexed coefficient has the sign predicted by the angular-momentum structure of a single-centered supersymmetric black hole in its domain Dabholkar, Murthy, and Zagier 2012.

The reproducible comparison is therefore not “index equals degeneracy” in general. It is:

$$\log\lvert B_6(Q,P;t_{\rm sc})\rvert =S_{\rm BH}(Q,P)+o(S_{\rm BH})$$

for charges with positive discriminant

$$\Delta=Q^2P^2-(Q\!\cdot\!P)^2>0,$$

in a single-center chamber $t_{\rm sc}$, with the prescribed contour and removal of hair factors. Charge-by-charge sign checks then test whether cancellations are mild in that sector.

Wall crossing as an adversarial control

For a primitive two-center decay $\Gamma=\Gamma_1+\Gamma_2$, crossing a marginal-stability wall changes the index by

$$\Delta\Omega(\Gamma) =(-1)^{\langle\Gamma_1,\Gamma_2\rangle-1} \left|\langle\Gamma_1,\Gamma_2\rangle\right| \Omega(\Gamma_1)\Omega(\Gamma_2),$$

up to the chosen index convention. The jump follows from the relative supersymmetric quantum mechanics of the centers Denef 2000. An attractor chamber may exclude these bound states even though another chamber contains them.

This control demonstrates both limitations at once: deformation protection is not chamber independence across a wall, and invariance of a suitably defined total index does not imply invariance of separate single-center and multicenter absolute counts. The evidence ceiling is a protected, chamber-specified statement; typical non-BPS degeneracies require other observables.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Dabholkar, Atish, Sameer Murthy, and Don Zagier. “Quantum Black Holes, Wall Crossing, and Mock Modular Forms.” Communications in Number Theory and Physics 6, 165–196 (2012). DOI. Open PDF.
• Denef, Frederik. “Supergravity Flows and D-Brane Stability.” Journal of High Energy Physics 2000, 8 (2000): 050. DOI. Open PDF.