Wild Wall Crossing and BPS Giants

TL;DR

The work uncovers wild BPS spectra in pure SU(3) N=2 gauge theory, where BPS degeneracies explode exponentially with charge in certain Coulomb-branch regions. It combines spectral networks and Kontsevich–Soibelman wall-crossing with quiver techniques to show that m-cohorts arise from walls with intersection ⟨γ, γ'⟩=m, and that for m≥3 these cohorts yield wild degeneracies tied to Kronecker m-quivers. The authors derive an explicit algebraic generating function P_m solving P_m = 1 + z P_m^{(m-1)^2}, connect this to KS/Gross–Pandharipande structures, and demonstrate exponential growth with precise asymptotics for Ω(n γ_c). They further explore physical implications via Denef’s multi-centered bound states, resolve a naive entropy paradox with a refined volume bound, and reveal intriguing spectral-moonshine features linking spin content to modular objects. The results suggest wild degeneracies are a general feature of higher-rank class S theories and raise open questions about duality frames, modular structures, and the semiclassical regime of BPS states.

Abstract

We show that the BPS spectrum of pure SU(3) four-dimensional super Yang-Mills with N=2 supersymmetry exhibits a surprising phenomenon: there are regions of the Coulomb branch where the growth of the BPS degeneracies with the charge is exponential. We show this using spectral networks and independently using wall-crossing formulae and quiver methods. The computations using spectral networks provide a very nontrivial example of how these networks determine the four-dimensional BPS spectrum. We comment on some physical implications of the wild spectrum: for example, exponentially many field-theoretic BPS states with large charge are gigantic. Finally, we exhibit some surprising, thus far unexplained, regularities of the BPS spectrum.

Figures (25)

• Figure 1: A hypothetical wall-crossing of two hypermultiplets with charges $\gamma,\, \gamma'$ such that $\langle \gamma, \gamma' \rangle =1$. Streets of type $12$ are shown in red, $23$ in blue, and $13$ in fuchsia; only two-way streets are depicted. Arrows denote street orientations according to the convention described in Section \ref{['sec:degen_net']}. Yellow crosses denote branch points. Arrows denote the direction of solitons of type $12,\,23,$ or $13$ (according to the street). The black dotted lines are identified to form the cylinder. $(A)$: The two hypermultiplet networks at a point $u^{-}$ just "before" the wall of marginal stability. $(B)$: The hypermultiplet networks at a point $u^{\text{wall}}$ on the wall of marginal stability and at phase $\vartheta = \operatorname{arg}\left[Z_{\gamma}(u^{\text{wall}}) \right] = \operatorname{arg}\left[Z_{\gamma'}(u^{\text{wall}}) \right] = \operatorname{arg}\left[Z_{\gamma + \gamma'}(u^{\text{wall}}) \right]$. $(C)$: Slightly "after" the wall at a point $u^{+}$, a BPS bound state of charge $\gamma + \gamma'$ is born and a two-way street of type $13$ "grows" as one proceeds away from the wall.
• Figure 2: A hypothetical wall-crossing of two hypermultiplets with charges $\gamma,\, \gamma'$ such that $\langle \gamma, \gamma' \rangle =3$. The story is similar to that described in the caption of Fig. \ref{['fig:1-herd_motivation']}.
• Figure 3: Left Frame: Two-way streets of a horse on some open disk $U$; the solid streets depicted are capable of being two-way; one-way streets are not shown. The sheets of the cover $\Sigma \rightarrow C$ are (locally) labeled from $1$ to $K \geq 3$. Red streets are of type $12$, blue streets are of type $23$, and fuchsia streets are of type $13$. We choose an orientation for this diagram such that all streets "flow up." Right Frame: A relatively simple example of a horse with one-way streets shown as partially transparent and two-way streets resolved (using the "British resolution", cf. Appendix \ref{['app:six-way']} or GMN5). One can imagine horses with increasingly intricate "backgrounds" of one-way streets.
• Figure 4: The first four herds on the cylinder. Solid streets are two-way; dotted, transparent streets are streets of Fig. \ref{['fig:horse']} that happen to be only one-way as indicated by Prop. \ref{['prop_Q']}. The black dotted lines are identified to form the cylinder and capital Latin letters are placed on either side to aid in the identification of streets. Top row (from left to right): The 1-herd (saddle) and 2-herd. The middle row shows a 3-herd and the bottom row shows a 4-herd.
• Figure 5: The spectral network ${\mathcal{W}}_\vartheta$ which occurs in the pure $SU(3)$ theory at the point \ref{['eq:MNpoint']} of the Coulomb branch. The phase $\vartheta$ has been chosen very close to the critical phase $\vartheta = \operatorname{arg} Z_{\gamma + \gamma'}$. Here we represent the cylinder $C$ as the periodically identified plane, i.e., the left and right sides of the figure should be identified. Streets which become two-way at $\vartheta = \operatorname{arg} Z_{\gamma + \gamma'}$ are shown in thick red, blue and fuchsia. We do not show the whole network but only a cutoff version of it, as described in GMN5.

Theorems & Definitions (19)

• Proposition 3.1
• proof
• Proposition 3.2
• proof : Proof (sketch)
• Corollary 3.3
• Proposition 3.4
• proof
• proof
• proof
• Corollary C.1