Exploring 5d BPS Spectra with Exponential Networks

TL;DR

The work develops a geometric framework to count 5d BPS states in M-theory on toric Calabi–Yau threefolds via framed 3d-5d wall-crossing, encoding the spectrum in the mirror curve Σ and constructing a nonabelianization map for exponential networks to extract 3d-5d BPS data. It extends spectral-network techniques to 3d-5d systems by lifting to an infinite-sheeted Σ, introduces shift-symmetric soliton data, and derives KS-type jumps (K-walls) that capture purely 5d boundstates. The C^3 example reproduces the MacMahon counting through a KK tower of D0/M-theory states, and matches 3d tt* CFIV indices, providing strong checks on the framework. The results offer a concrete pathway to compute enumerative invariants of toric Calabi–Yau threefolds via Lagrangian defects and exponential networks, with potential extensions to more complex geometries and knot conormals.

Abstract

We develop geometric techniques for counting BPS states in five-dimensional gauge theories engineered by M theory on a toric Calabi-Yau threefold. The problem is approached by studying framed 3d-5d wall-crossing in presence of a single M5 brane wrapping a special Lagrangian submanifold $L$. The spectrum of 3d-5d BPS states is encoded by the geometry of the manifold of vacua of the 3d-5d system, which further coincides with the mirror curve describing moduli of the Lagrangian brane. Information about the BPS spectrum is extracted from the geometry of the mirror curve by construction of a nonabelianization map for exponential networks. For the simplest Calabi-Yau, $\mathbb{C}^3$ we reproduce the count of 5d BPS states encoded by the Mac Mahon function in the context of topological strings, and match predictions of 3d $tt^*$ geometry for the count of 3d-5d BPS states. We comment on applications of our construction to the study of enumerative invariants of toric Calabi-Yau threefolds.

Figures (15)

• Figure 1: An open path on $\Sigma$ connecting two vacua $\sigma_i(t), \sigma_j(t)$. Calibrated paths correspond to BPS states $T[L]$. Exponential networks arise from projecting BPS paths to the $t$-plane.
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• Figure 4: Intersecting $\mathcal{E}$-walls of types $(ij,n)$ and $(ji,m)$ give origin to an infinite family of new walls.
• Figure 5: Behavior of an $ii$-wall for $\vartheta<0$ (left), $\vartheta=0$ (center), and $\vartheta>0$ (right). The red dot indicates the starting point of integration $x_0$, which for simplicity is kept constant in $\vartheta$, the black dot indicates $x=0$. This behavior is universal, it does not depend on the geometry of the mirror curve.