Geometric engineering of (framed) BPS states

TL;DR

This work develops a comprehensive geometric-engineering framework for N=2 $\mathrm{SU}(N)$ gauge theories, building BPS spectra from toric CY3folds $X_N$ via exceptional collections and quivers with potential $(Q,W)$. It introduces an intrinsic stability structure on a gauge-theory subcategory $\mathcal{G}$, relates BPS degeneracies to motivic DT invariants, and proves an absence-of-walls conjecture connecting large-radius BPS states to the limit weak-coupling spectrum, supported by explicit SU(2) and SU(3) analyses. The framework extends to line defects through framed quivers, providing a recursive DT-based method to compute framed and unframed BPS spectra and establishing no-exotics results in several cases. The results hinge on rich interactions among geometric engineering, Bridgeland stability, mirror symmetry, and wall-crossing, with strong evidence across multiple chambers and a detailed mapping between large-radius and field-theory limits, including strong-coupling mutations and deceptive chambers. The methodology yields a versatile toolkit for exploring BPS spectra in toric CY geometries and offers precise conjectures and algorithms that can be applied to a broad class of local Calabi–Yau threefolds.

Abstract

BPS quivers for N=2 SU(N) gauge theories are derived via geometric engineering from derived categories of toric Calabi-Yau threefolds. While the outcome is in agreement of previous low energy constructions, the geometric approach leads to several new results. An absence of walls conjecture is formulated for all values of N, relating the field theory BPS spectrum to large radius D-brane bound states. Supporting evidence is presented as explicit computations of BPS degeneracies in some examples. These computations also prove the existence of BPS states of arbitrarily high spin and infinitely many marginal stability walls at weak coupling. Moreover, framed quiver models for framed BPS states are naturally derived from this formalism, as well as a mathematical formulation of framed and unframed BPS degeneracies in terms of motivic and cohomological Donaldson-Thomas invariants. We verify the conjectured absence of BPS states with "exotic" SU(2)_R quantum numbers using motivic DT invariants. This application is based in particular on a complete recursive algorithm which determine the unframed BPS spectrum at any point on the Coulomb branch in terms of noncommutative Donaldson-Thomas invariants for framed quiver representations.

Figures (6)

• Figure 1: Schematic representation of the complex structure moduli space for general $N\geq 2$.
• Figure 2: The toric polytope for $X_3$ and the singular threefold $Y_0/\mathbb {Z}_3$. The polytope (2.a) for $X_N$ is similar, except it will contain $N-1$ inner points on the vertical axis.
• Figure 3: The magnitudes of the periods $\Pi_c$, $\Pi_i$ and $\Pi_D$ along the trajectory from the large volume limit to the field theory limit, for $2\pi^2\, u/M_0^2=-10^4$, $c_0=1$ and $\xi=10^{-4}$.
• Figure 4: The arguments of the periods $\Pi_c$, $\Pi_i$ and $\Pi_D$ along the trajectory from the large volume limit to the field theory limit, for $2\pi^2\, u/M_0^2=-10^4$, $c_0=1$ and $\xi=10^{-4}$.
• Figure 5: Schematic representation of the BPS charge vectors in equation \ref{['eq:pmcentralcharges']} for $SU(3)$ with positive roots $\alpha_1,\alpha_2, \alpha_1+\alpha_2$.

Theorems & Definitions (8)

• Definition 8.1
• Definition 8.2
• Definition 8.3
• Proposition 8.4
• Conjecture 8.5
• Corollary 8.6
• Definition 8.7
• Definition 8.8