Abstract
We study BPS spectra of D-branes on local Calabi-Yau threefolds mathcal {O}(-p)oplus mathcal {O}(p-2)rightarrow mathbb {P}^1 with p=0,1, corresponding to mathbb {C}^3/mathbb {Z}_{2} and the resolved conifold. Nonabelianization for exponential networks is applied to compute directly unframed BPS indices counting states with D2 and D0 brane charges. Known results on these BPS spectra are correctly reproduced by computing new types of BPS invariants of 3d-5d BPS states, encoded by nonabelianization, through their wall-crossing. We also develop the notion of exponential BPS graphs for the simplest toric examples, and show that they encode both the quiver and the potential associated to the Calabi-Yau via geometric engineering.
Highlights
In this paper we continue developing our geometric approach for counting BPS states in five dimensional gauge theories with eight supercharges compactified on a circle of finite radius
Continuing from our previous work [18], we focus on BPS spectra of 5d gauge theories engineered from M-theory on a Calabi-Yau threefold X
In our work we focus on a different kind of BPS states of T [L], corresponding to field configurations that reduce to kinks upon shrinking the S1
Summary
In this paper we continue developing our geometric approach for counting BPS states in five dimensional gauge theories with eight supercharges compactified on a circle of finite radius. We do not explore this construction in this paper, the physical principles underlying this statement carry over directly to 3d-5d systems, and we fully expect that exponential BPS graphs should likewise encode motivic spectrum generators for BPS states of toric CY threefolds This observation is especially important for effective applications to enumerative geometry, since the number of BPS states can quickly grow out of control. Another salient novelty introduced in this work is the concept of “exponential BPS graph” These are (comparatively) simple ribbon graphs embedded in a Riemann surface (in this paper, it is just C∗), whose topology is expected to encode the whole spectrum of generalized DT invariants in degree zero (that is, with zero unit of D6 charge). The appendices contain some of the materials to support our computations in the main body of this paper
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