BPS States, Crystals, and Matrices

Figure 11

Toric diagram for the resolution of ℂ³/ℤ^𝑁+1 geometry has 𝑁+1 vertices of the same type ⊕. (a): toric diagram for 𝑁=1 and arrow representation of 𝐴₊ and 𝑊₁. In the noncommutative chamber, this leads to the same plane partition crystal as in Figure 10, however colored now in yellow and red. (b): for the chamber with positive 𝑅 and 2<𝐵<3, the crystal develops two additional corners and its generating function reads 𝑍_2|1=⟨Ω₊|(𝑊₁)²|Ω₋⟩. (c): for negative 𝑅 and positive 𝑛−1<𝐵<𝑛 the crystal is finite along two axes and develops 𝑛−1 yellow corners; its generating function for the case of 𝑛=5 shown in the picture reads 𝑍_5|1=⟨0|(𝑊₁)⁵|0⟩ (two external arrows, corresponding to Γ₋ acting on ⟨0| and Γ₊ acting on |0⟩, are suppressed.).

(a)

(b)

(c)