Quiver Asymptotics and Amoeba: Instantons on Toric Divisors of Calabi-Yau Threefolds

Abstract: The BPS bound states of D4-D2-D0 branes on the non-compact divisors of Calabi-Yau threefolds and the instantons in the dual quiver gauge theories are previously studied using two-dimensional crystal melting model and dimer model. Using the tropical geometry associated with the toric quiver, we study the asymptotic of the quiver gauge theory to compute some of their thermodynamic observables and extract the phase structure. We obtain that the thermodynamic observables such as free energy, entropy and growth rate are explicitly obtained from the limit shape of the crystal model, the boundary of the Amoeba and its Harnack curve characterization. Furthermore, we observe that there is a Hagedorn phase transition in the instanton sector inferred from the Gumbel distribution of the fluctuations in the crystal model. We present explicit computations of the results in some concrete examples of $\mathbb{C}^3$, conifold, local $\mathbb{P}^1\times \mathbb{P}^1$ and local $\mathbb{P}^2$ quivers.