Abstract
The BPS bound states of D4-D2-D0 branes on the noncompact 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, [T. Nishinaka et al., J. High Energy Phys. 5 (2014) 139]. 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 derived 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 concrete examples of ${\mathbb{C}}^{3}$, conifold, local ${\mathbb{P}}^{1}\ifmmode\times\else\texttimes\fi{}{\mathbb{P}}^{1}$ and local ${\mathbb{P}}^{2}$ quivers.
Highlights
The D-brane configurations and their corresponding BPS states in quiver gauge theories have been a fruitful field of study in nonperturbative effects in gauge/string duality
In our previous works [5,6], we introduced the applications of the statistical dimer model and associated Mahler measure, as well as the hyperbolic geometry, in the asymptotic analysis of the BPS sector and the computations of the BPS growth rate, entropy and free energy of the toric quivers and the particular class of isoradial ones
We describe the main problem of this study, the asymptotic counting of the BPS states of the quiver gauge theory, and the instantons of the topologically twisted N 1⁄4 4 super Yang-Mills theory, the low energy effective theory in the world-volume of the D4 brane
Summary
The D-brane configurations and their corresponding BPS states in quiver gauge theories have been a fruitful field of study in nonperturbative effects in gauge/string duality. Our method and results are implemented in some concrete examples In this part, we describe the main problem of this study, the asymptotic counting of the BPS states of the quiver gauge theory, and the instantons of the topologically twisted N 1⁄4 4 super Yang-Mills theory, the low energy effective theory in the world-volume of the D4 brane. After a brief review of the related gauge theory and D-brane bound states, we explain the construction of the crystal model as the main method of the asymptotic analysis of the instanton sector. Based on the counting tools of the crystal models and by introducing asymptotic methods from tropical geometry, large deviation technique and number theory, we propose our new approach toward the thermodynamics and phase structure of the instanton sector,
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