### Amoebas and InstantonsAmoebas and Instantons

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 Author Maeda, Takashi ♦ Nakatsu, Toshio Source arXiv.org Content type Text File Format PDF Date of Submission 2006-01-31 Language English
 Subject Domain (in DDC) Computer science, information & general works ♦ Natural sciences & mathematics ♦ Mathematics ♦ Physics Subject Keyword High Energy Physics - Theory ♦ Mathematics - Algebraic Geometry ♦ math ♦ physics:hep-th Abstract We study a statistical model of random plane partitions. The statistical model has interpretations as five-dimensional $\mathcal{N}=1$ supersymmetric SU(N) Yang-Mills on $\mathbb{R}^4\times S^1$ and as K\"ahler gravity on local SU(N) geometry. At the thermodynamic limit a typical plane partition called the limit shape dominates in the statistical model. The limit shape is linked with a hyperelliptic curve, which is a five-dimensional version of the SU(N) Seiberg-Witten curve. Amoebas and the Ronkin functions play intermediary roles between the limit shape and the hyperelliptic curve. In particular, the Ronkin function realizes an integration of thermodynamical density of the main diagonal partitions, along one-dimensional slice of it and thereby is interpreted as the counting function of gauge instantons. The radius of $S^1$ can be identified with the inverse temperature of the statistical model. The large radius limit of the five-dimensional Yang-Mills is the low temperature limit of the statistical model, where the statistical model is frozen to a ground state that is associated with the local SU(N) geometry. We also show that the low temperature limit corresponds to a certain degeneration of amoebas and the Ronkin functions known as tropical geometry. Description Reference: Int.J.Mod.Phys.A22:937-984,2007 Educational Use Research Learning Resource Type Article Page Count 58