In the second part, we discuss the wall crossing phenomena. This suggests a remarkable interpretation that an atom in the crystal is a discretization of the classical geometry, giving an important clue as such to the geometry at the Planck scale. We also show that smooth geometry of the Calabi-Yau manifold emerges in the thermodynamic limit of the crystal. Each of the BPS states contributing to the BPS index is in one-to-one correspondence with a configuration of a molten crystal, and the statistical partition function of the melting crystal gives the BPS partition function. In the first part, we discuss the statistical mechanical model of crystal melting counting BPS states. This paper is divided into two parts, which are closely related to each other. This paper summarizes recent developments in the theory of Bogomol'nyi-Prasad-Sommerfield (BPS) state counting and the wall crossing phenomena, emphasizing in particular the role of the statistical mechanical model of crystal melting. Crystal Melting and Wall Crossing Phenomena
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March 2023
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