Crystal Melting and Wall Crossing Phenomena

Abstract

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. This paper is divided into two parts, which are closely related to each other. In the first part, we discuss the statistical mechanical model of crystal melting counting BPS states. Each of the BPS state 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. We also show that smooth geometry of the Calabi-Yau manifold emerges in the thermodynamic limit of the crystal. This suggests a remarkable interpretation that an atom in the crystal is a discretization of the classical geometry, giving an important clue as to the geometry at the Planck scale.In the second part we discuss the wall crossing phenomena. Wall crossing phenomena states that the BPS index depends on the value of the moduli of the Calabi-Yau manifold, and jumps along real codimension one subspaces in the moduli space. We show that by using type IIA/M-theory duality, we can provide a simple and an intuitive derivation of the wall crossing phenomena, furthermore clarifying the connection with the topological string theory. This derivation is consistent with another derivation from the wall crossing formula, motivated by multi-centered BPS extremal black holes. We also explain the representation of the wall crossing phenomena in terms of crystal melting, and the generalization of the counting problem and the wall crossing to the open BPS invariants.

PhD thesis, 129 pages, 39 figures, comments welcome; v2: typos corrected, references added, now in IJMPA format; v3: figures corrected

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• Shifted Quiver Yangians and Representations from BPS Crystals
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• Exact Stringy Microstates from Gauge Theories
• Gauge/Bethe correspondence from quiver BPS algebras
• Wall Crossing As Seen By Matrix Models
• Quiver Yangians and Crystal Melting: A Concise Summary
• The Origin of Calabi-Yau Crystals in BPS States Counting
• Crystal Melting, BPS Quivers and Plethystics
• BPS States Meet Generalized Cohomology
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• On Supersymmetric Interface Defects, Brane Parallel Transport, Order-Disorder Transition and Homological Mirror Symmetry
• Wall-Crossing Effects on Quiver BPS Algebras
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• Algorithms for representations of quiver Yangian algebras
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• BPS states, crystals and matrices
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• Topological string amplitudes and Seiberg-Witten prepotentials from the counting of dimers in transverse flux