Abstract
It has recently been shown that the statistical mechanics of crystal melting maps to A-model topological string amplitudes on non-compact Calabi-Yau spaces. In this note we establish a one to one correspondence between two and three dimensional crystal melting configurations and certain BPS black holes given by branes wrapping collapsed cycles on the orbifolds 2/n and 3/n × n in the large n limit. The ranks of gauge groups in the associated gauged quiver quantum mechanics determine the profiles of crystal melting configurations and the process of melting maps to flop transitions which leave the background Calabi-Yau invariant. We explain the connection between these two realizations of crystal melting and speculate on the underlying physical meaning.
Highlights
An important feature of string theory is how geometric data in a string compactification appear in the associated low energy effective theory
In subsection 3.1 we present a naive analysis of geometric transitions which preserve the Calabi-Yau geometry and find far too many black hole charge configurations in comparison with crystal melting configurations
Returning to the problem of crystal melting, we show in subsection 4.3 that the BPS black holes generated by these transformations are in one to one correspondence with crystal melting configurations
Summary
An important feature of string theory is how geometric data in a string compactification appear in the associated low energy effective theory. In this note we combine these themes and show that the crystal melting configurations of topological string theory are in one to one correspondence with certain BPS black holes given by branes wrapped on collapsed cycles in the supersymmetric orbifolds C2/Zn and C3/Zn × Zn in the limit of large n. The black holes in this spectrum are generated by all possible flop transitions which leave the background Calabi-Yau geometry invariant, and the stability of these configurations is determined by the rigid structure of the partially melted crystal In both cases the duality group generated by these transitions corresponds to the Weyl group of an infinite dimensional algebra. Original motivation for this note was to give a physical interpretation of the infinite honeycomb lattice dimer model of crystal melting purely as a quiver gauge theory of the type shown in figure (4). 2d tableaux was interpreted as the non-perturbative creation of baby universes with Bertotti-Robinson cosmology AdS2 × S2
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