Crystal melting and black holes

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 C 2 /Z n and C 3 /Z n × Z 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.

Beyond Aztec Castles: Toric Cascades in the dP 3 Quiver

“…In this paper, we focus on a specific example of such a theory, which is associated to the cone over the del Pezzo surface of degree 6 (CP 2 blown up at three points) which we refer to as dP 3 in this paper following the conventions of the physics literature [BP,FHHU01]. As previously studied, via the AdS/CFT correspondence, the corresponding quiver gauge theory is built using a highly symmetric six-vertex quiver (illustrated in Figure 1 (Middle)) and the potential About a decade ago, using methods from dimer theory first developed by mathematicians and physicists alike in the context of statistical mechanics [HV07,KOS06], physicists described how to associate a brane tiling to such a quiver gauge theory [FHK + ,FHKV08]. See [GK12] for a more recent mathematical treatment.…”

W1+∞algebra as a symmetry behind the Alday-Gaiotto-Tachikawa relation

“…W 1+∞ symmetry has been applied to many topics, for example, the quantum Hall effect [24], matrix model [25] (see also a recent development in the context of AGT [26]), topological string [27] and crystal melting [28]. We hope that it is a good time now to develop the representation theory, such as the correlation function, to more detail.…”

Generalized MacMahon Gd(q) as q-deformed CFT2 correlation function

“…These Ψ ± operators will be denoted later as 15) but to keep the notations simpler, we will momentary use Ψ ± (z) and come later to the Γ…”

“…Recently, these functions have known a revival of interest in connection with topological string theory [8,9,10]; in particular in the study of BPS black holes, given by branes wrapping collapsed cycles in CalabiYau orbifolds, and in the infinite n limit of quiver gauge theories [11,12,13,14,15]. MacMahon functions G 2 (q) and G 3 (q) are also used in the explicit computation of the amplitudes of A-model topological string on local Calabi-Yau manifolds [16,17,5,18].…”