Crystal Melting, BPS Quivers and Plethystics

Abstract

We study the refined and unrefined crystal/BPS partition functions of D6-D2-D0 brane bound states for all toric Calabi-Yau threefolds without compact 4-cycles and some non-toric examples. They can be written as products of (generalized) MacMahon functions. We check our expressions and use them as vacuum characters to study the gluings. We then consider the wall crossings and discuss possible crystal descriptions for different chambers. We also express the partition functions in terms of plethystic exponentials. For C 3 and tripled affine quivers, we find their connections to nilpotent Kac polynomials. Similarly, the partition functions of D4-D2-D0 brane bound states can be obtained by replacing the (generalized) MacMahon functions with the inverse of (generalized) Euler functions.

A Note on Quiver Yangians and $\mathcal{R}$-Matrices

“…The realization of quiver Yangians should also incorporate the wall crossing phenomena [13,14]. The crystal configurations for different chambers have also been studied such as in [15][16][17]. In particular, the quiver Yangians were extended to the shifted quiver Yangians in [18] which provides a nice framework for the study of wall crossing and closed/open BPS states counting problems.…”