The goal of this short article is to summarize some of the recent developments in quiver Yangians and crystal meltings. This article is based on a lecture delivered by the author at International Congress on Mathematical Physics (ICMP), Geneva, 2021.
The signs in front are chosen such that φa⇒b(u)φb⇒a(−u) = 1, which is needed for the consistency of the relations; see Ref. 8 for details. Alternatively, we can disregard the signs by choosing an ordering between the vertices.
We obtain a two-dimensional projection of the crystal when we consider flavor symmetries parameterized by hX, hY, hZ with hX + hY + hZ = 0. To obtain a three-dimensional crystal, we also need to consider an R-symmetry, which amounts to lifting the condition hX + hY + hZ = 0 so that we have three parameters hX, hY, hZ. The atom located at (i, j, k) has hX charge ihX, hY charge jhY, and hZ charge khZ.
In general, we can play such a game for for each quiver vertex a.