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.
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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.
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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.
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In general, we can play such a game for for each quiver vertex a.
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In general, in addition to (2), there are extra relations—Serre relations—satisfied by the generators. When the Serre relations are included, we have the reduced quiver Yangian.
© 2023 Author(s). Published under an exclusive license by AIP Publishing.
2023
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