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.

1.
M.
Rapcak
, “
Branes, quivers and BPS algebras
,” arXiv:2112.13878 [hep-th] (
2021
).
2.
W.
Li
and
M.
Yamazaki
, “
Quiver Yangian from crystal melting
,”
J. High Energy Phys.
2020
(
11
),
035
; arXiv:2003.08909 [hep-th].
3.
D.
Galakhov
and
M.
Yamazaki
, “
Quiver Yangian and supersymmetric quantum mechanics
,”
Commun. Math Phys.
396
(
2
),
713
713
(
2022
); arXiv:2008.07006 [hep-th] (
2020
).
4.
D.
Galakhov
,
W.
Li
, and
M.
Yamazaki
, “
Shifted quiver Yangians and representations from BPS crystals
,”
J. High Energy Phys.
2021
(
8
),
146
; arXiv:2106.01230 [hep-th] (
2021
).
5.
G.
Noshita
and
A.
Watanabe
, “
A note on quiver quantum toroidal algebra
,”
J. High Energy Phys.
2022
(
5
),
11
; arXiv:2108.07104 [hep-th] (
2021
).
6.
D.
Galakhov
,
W.
Li
, and
M.
Yamazaki
, “
Toroidal and elliptic quiver BPS algebras and beyond
,”
J. High Energy Phys.
2022
(
02
),
024
; arXiv:2108.10286 [hep-th].
7.
G.
Noshita
and
A.
Watanabe
, “
Shifted quiver quantum toroidal algebra and subcrystal representations
,”
J. High Energy Phys.
2022
(
5
),
122
; arXiv:2109.02045 [hep-th] (
2021
).
8.
V. G.
Drinfeld
, “
Hopf algebras and the quantum Yang-Baxter equation
,”
Dokl. Akad. Nauk SSSR
283
(
5
),
1060
1064
(
1985
).
9.
V. G.
Drinfeld
, “
Quantum groups
,” in
Proceedings of the International Congress of Mathematics
,
1986
.
10.
K. D.
Kennaway
, “
Brane tilings
,”
Int. J. Mod. Phys. A
22
,
2977
3038
(
2007
); arXiv:0706.1660 [hep-th].
11.
M.
Yamazaki
, “
Brane tilings and their applications
,”
Fortschr. Phys.
56
,
555
686
(
2008
); arXiv:0803.4474 [hep-th].
12.

The signs in front are chosen such that φab(u)φba(−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.

13.
L.
Bezerra
and
E.
Mukhin
, “
Quantum toroidal algebra associated with glm|n
,”
Algebr. Represent. Theory
24
,
541
564
(
2021
); arXiv:1904.07297 [math.QA] (
2019
).
14.
L.
Bezerra
and
E.
Mukhin
, “
Braid actions on quantum toroidal superalgebras
,” arXiv:1912.08729 [math.QA] (
2019
).
15.
A.
Okounkov
,
N.
Reshetikhin
, and
C.
Vafa
, “
Quantum Calabi-Yau and classical crystals
,”
Prog. Math.
244
,
597
(
2006
); arXiv:hep-th/0309208.
16.
A.
Iqbal
,
C.
Vafa
,
N.
Nekrasov
, and
A.
Okounkov
, “
Quantum foam and topological strings
,”
J. High Energy Phys.
2008
,
011
; arXiv:hep-th/0312022.
17.

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.

18.
H.
Ooguri
and
M.
Yamazaki
, “
Crystal melting and toric Calabi-Yau manifolds
,”
Commun. Math. Phys.
292
,
179
199
(
2009
); arXiv:0811.2801 [hep-th].
19.
S.
Mozgovoy
and
M.
Reineke
, “
On the noncommutative Donaldson-Thomas invariants arising from brane tilings
,”
Adv. Math.
223
,
1521
1544
(
2010
).
20.
M.
Yamazaki
, “
Crystal melting and wall crossing phenomena
,”
Int. J. Mod. Phys. A
26
,
1097
1228
(
2011
); arXiv:1002.1709 [hep-th].
21.
B.
Feigin
,
M.
Jimbo
,
T.
Miwa
, and
E.
Mukhin
, “
Quantum toroidal gl1 algebra: Plane partitions
,”
Kyoto J. Math.
52
,
621
659
(
2012
).
22.
A.
Tsymbaliuk
, “
The affine Yangian of gl1 revisited
,”
Adv. Math.
304
,
583
645
(
2017
); arXiv:1404.5240 [math.RT].
23.
T.
Procházka
, “W
-symmetry, topological vertex and affine Yangian
,”
J. High Energy Phys.
2016
(
10
),
077
; arXiv:1512.07178 [hep-th].
24.
B.
Feigin
,
M.
Jimbo
,
T.
Miwa
, and
E.
Mukhin
, “
Representations of quantum toroidal gln
,” arXiv:1204.5378 [math.QA].
25.
B.
Feigin
,
M.
Jimbo
,
T.
Miwa
, and
E.
Mukhin
, “
Branching rules for quantum toroidal gln
,”
Adv. Math.
300
,
229
274
(
2016
); arXiv:1309.2147 [math.QA].
26.
M. R.
Gaberdiel
,
R.
Gopakumar
,
W.
Li
, and
C.
Peng
, “
Higher spins and Yangian symmetries
,”
J. High Energy Phys.
2017
(
04
),
152
; arXiv:1702.05100 [hep-th].
27.
M.
Rapcak
,
Y.
Soibelman
,
Y.
Yang
, and
G.
Zhao
, “
Cohomological Hall algebras, vertex algebras and instantons
,”
Commun. Math. Phys.
376
,
1803
1873
(
2019
); arXiv:1810.10402 [math.QA].
28.
M.
Rapcak
,
Y.
Soibelman
,
Y.
Yang
, and
G.
Zhao
, “
Cohomological Hall algebras and perverse coherent sheaves on toric Calabi-Yau 3-folds
,” arXiv:2007.13365 [math.QA] (
2020
).
29.

In general, we can play such a game for ψ0(a) for each quiver vertex a.

30.
K.
Nagao
and
H.
Nakajima
, “
Counting invariant of perverse coherent sheaves and its wall-crossing
,”
Int. Math. Res. Not.
2011
,
3885
3938
; arXiv:0809.2992 [math.AG].
31.
D. L.
Jafferis
and
G. W.
Moore
, “
Wall crossing in local Calabi Yau manifolds
,” arXiv:0810.4909 [hep-th] (
2008
).
32.
W.-y.
Chuang
and
D. L.
Jafferis
, “
Wall crossing of BPS states on the conifold from seiberg duality and pyramid partitions
,”
Commun. Math. Phys.
292
,
285
301
(
2009
); arXiv:0810.5072 [hep-th].
33.
K.
Nagao
, “
Refined open non-commutative Donaldson-Thomas invariants for small crepant resolutions
,” arXiv:0907.3784 [math.AG] (
2009
).
34.
K.
Nagao
and
M.
Yamazaki
, “
The non-commutative topological vertex and wall crossing phenomena
,”
Adv. Theor. Math. Phys.
14
,
1147
1181
(
2010
); arXiv:0910.5479 [hep-th].
35.
P.
Sulkowski
, “
Wall-crossing, open BPS counting and matrix models
,”
J. High Energy Phys.
2011
(
03
),
089
;
36.
D.
Gaiotto
and
M.
Rapčák
, “
Vertex algebras at the corner
,”
J. High Energy Phys.
2019
(
01
),
160
; arXiv:1703.00982 [hep-th].
37.

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.

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