Counting molten crystals

Counting molten crystals¶

We follow arXiv:0809.0117 and arXiv:2012.14358.

Given a brane tiling we associate with it a quiver with potential $(Q,W)$. For any vertex $i\in Q_0$, we consider the set $\Delta_i$ of paths starting at $i$ up to an equivalence relation induced by the potential $W$. This set has a poset structure and can be interpreted as a crystal embedded in $\mathbb R^3$. For example, a brane tiling for $\mathbb C^3$ corresponds to a quiver with one vertex $0$, three loops $x,y,z$ and potential $W=x[y,z]$. The corresponding Jacobian algebra is $\mathbb C[x,y,z]$ which has a basis parametrized by $\Delta_0=\mathbb N^3$, our crystal.

We define molten crystals as complements of finite ideals $I\subset \Delta_i$, meaning a subset $I$ such that $u\le v$ and $v\in I$ implies $u\in I$. For any path $u\in\Delta_i$, let $t(u)\in Q_0$ be the target vertex of $u$. Define the dimension vector of $I$ to be $\dim I=\sum_{u\in I}e_{t(u)}\in\mathbb N^{Q_0}$. We are interested in computing the partition function $$Z_{\Delta_i}(x)=\sum_{I\subset\Delta_i}x^{\dim I}$$ which is closely related to numerical DT invariants of $(Q,W)$.

In [16]:

import os, sys
module_path = os.path.abspath(os.path.join('..'))
if module_path not in sys.path:
    sys.path.append(module_path)
from msinvar import *
from msinvar.brane_tilings import *
set_plots()

C3¶

In [17]:

Q=CyclicQuiver(1);
PQ=Q.translation_PQ(1); PQ # construct quiver with potential for C^3

Out[17]:

In [18]:

Q1=BTQuiver(potential=PQ._potential)
I=partition_func(Q1, 0, 10); show(I)

$\newcommand{\Bold}[1]{\mathbf{#1}}1 + x + 3 x^{2} + 6 x^{3} + 13 x^{4} + 24 x^{5} + 48 x^{6} + 86 x^{7} + 160 x^{8} + 282 x^{9} + 500 x^{10}$

In [19]:

show(I.Log()) #Ideals in N^3 are plane partitions and are counted by the MacMahon formula

$\newcommand{\Bold}[1]{\mathbf{#1}}x + 2 x^{2} + 3 x^{3} + 4 x^{4} + 5 x^{5} + 6 x^{6} + 7 x^{7} + 8 x^{8} + 9 x^{9} + 10 x^{10}$

C3/Z3 with action (1,1,1)¶

In [10]:

Q=CyclicQuiver(3);
PQ=Q.translation_PQ(1); PQ

Out[10]:

In [11]:

Q1=BTQuiver(potential=PQ._potential)
I=partition_func(Q1, 0, 10); show(I)

$\newcommand{\Bold}[1]{\mathbf{#1}}1 + x_{0} + 3 x_{0} x_{1} + 3 x_{0} x_{1}^{2} + 3 x_{0} x_{1} x_{2} + x_{0} x_{1}^{3} + 3 x_{0}^{2} x_{1} x_{2} + 9 x_{0} x_{1}^{2} x_{2} + 9 x_{0}^{2} x_{1}^{2} x_{2} + 6 x_{0} x_{1}^{3} x_{2} + 9 x_{0} x_{1}^{2} x_{2}^{2} + 9 x_{0}^{2} x_{1}^{3} x_{2} + 21 x_{0}^{2} x_{1}^{2} x_{2}^{2} + 15 x_{0} x_{1}^{3} x_{2}^{2} + 3 x_{0} x_{1}^{2} x_{2}^{3} + 3 x_{0}^{2} x_{1}^{4} x_{2} + 12 x_{0}^{3} x_{1}^{2} x_{2}^{2} + 39 x_{0}^{2} x_{1}^{3} x_{2}^{2} + 12 x_{0}^{2} x_{1}^{2} x_{2}^{3} + 20 x_{0} x_{1}^{3} x_{2}^{3} + 36 x_{0}^{3} x_{1}^{3} x_{2}^{2} + 18 x_{0}^{2} x_{1}^{4} x_{2}^{2} + 18 x_{0}^{3} x_{1}^{2} x_{2}^{3} + 73 x_{0}^{2} x_{1}^{3} x_{2}^{3} + 15 x_{0} x_{1}^{3} x_{2}^{4} + 36 x_{0}^{3} x_{1}^{4} x_{2}^{2} + 12 x_{0}^{4} x_{1}^{2} x_{2}^{3} + 108 x_{0}^{3} x_{1}^{3} x_{2}^{3} + 45 x_{0}^{2} x_{1}^{4} x_{2}^{3} + 75 x_{0}^{2} x_{1}^{3} x_{2}^{4} + 6 x_{0} x_{1}^{3} x_{2}^{5} + 12 x_{0}^{3} x_{1}^{5} x_{2}^{2} + 3 x_{0}^{5} x_{1}^{2} x_{2}^{3} + 73 x_{0}^{4} x_{1}^{3} x_{2}^{3} + 159 x_{0}^{3} x_{1}^{4} x_{2}^{3} + 150 x_{0}^{3} x_{1}^{3} x_{2}^{4} + 60 x_{0}^{2} x_{1}^{4} x_{2}^{4} + 42 x_{0}^{2} x_{1}^{3} x_{2}^{5} + x_{0} x_{1}^{3} x_{2}^{6}$

In [12]:

I=partition_func1(Q1, 0, 10); show(I)

$\newcommand{\Bold}[1]{\mathbf{#1}}1 + x + 3 x^{2} + 6 x^{3} + 13 x^{4} + 24 x^{5} + 48 x^{6} + 86 x^{7} + 160 x^{8} + 282 x^{9} + 500 x^{10}$

In [13]:

show(I.Log())

$\newcommand{\Bold}[1]{\mathbf{#1}}x + 2 x^{2} + 3 x^{3} + 4 x^{4} + 5 x^{5} + 6 x^{6} + 7 x^{7} + 8 x^{8} + 9 x^{9} + 10 x^{10}$

C3/Z4 with action (1,1,2)¶

In [14]:

Q=CyclicQuiver(4);
PQ=Q.translation_PQ(1); PQ

Out[14]:

In [15]:

Q1=BTQuiver(potential=PQ._potential)
I=partition_func(Q1, 0, 5); show(I)

$\newcommand{\Bold}[1]{\mathbf{#1}}1 + x_{0} + 2 x_{0} x_{1} + x_{0} x_{2} + x_{0} x_{1}^{2} + x_{0}^{2} x_{2} + 4 x_{0} x_{1} x_{2} + 2 x_{0}^{2} x_{1} x_{2} + 4 x_{0} x_{1}^{2} x_{2} + x_{0}^{2} x_{2}^{2} + 2 x_{0} x_{1} x_{2}^{2} + 4 x_{0} x_{1} x_{2} x_{3} + x_{0}^{2} x_{1}^{2} x_{2} + x_{0}^{3} x_{2}^{2} + 4 x_{0}^{2} x_{1} x_{2}^{2} + 6 x_{0} x_{1}^{2} x_{2}^{2} + 4 x_{0}^{2} x_{1} x_{2} x_{3} + 4 x_{0} x_{1}^{2} x_{2} x_{3} + 4 x_{0} x_{1} x_{2}^{2} x_{3}$

C3/Z4 with action (1,0,3)¶

In [36]:

Q=CyclicQuiver(4);
PQ=Q.translation_PQ(0); PQ

Out[36]:

In [37]:

Q1=BTQuiver(potential=PQ._potential)
I=partition_func(Q1, 0, 5); show(I)

$\newcommand{\Bold}[1]{\mathbf{#1}}1 + x_{0} + x_{0}^{2} + x_{0} x_{1} + x_{0} x_{3} + x_{0}^{3} + x_{0}^{2} x_{1} + x_{0} x_{1} x_{2} + x_{0}^{2} x_{3} + x_{0} x_{1} x_{3} + x_{0} x_{2} x_{3} + x_{0}^{4} + x_{0}^{3} x_{1} + x_{0}^{2} x_{1}^{2} + x_{0}^{2} x_{1} x_{2} + x_{0}^{3} x_{3} + 2 x_{0}^{2} x_{1} x_{3} + x_{0}^{2} x_{2} x_{3} + 4 x_{0} x_{1} x_{2} x_{3} + x_{0}^{2} x_{3}^{2} + x_{0}^{5} + x_{0}^{4} x_{1} + x_{0}^{3} x_{1}^{2} + x_{0}^{3} x_{1} x_{2} + x_{0}^{2} x_{1}^{2} x_{2} + x_{0}^{4} x_{3} + 2 x_{0}^{3} x_{1} x_{3} + x_{0}^{2} x_{1}^{2} x_{3} + x_{0}^{3} x_{2} x_{3} + 8 x_{0}^{2} x_{1} x_{2} x_{3} + x_{0} x_{1}^{2} x_{2} x_{3} + x_{0} x_{1} x_{2}^{2} x_{3} + x_{0}^{3} x_{3}^{2} + x_{0}^{2} x_{1} x_{3}^{2} + x_{0}^{2} x_{2} x_{3}^{2} + x_{0} x_{1} x_{2} x_{3}^{2}$

C3/Z5 with action (1,1,3)¶

In [39]:

Q=CyclicQuiver(5);
PQ=Q.translation_PQ(1); PQ

Out[39]:

In [40]:

Q1=BTQuiver(potential=PQ._potential)
I=partition_func(Q1, 0, 5); show(I)

$\newcommand{\Bold}[1]{\mathbf{#1}}1 + x_{0} + 2 x_{0} x_{1} + x_{0} x_{3} + x_{0} x_{1}^{2} + 2 x_{0} x_{1} x_{2} + 3 x_{0} x_{1} x_{3} + 3 x_{0} x_{1}^{2} x_{2} + 3 x_{0} x_{1}^{2} x_{3} + 4 x_{0} x_{1} x_{2} x_{3} + 3 x_{0} x_{1} x_{3} x_{4} + 3 x_{0} x_{1}^{2} x_{2}^{2} + x_{0} x_{1}^{3} x_{3} + 7 x_{0} x_{1}^{2} x_{2} x_{3} + 2 x_{0} x_{1} x_{2} x_{3}^{2} + 6 x_{0} x_{1}^{2} x_{3} x_{4} + 5 x_{0} x_{1} x_{2} x_{3} x_{4}$

Examples from a Database¶

We use quivers with potentials from a database by Boris Pioline.

In [55]:

for l in BTD:
    Q=BTQuiver(potential=l[1])
    print(l[0],":",Q); show(Q)
    I=partition_func1(Q, 1, 15); show(I)
    print('\n')

C^3 : Quiver with 1 vertices, 3 arrows and potential with 2 terms

$\newcommand{\Bold}[1]{\mathbf{#1}}1 + x + 3 x^{2} + 6 x^{3} + 13 x^{4} + 24 x^{5} + 48 x^{6} + 86 x^{7} + 160 x^{8} + 282 x^{9} + 500 x^{10} + 859 x^{11} + 1479 x^{12} + 2485 x^{13} + 4167 x^{14} + 6879 x^{15}$


Conifold=Y10 : Quiver with 2 vertices, 4 arrows and potential with 2 terms

$\newcommand{\Bold}[1]{\mathbf{#1}}1 + x + 2 x^{2} + 5 x^{3} + 10 x^{4} + 18 x^{5} + 32 x^{6} + 59 x^{7} + 106 x^{8} + 181 x^{9} + 305 x^{10} + 518 x^{11} + 867 x^{12} + 1418 x^{13} + 2301 x^{14} + 3724 x^{15}$


C^2xC/2 : Quiver with 2 vertices, 6 arrows and potential with 4 terms


C^2xC/3 : Quiver with 3 vertices, 9 arrows and potential with 6 terms


PdP6=C^3/2x2 : Quiver with 4 vertices, 12 arrows and potential with 8 terms


SPP=L121 : Quiver with 3 vertices, 7 arrows and potential with 4 terms

$\newcommand{\Bold}[1]{\mathbf{#1}}1 + x + 3 x^{2} + 6 x^{3} + 11 x^{4} + 22 x^{5} + 42 x^{6} + 74 x^{7} + 133 x^{8} + 233 x^{9} + 397 x^{10} + 677 x^{11} + 1138 x^{12} + 1877 x^{13} + 3083 x^{14} + 5007 x^{15}$


P2=C^3/(1,1,1) : Quiver with 3 vertices, 9 arrows and potential with 6 terms


F0.1=P1xP1 : Quiver with 4 vertices, 12 arrows and potential with 8 terms

$\newcommand{\Bold}[1]{\mathbf{#1}}1 + x + 4 x^{2} + 6 x^{3} + 8 x^{4} + 13 x^{5} + 24 x^{6} + 42 x^{7} + 72 x^{8} + 139 x^{9} + 281 x^{10} + 548 x^{11} + 1088 x^{12} + 2202 x^{13} + 4586 x^{14} + 9650 x^{15}$


F0.2=P1xP1 : Quiver with 4 vertices, 8 arrows and potential with 4 terms


F1=dP1=Y21=L312 : Quiver with 4 vertices, 10 arrows and potential with 6 terms

$\newcommand{\Bold}[1]{\mathbf{#1}}1 + x + 2 x^{2} + 4 x^{3} + 9 x^{4} + 18 x^{5} + 35 x^{6} + 70 x^{7} + 143 x^{8} + 274 x^{9} + 519 x^{10} + 961 x^{11} + 1744 x^{12} + 3081 x^{13} + 5320 x^{14} + 9007 x^{15}$


F2=C^3/(1,1,2) : Quiver with 4 vertices, 12 arrows and potential with 8 terms


dP2.1 : Quiver with 5 vertices, 13 arrows and potential with 8 terms

$\newcommand{\Bold}[1]{\mathbf{#1}}1 + x + 2 x^{2} + 3 x^{3} + 7 x^{4} + 14 x^{5} + 31 x^{6} + 64 x^{7} + 135 x^{8} + 296 x^{9} + 644 x^{10} + 1362 x^{11} + 2775 x^{12} + 5483 x^{13} + 10489 x^{14} + 19393 x^{15}$


dP2.2 : Quiver with 5 vertices, 11 arrows and potential with 6 terms

$\newcommand{\Bold}[1]{\mathbf{#1}}1 + x + 2 x^{2} + 6 x^{3} + 13 x^{4} + 28 x^{5} + 52 x^{6} + 94 x^{7} + 169 x^{8} + 291 x^{9} + 488 x^{10} + 788 x^{11} + 1253 x^{12} + 1979 x^{13} + 3084 x^{14} + 4760 x^{15}$


PdP2 : Quiver with 5 vertices, 13 arrows and potential with 8 terms

$\newcommand{\Bold}[1]{\mathbf{#1}}1 + x + 3 x^{2} + 7 x^{3} + 16 x^{4} + 35 x^{5} + 68 x^{6} + 131 x^{7} + 240 x^{8} + 427 x^{9} + 742 x^{10} + 1247 x^{11} + 2067 x^{12} + 3363 x^{13} + 5402 x^{14} + 8582 x^{15}$


dP3.1 : Quiver with 6 vertices, 12 arrows and potential with 6 terms

$\newcommand{\Bold}[1]{\mathbf{#1}}1 + x + 2 x^{2} + 4 x^{3} + 7 x^{4} + 14 x^{5} + 26 x^{6} + 44 x^{7} + 73 x^{8} + 123 x^{9} + 204 x^{10} + 338 x^{11} + 554 x^{12} + 883 x^{13} + 1397 x^{14} + 2201 x^{15}$


dP3.2 : Quiver with 6 vertices, 14 arrows and potential with 8 terms

$\newcommand{\Bold}[1]{\mathbf{#1}}1 + x + 3 x^{2} + 7 x^{3} + 17 x^{4} + 34 x^{5} + 63 x^{6} + 107 x^{7} + 176 x^{8} + 279 x^{9} + 438 x^{10} + 671 x^{11} + 1020 x^{12} + 1531 x^{13} + 2300 x^{14} + 3443 x^{15}$


dP3.3 : Quiver with 6 vertices, 14 arrows and potential with 8 terms

$\newcommand{\Bold}[1]{\mathbf{#1}}1 + x + 4 x^{2} + 10 x^{3} + 22 x^{4} + 43 x^{5} + 72 x^{6} + 111 x^{7} + 168 x^{8} + 251 x^{9} + 374 x^{10} + 567 x^{11} + 858 x^{12} + 1295 x^{13} + 1963 x^{14} + 2982 x^{15}$


dP3.4 : Quiver with 6 vertices, 18 arrows and potential with 12 terms

$\newcommand{\Bold}[1]{\mathbf{#1}}1 + x + 6 x^{2} + 21 x^{3} + 56 x^{4} + 126 x^{5} + 246 x^{6} + 432 x^{7} + 696 x^{8} + 1059 x^{9} + 1552 x^{10} + 2250 x^{11} + 3258 x^{12} + 4761 x^{13} + 7003 x^{14} + 10410 x^{15}$


L131 : Quiver with 4 vertices, 10 arrows and potential with 6 terms

$\newcommand{\Bold}[1]{\mathbf{#1}}1 + x + 3 x^{2} + 6 x^{3} + 12 x^{4} + 23 x^{5} + 43 x^{6} + 78 x^{7} + 141 x^{8} + 246 x^{9} + 426 x^{10} + 725 x^{11} + 1221 x^{12} + 2029 x^{13} + 3339 x^{14} + 5442 x^{15}$


L152 : Quiver with 6 vertices, 16 arrows and potential with 10 terms

$\newcommand{\Bold}[1]{\mathbf{#1}}1 + x + 3 x^{2} + 7 x^{3} + 15 x^{4} + 32 x^{5} + 62 x^{6} + 121 x^{7} + 223 x^{8} + 406 x^{9} + 718 x^{10} + 1242 x^{11} + 2110 x^{12} + 3507 x^{13} + 5749 x^{14} + 9275 x^{15}$


C^3/(1,1,3) : Quiver with 5 vertices, 15 arrows and potential with 10 terms


Y23=L153 : Quiver with 6 vertices, 16 arrows and potential with 10 terms

$\newcommand{\Bold}[1]{\mathbf{#1}}1 + x + 3 x^{2} + 8 x^{3} + 18 x^{4} + 36 x^{5} + 68 x^{6} + 121 x^{7} + 210 x^{8} + 356 x^{9} + 594 x^{10} + 976 x^{11} + 1588 x^{12} + 2547 x^{13} + 4062 x^{14} + 6414 x^{15}$


C^3/(1,1,4) : Quiver with 6 vertices, 18 arrows and potential with 12 terms


PdP3a=C^3/(1,2,3) : Quiver with 6 vertices, 18 arrows and potential with 12 terms


Y30 : Quiver with 6 vertices, 12 arrows and potential with 6 terms


Y31 : Quiver with 6 vertices, 14 arrows and potential with 8 terms

$\newcommand{\Bold}[1]{\mathbf{#1}}1 + x + 3 x^{2} + 5 x^{3} + 9 x^{4} + 15 x^{5} + 25 x^{6} + 43 x^{7} + 75 x^{8} + 125 x^{9} + 211 x^{10} + 357 x^{11} + 605 x^{12} + 1018 x^{13} + 1720 x^{14} + 2888 x^{15}$

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