Brane tilings and crystal counting

Brane tilings and molten crystal counting

This code is based on arXiv:0809.0117. See also arXiv:2012.14358

Given a brane tiling (a bipartite graph embedded in a torus) 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 compute 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).

EXAMPLES:

sage: from msinvar import *
sage: CQ=CyclicQuiver(1)
sage: PQ=CQ.translation_PQ(); PQ
Ginzburg PQ: Quiver with 1 vertices, 3 arrows and potential with 2 terms
sage: Q=BTQuiver(PQ)
sage: Z=Q.partition_func(0, 8); Z
1 + x + 3*x^2 + 6*x^3 + 13*x^4 + 24*x^5 + 48*x^6 + 86*x^7 + 160*x^8
sage: Z.Log()
x + 2*x^2 + 3*x^3 + 4*x^4 + 5*x^5 + 6*x^6 + 7*x^7 + 8*x^8

sage: BTD[1]
['Conifold=Y10',
 'Phi[1,2,1]*Phi[2,1,1]*Phi[1,2,2]*Phi[2,1,2]-Phi[1,2,1]*Phi[2,1,2]*Phi[1,2,2]*Phi[2,1,1]']
sage: Q=BTQuiver(potential=BTD[1][1]); Q
Quiver with 2 vertices, 4 arrows and potential with 2 terms
sage: Z=Q.NCDT(1, 5); Z
1 + x0 - 2*x0*x1 - 4*x0^2*x1 + x0*x1^2 - 2*x0^3*x1 + 8*x0^2*x1^2 + 14*x0^3*x1^2 - 4*x0^2*x1^3
sage: Z.Log()
x0 - x0^2 - 2*x0*x1 - 2*x0^2*x1 + x0*x1^2 + 6*x0^2*x1^2 + 3*x0^3*x1^2 - 2*x0^2*x1^3

sage: CQ=CyclicQuiver(3)
sage: PQ=CQ.translation_PQ(1); PQ
Translation PQ: Quiver with 3 vertices, 9 arrows and potential with 6 terms
sage: Q=BTQuiver(PQ,prec=[1,1,1])
sage: Q.intAtt_from_crystals().dict() # numerical integer attractor inv. 
{(0, 0, 1): 1, (0, 1, 0): 1, (1, 0, 0): 1, (1, 1, 1): -3}

class msinvar.brane_tilings.BTQuiver(*args, **kw)

Bases: msinvar.quivers.Quiver

Quiver with potential associated with a brane tiling.

Contains the method wt() such that two paths are equivalent (equal in the Jacobian algebra if and only if their weights are equal).

wt(a)

Weight of an arrow or a path.

path2vec(p)

Multiplicities of all arrows in a path.

ideal_dim(I)

Dimension vector of an ideal I that consists of atoms (paths).

add_new_arrows(P, n)

Add new arrows to the atoms in the poset P.

• P – poset of atoms,

• n – index in P, where atoms that require new arrows start.

create_path_poset(i, N=10)

Creat the poset of atoms (paths) that start at the vertex i and have length <= N.

crystal_dict(i, N=5)

NCDT(i, N=5)

partition_func(i, N=5)

partition_func1(i, N=5)

ratAtt_from_crystals()

See ratAtt_from_crystals().

intAtt_from_crystals()

See intAtt_from_crystals().

class msinvar.brane_tilings.Atom(Q, t, wt=None)

Bases: object

Atom – equivalence class of a path in the Jacobian algebra.

• Q – a BTQuiver,

• t – a target,

• wt – a weight.

add_arrow(a)

equal(u)

msinvar.brane_tilings.crystal_dict(Q, i, N=5)

Dictionary counting ideals (molten crystals) in the crystal of atoms that start at the vertex i and have length <= N.

We keep track of the dimension vectors of ideals.

msinvar.brane_tilings.NCDT(Q, i, N=5)

Numerical NCDT invariants counting (with a sign) ideals (molten crystals) in the crystal of atoms that start at the vertex i and have length <= N.

We keep track of the dimension vectors of ideals.

msinvar.brane_tilings.partition_func(Q, i, N=5)

Partition function of ideals (molten crystals) in the crystal of atoms that start at the vertex i and have length <= N.

We keep track of the dimension vectors of ideals.

msinvar.brane_tilings.partition_func1(Q, i, N=5)

Partition function of ideals (molten crystals) in the crystal of atoms that start at the vertex i and have length <= N.

We keep track only of the sizes of ideals.

msinvar.brane_tilings.BT_example(n=None)

Brane tilings from the database by Boris Pioline.

n – number of the brane tiling. If None, we return the whole list.

msinvar.brane_tilings.ratAtt_from_crystals(Q)

Numerical rational attractor invariants obtained by recursion from the NCDT invariants (computed by counting crystals) and the flow tree formula.

This algorithm works for any brane tiling.

EXAMPLE:

sage: from msinvar import *
sage: Q=BT_example(6); Q
P2=C^3/(1,1,1): Quiver with 3 vertices, 9 arrows and potential with 6 terms
sage: Q.prec([1,1,1])
sage: Q.ratAtt_from_crystals().dict()  
{(0, 0, 1): 1, (0, 1, 0): 1, (1, 0, 0): 1, (1, 1, 1): -3}

msinvar.brane_tilings.intAtt_from_crystals(Q)

Numerical integer attractor invariants obtained by recursion from the NCDT invariants (computed by counting crystals) and the flow tree formula.

This algorithm works for any brane tiling.

EXAMPLE:

sage: from msinvar import *
sage: Q=BT_example(6); Q
P2=C^3/(1,1,1): Quiver with 3 vertices, 9 arrows and potential with 6 terms
sage: Q.prec([1,1,1])
sage: Q.intAtt_from_crystals().dict()  
{(0, 0, 1): 1, (0, 1, 0): 1, (1, 0, 0): 1, (1, 1, 1): -3}