Flow tree formulas

This module implements attractor tree formula (see arXiv:1910.03098 and arXiv:2101.07636) and flow tree formula (see arXiv:1804.06928 and arXiv:2102.11200).

EXAMPLES:

sage: from msinvar import *
sage: Q=KroneckerQuiver(2, prec=[2,2])
sage: z=Stability([1,0])
sage: OmbAtt=Q.ratAtt_default() #rational attractor invariant
sage: OmbAtt.dict()
{(0, 1): 1, (0, 2): (-y)/(-2*y^2 - 2), (1, 0): 1, (2, 0): (-y)/(-2*y^2 - 2)}

First we apply attractor tree formula to find rational DT invariants for the above stability z:

sage: Omb1=attr_tree_formula(Q, z, OmbAtt)
sage: Omb1.simp().dict()
{(0, 1): 1,
 (0, 2): 1/2*y/(y^2 + 1),
 (1, 0): 1,
 (1, 1): (-y^2 - 1)/y,
 (1, 2): 1,
 (2, 0): 1/2*y/(y^2 + 1),
 (2, 1): 1,
 (2, 2): (-1/2*y^4 - 1/2)/(y^3 + y)}

Next we apply flow tree formula:

sage: Omb2=flow_tree_formula(Q, z, OmbAtt)
sage: Omb2.simp().dict()
{(0, 1): 1,
 (0, 2): 1/2*y/(y^2 + 1),
 (1, 0): 1,
 (1, 1): (-y^2 - 1)/y,
 (1, 2): 1,
 (2, 0): 1/2*y/(y^2 + 1),
 (2, 1): 1,
 (2, 2): (-1/2*y^4 - 1/2)/(y^3 + y)}

Finally, we apply the wall-crossing formula to determine the same invariant from the total invariant (stacky invariant for the trivial stability):

sage: Omb3=Q.rat_from_total(z, Q.total())
sage: Omb3.dict()
{(0, 1): 1,
 (0, 2): 1/2*y/(y^2 + 1),
 (1, 0): 1,
 (1, 1): (-y^2 - 1)/y,
 (1, 2): 1,
 (2, 0): 1/2*y/(y^2 + 1),
 (2, 1): 1,
 (2, 2): (-1/2*y^4 - 1/2)/(y^3 + y)}

msinvar.flow_trees.attr_tree_formula(W, z, I, t=0)

Attractor tree formula, conjectured in arXiv:1910.03098 and proved in arXiv:2101.07636.

• W – wall-crossing structure WCS.

• z – stability parameter.

• I – rational attractor invariant \(\bar\Omega_*\).

• t – rational number.

msinvar.flow_trees.attr_tree_transform(W, z, t=0)

msinvar.flow_trees.transform_g(f, t=0)

Auxiliary transform (collection) g from arXiv:2101.07636.

msinvar.flow_trees.flow_tree_formula(W, z, I, d=None, quant=True)

Flow tree formula, conjectured in arXiv:1804.06928 and proved in arXiv:2102.11200. We use its modification from arXiv:2101.07636.

• W – wall-crossing structure WCS.

• z – stability parameter.

• I – rational attractor invariant \(\bar\Omega_*\).

• quant – quantized or not.

msinvar.flow_trees.permIndex(l, te, M, kp)

Permutation index used in the flow tree formula.

msinvar.flow_trees.simp(f)