Invariants of quivers with potentials

Compute refined invariants for some quivers with potentials.

1. Compute (total refined) invariants for the cyclic quiver C_n with the potential consisting of one term (the cycle).

2. Compute (total refined) invariants for the McKay quiver of C^3/Z_r with the induced potential (consisting of 3-cycles), where the action is given by (1, k, -k-1) for any 0<=k<r. This potential quiver can be obtained as a translation potential quiver of a cyclic quiver, see arXiv:1911.01788. Some of the results are presented in arXiv:2012.14358.

EXAMPLES:

sage: from msinvar.quivers import CyclicQuiver
sage: from msinvar.potential_quiver_invar import *
sage: r, k = 3, 1
sage: CQ = CyclicQuiver(r); CQ
Cyclic quiver: Quiver with 3 vertices and 3 arrows
sage: CQ.prec([2]*r) # precision vector
sage: CQ.intAtt().dict() #Attractor invar for a cyclic quiver without potential
{(0, 0, 1): 1, (0, 1, 0): 1, (1, 0, 0): 1, (1, 1, 1): -y}

sage: total=cyclic_potential_total(CQ) # Total invar for a cyclic quiver with potential
sage: CQ.intAtt(total).dict() #Attractor invar for a cyclic quiver with potential 
{(0, 0, 1): 1, (0, 1, 0): 1, (1, 0, 0): 1}

sage: PQ = CQ.translation_PQ(1); PQ # translation quiver with potential
Translation PQ: Quiver with 3 vertices, 9 arrows and potential with 6 terms
sage: PQ.prec([2,2,2])
sage: total=translation_PQ_total(PQ)
sage: PQ.intAtt(total).simp().dict() #Attractor invar for the translation quiver with potential
{(0, 0, 1): 1,
 (0, 1, 0): 1,
 (1, 0, 0): 1,
 (1, 1, 1): (-2*y^2 - 1)/y,
 (2, 2, 2): (-2*y^2 - 1)/y}

msinvar.potential_quiver_invar.interaction_invariant(S, si, dim, Q)

Return expression of the form:

\[\sum_{m:S\to N} (-y)^{-\sum_{i,j\in S} m_im_j\sigma(i,j)} /(1/y^2)_m\cdot x^{\sum_{i\in S} m_i \dim(i)}\]

1. Q is a Quiver, y=Q.y, r=Q.rank.

2. S is a list.

3. \(\sigma:S\times S\to Z\) is a map (interaction form).

4. \(\dim:S\to Z^r\) is a map (dimension vectors).

5. \((q)_m=\prod_{i\in S}(q)_{m_i},\ (q)_k=(1-q)...(1-q^k)\).

6. The sum runs over m such that \(\sum_i m_i \dim(i)\le\) Q.prec().

msinvar.potential_quiver_invar.translation_PQ_total(PQ, prec=None)

Return total invariant (stacky invariant for the trivial stability) for the translation potential quiver, assuming that the base quiver has implemented methods ind_list(), ind_dim(), ind_hom().

PQ – translation potential quiver or its wall-crossing structure.

See msinvar.quivers.TranslationPQ.

EXAMPLES:

sage: from msinvar import *
sage: from msinvar.potential_quiver_invar import *
sage: Q=CyclicQuiver(3); Q
Cyclic quiver: Quiver with 3 vertices and 3 arrows
sage: PQ=Q.translation_PQ(1); PQ
Translation PQ: Quiver with 3 vertices, 9 arrows and potential with 6 terms
sage: I=translation_PQ_total(PQ, [2,2,2])
sage: PQ.intAtt(I).simp().dict()
{(0, 0, 1): 1,
 (0, 1, 0): 1,
 (1, 0, 0): 1,
 (1, 1, 1): (-2*y^2 - 1)/y,
 (2, 2, 2): (-2*y^2 - 1)/y}

msinvar.potential_quiver_invar.cyclic_potential_total(CQ, prec=None)

Return total stacky invariants (for the trivial stability) for a cyclic quiver with the cyclic potential.

CQ is the cyclic quiver or its wall-crossing structure.

class msinvar.potential_quiver_invar.QuiverExample1

Bases: msinvar.quivers.Quiver

Quiver 1->0, 1->2 with methods for indecomposables and a non-trivial automorphism.

EXAMPLES:

sage: from msinvar.potential_quiver_invar import *
sage: Q=QuiverExample1(); Q
Quiver with 3 vertices and 2 arrows
sage: PQ=Q.translation_PQ(); PQ
Translation PQ: Quiver with 3 vertices, 7 arrows and potential with 4 terms
sage: PQ.prec([3,3,3])
sage: I=PQ.translation_PQ_total()
sage: PQ.intAtt(I).dict()
{(0, 0, 1): 1,
 (0, 1, 0): -y,
 (0, 1, 1): 1,
 (1, 0, 0): 1,
 (1, 0, 1): -y,
 (1, 1, 0): 1,
 (1, 1, 1): -y,
 (1, 2, 1): -y}

ind_list(*args)

Return the list of all indecomposable representations, paramertized by pairs (i,j) with i<=j.

ind_dim(a)

Dimension vector of an indecomposable representation a.

ind_hom(a, b)

Dimension of the Hom-space between indecomposable representations a and b.

tau(a)

Non-trivial automorphism of the quiver.

ind_tau(a)

Bijection on indecomposables induced by tau().

translation_PQ()

See translation_PQ()