4-node and 5-node and cyclic quivers

J. Manschot, B. Pioline, A.Sen, Feb 2013. Updated for CoulombHiggs 2.0, Sep 2013, for CoulombHiggs 6.0, Dec 2020

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<<CoulombHiggs`

Example 5.1.1 in “From Black holes to quivers”

Mat={{0,3,-5,-4},{-3,0,4,4},{5,-4,0,7},{4,-4,-7,0}};
Cvec={21/10,3,-11/10,-4};
Nvec={1,1,1,1};
OmR[{1,1,1,1}]=OmS[{1,1,1,1},t]+(-1)^(kk+1)Sum[y^(-7kk+1+2i),{i,0,7kk-1}](OmS[{1,1,1,0},t]+Sum[y^(-kk+1+2i),{i,0,kk-1}]^2);

$QuiverMultiplier=kk;
$QuiverMultiplierAssumption={kk∈Integers,kk>0};
$QuiverNoLoop=False;
QPoinca=CoulombBranchFormula[Mat,Cvec,Nvec]

Simplify[Table[Series[DropFugacity[QPoinca]- OmR[{1,1,1,1}],{y,0,5}],{kk,1,3}]]

Example 5.1.2 in “From Black holes to quivers”

Mat={{0,15,-5,-10},{-15,0,20,2},{5,-20,0,35},{10,-2,-35,0}};
Cvec={2,3,-6,1};
Nvec={1,1,1,1};
OmR[{1,1,1,1}]=OmS[{1,1,1,1},t]-(1+(-1)^kk)/2(y^(3kk)-y^(-3kk)-3/2kk(y^2-1/y^2))/(y-1/y)^3+(1-(-1)^kk)/2(y^(3kk)-y^(-3kk)-3kk(y-1/y))/(y-1/y)^3;

$QuiverMultiplier=kk;
$QuiverMultiplierAssumption={kk∈Integers,kk>0};
QPoinca=CoulombBranchFormula[Mat,Cvec,Nvec]

Simplify[Table[Series[DropFugacity[QPoinca]-OmR[Nvec],{y,0,5}],{kk,1,10}]]

Example 5.2 in “From Black holes to quivers”

Mat=(({{0, 1, 0, 0, -1}, {-1, 0, 1, 0, 0}, {0, -1, 0, 1, -1}, {0, 0, -1, 0, 1}, {1, 0, 1, -1, 0}}));Cvec={1,1,1,1,-4};Nvec={1,1,1,1,1};
OmR[{1,1,1,1,1}]=OmS[{1,1,1,1,1},t]+(1+(-1)^kk)/2( (y^(kk/2)-y^(-kk/2))^2/(y-1/y)^2 OmS[{0,0,1,1,1},t]+(y^(2kk)-2 y^kk+2-2y^(-kk)+y^(-2kk)-kk^2/2(y-1/y)^2)/(y-1/y)^4)+(1-(-1)^kk)/2(-(y^kk+y^(-kk)-y-1/y)/(y-1/y)^2 OmS[{0,0,1,1,1},t]+(y^(2kk)+2+y^(-2kk)-(y+1/y)(y^kk+y^(-kk))-(kk^2-1)/2(y-1/y)^2)/(y-1/y)^4);

$QuiverMultiplier=kk;
$QuiverMultiplierAssumption={kk∈Integers,kk>0};
QPoinca=CoulombBranchFormula[Mat,Cvec,Nvec]

Simplify[Table[Series[DropFugacity[QPoinca]-OmR[Nvec],{y,0,5}],{kk,1,8}]]

Cyclic Abelian quivers

Avec={1,1,1,1};
Cvec={1,1,1,-3};

Avec={26,97,57,82,44,13};
Cvec={1,8,27,64,125,-225};

n=Length[Avec];
Mat=CyclicQuiverDSZ[Avec];

$QuiverMultiplier=kk;
$QuiverMultiplierAssumption={kk∈Integers,kk>0};
QPoinca=CoulombBranchFormula[Mat,Cvec,Nvec]

QHiggs=OmS[Nvec,t]+(-1)^($QuiverMultiplier Plus@@Avec)(y-1/y)^(1-n)y^(2$QuiverMultiplier Avec[[n]]-$QuiverMultiplier Plus@@Avec)Product[1-y^(2$QuiverMultiplier Avec[[i]]),{i,n-1}]

Table[Series[DropFugacity[QPoinca]-QHiggs,{y,0,0}],{kk,1,10}]/.OmS[gam_]->0

Simplify[CyclicQuiverOmS[Avec,y]]