Kronecker quiver

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

SetDirectory[NotebookDirectory[]];
<<CoulombHiggs`

ClearAll[m];
$QuiverNoLoop=True;
$QuiverMultiplier=m;
$QuiverMultiplierAssumption={m∈Integers,m>0};
Mat={{0,1},{-1,0}};

Recorded results for low dimension vector

OmR[{1,1},y_]:=(-1)^(m+1)(((y^(1-m)) (-1+(y^(2 m))))/(-1+(y^2)));
OmR[{1,2},y_]:=(((y^2) (-1-(y^2)+(y^(2-2 m))+(y^(2 m))))/((((-1+(y^2)))^2) (1+(y^2))));
OmR[{2,1},y_]:=(((y^2) (-1-(y^2)+(y^(2-2 m))+(y^(2 m))))/((((-1+(y^2)))^2) (1+(y^2))));
OmR[{2,2},y_]:=-(((y^(3-4 m)) (-1+(y^(4 m))) (2 (y^4)-(1+(((-1))^m)) (y^(2 m))+2 (y^(4 m))+2 (-1+(((-1))^m)) (y^(2+2 m))-(1+(((-1))^m)) (y^(4+2 m))))/(2 (((-1+(y^2)))^3) (((1+(y^2)))^2)));
OmR[{1,3},y_]:=(-1)^(m+1)(((y^(3-3 m)) (-1+(y^m)) (1+(y^m)) (-y+(y^m)) (y+(y^m)) (-(y^2)+(y^m)) ((y^2)+(y^m)))/((((-1+(y^2)))^3) (1+(y^2)) (1+(y^2)+(y^4))));
OmR[{1,4},y_]:=((y^4) ((y^2)+(y^4)+2 (y^6)+(y^8)+(y^10)+(y^(12-4 m))-(y^(6-2 m))-(y^(8-2 m))-(y^(10-2 m))-(y^(12-2 m))-(y^(2 m))+(y^(4 m))-(y^(2+2 m))-(y^(4+2 m))-(y^(6+2 m))))/((((-1+(y^2)))^4) (((1+(y^2)))^2) (1+(y^4)) (1+(y^2)+(y^4)));
OmR[{2,3},y_]:=(((y^4) ((y^(8-6 m))+(y^(6 m))+(((1+(y^2)))^4)-(y^(2 m)) (2+3 (y^2)+3 (y^4)+(y^6))-(y^(2-2 m)) (1+3 (y^2)+3 (y^4)+2 (y^6))))/((((-1+(y^2)))^4) (((1+(y^2)))^2) (1+(y^2)+(y^4))));
OmR[{2,4},y_]:=-((y^(5-8 m)) (((-1+(y^(2 m))))^2) (-(y^14)+(y^(8 m))+2 (y^(10 m))+(y^(12 m))+(y^(6 (2+m)))+(y^(4 (3+m)))-2 (y^(2 (7+m)))+2 (y^(6+4 m))+2 (y^(8+4 m))+3 (y^(10+4 m))-(y^(14+4 m))-(y^(2+6 m))-3 (y^(4+6 m))+3 (y^(10+6 m))-(y^(2+8 m))-3 (y^(4+8 m))-2 (y^(6+8 m))-2 (y^(8+8 m))))/((((-1+(y^2)))^5) (((1+(y^2)))^3) (1+(y^4)) (1+(y^2)+(y^4)));
OmR[{2,5},y_]:=((y^(6-10 m)) (-1+(y^m)) (1+(y^m)) (-y+(y^m)) (y+(y^m)) (-(y^2)+(y^m)) ((y^2)+(y^m)) (-(y^16)+(y^(14 m))-(y^(4 (3+m)))+(y^(4+10 m))+(y^(12 m)) (1+(y^2)+(y^4))-(y^(2 (6+m))) (1+(y^2)+(y^4))-(y^(8 m)) (1+3 (y^2)+4 (y^4)+5 (y^6)+5 (y^8)+4 (y^10)+2 (y^12)+(y^14))+(y^(2+6 m)) (1+(y^2) (1+(-1+y) y) (1+(y^2)) (1+y+(y^2)) (2+(y^4)+(y^6)))))/((((-1+(y^2)))^6) (((1+(y^2)))^3) (1+(y^4)) (1+(y^2)+(y^4)) (1+(y^2)+(y^4)+(y^6)+(y^8)));
OmR[{3,3},y_]:=(((((-y))^(5-9 m)) ((((y^2)-(y^(2 m))))^2) (((-1+(y^(2 m))))^2) (1+(y^(2 m))+(y^(4 m))) (-(y^8)+(y^(4 m))+(y^(6 m))-2 (y^(6+2 m))-(y^(8+2 m))+2 (y^(2+4 m))))/((((-1+(y^2)))^5) (((1+(y^2)))^2) (((1+(y^2)+(y^4)))^2)));
OmR[{3,4},y_]:=((y^6) (-1-7 (y^2)-21 (y^4)-39 (y^6)-52 (y^8)-52 (y^10)-39 (y^12)-21 (y^14)-7 (y^16)-(y^18)+(y^(18-12 m))-(y^(8-6 m))-2 (y^(10-6 m))-4 (y^(12-6 m))-4 (y^(14-6 m))-3 (y^(16-6 m))-2 (y^(18-6 m))-(y^(6-4 m))-(y^(8-4 m))-2 (y^(10-4 m))-(y^(12-4 m))-2 (y^(14-4 m))-(y^(16-4 m))-(y^(18-4 m))+(y^(2-2 m))+6 (y^(4-2 m))+15 (y^(6-2 m))+26 (y^(8-2 m))+32 (y^(10-2 m))+30 (y^(12-2 m))+21 (y^(14-2 m))+10 (y^(16-2 m))+3 (y^(18-2 m))+3 (y^(2 m))-(y^(4 m))-2 (y^(6 m))+(y^(12 m))-(y^(4 (3+m)))+26 (y^(2 (5+m)))+15 (y^(2 (6+m)))+6 (y^(2 (7+m)))+(y^(2 (8+m)))+10 (y^(2+2 m))+21 (y^(4+2 m))+30 (y^(6+2 m))+32 (y^(8+2 m))-(y^(2+4 m))-2 (y^(4+4 m))-(y^(6+4 m))-2 (y^(8+4 m))-(y^(10+4 m))-3 (y^(2+6 m))-4 (y^(4+6 m))-4 (y^(6+6 m))-2 (y^(8+6 m))-(y^(10+6 m))))/((((-1+(y^2)))^6) (((1+(y^2)))^3) (1+(y^4)) (((1+(y^2)+(y^4)))^2));
OmR[{3,5},y_]:=(-1)^(m+1)(-(y^(33-15 m))+(y^(25-9 m))+(y^(27-9 m))+(y^(29-9 m))+(y^(31-9 m))+(y^(33-9 m))+(y^(17-7 m))+2 (y^(19-7 m))+5 (y^(21-7 m))+7 (y^(23-7 m))+9 (y^(25-7 m))+8 (y^(27-7 m))+7 (y^(29-7 m))+4 (y^(31-7 m))+2 (y^(33-7 m))-(y^(15-5 m))-3 (y^(17-5 m))-6 (y^(19-5 m))-10 (y^(21-5 m))-13 (y^(23-5 m))-14 (y^(25-5 m))-12 (y^(27-5 m))-8 (y^(29-5 m))-4 (y^(31-5 m))-(y^(33-5 m))-(y^(11-3 m))-5 (y^(13-3 m))-13 (y^(15-3 m))-25 (y^(17-3 m))-38 (y^(19-3 m))-47 (y^(21-3 m))-49 (y^(23-3 m))-43 (y^(25-3 m))-31 (y^(27-3 m))-18 (y^(29-3 m))-8 (y^(31-3 m))-2 (y^(33-3 m))+2 (y^(9-m))+12 (y^(11-m))+34 (y^(13-m))+69 (y^(15-m))+108 (y^(17-m))+140 (y^(19-m))+150 (y^(21-m))+136 (y^(23-m))+102 (y^(25-m))+63 (y^(27-m))+29 (y^(29-m))+9 (y^(31-m))+(y^(33-m))+14 (y^(5 (3+m)))-2 (y^(7 (3+m)))+43 (y^(3 (5+m)))+(y^(5 (5+m)))-(y^(7+m))+38 (y^(3 (7+m)))-9 (y^(9+m))+5 (y^(3 (9+m)))-29 (y^(11+m))-63 (y^(13+m))-102 (y^(15+m))-136 (y^(17+m))-150 (y^(19+m))-140 (y^(21+m))-108 (y^(23+m))-69 (y^(25+m))-34 (y^(27+m))-12 (y^(29+m))-2 (y^(31+m))+2 (y^(7+3 m))+8 (y^(9+3 m))+18 (y^(11+3 m))+31 (y^(13+3 m))+49 (y^(17+3 m))+47 (y^(19+3 m))+25 (y^(23+3 m))+13 (y^(25+3 m))+(y^(29+3 m))+(y^(7+5 m))+4 (y^(9+5 m))+8 (y^(11+5 m))+12 (y^(13+5 m))+13 (y^(17+5 m))+10 (y^(19+5 m))+6 (y^(21+5 m))+3 (y^(23+5 m))-2 (y^(7+7 m))-4 (y^(9+7 m))-7 (y^(11+7 m))-8 (y^(13+7 m))-9 (y^(15+7 m))-7 (y^(17+7 m))-5 (y^(19+7 m))-(y^(23+7 m))-(y^(7+9 m))-(y^(9+9 m))-(y^(11+9 m))-(y^(13+9 m))-(y^(15+9 m))+(y^(7+15 m)))/((((-1+(y^2)))^7) (((1+(y^2)))^3) (1+(y^4)) (((1+(y^2)+(y^4)))^2) (1+(y^2)+(y^4)+(y^6)+(y^8)));
OmR[{4,4},y_]:=-((y^(7-16 m)) ((((y^2)-(y^(2 m))+(y^(4 m))-(y^(2+2 m))))^2) (-(y^20)-(y^(10 m))-(y^(12 m))+(y^(16 m))+3 (y^(18 m))+3 (y^(20 m))+2 (y^(22 m))+(y^(24 m))+2 (y^(10 (1+m)))+4 (y^(12 (1+m)))+4 (y^(14 (1+m)))-(y^(16 (1+m)))+3 (y^(6 (2+m)))+3 (y^(8 (2+m)))-4 (y^(6 (3+m)))-3 (y^(4 (4+m)))-3 (y^(4 (5+m)))-2 (y^(2 (9+m)))-2 (y^(20+2 m))-4 (y^(18+4 m))+(y^(8+6 m))+2 (y^(10+6 m))-3 (y^(16+6 m))-3 (y^(20+6 m))+(y^(4+8 m))+2 (y^(6+8 m))+8 (y^(8+8 m))+10 (y^(10+8 m))+12 (y^(12+8 m))+6 (y^(14+8 m))-2 (y^(18+8 m))-(y^(20+8 m))-2 (y^(2+10 m))-5 (y^(4+10 m))-4 (y^(6+10 m))-(y^(8+10 m))+5 (y^(12+10 m))+4 (y^(14+10 m))+2 (y^(16+10 m))-2 (y^(2+12 m))-5 (y^(4+12 m))-4 (y^(6+12 m))-4 (y^(8+12 m))+4 (y^(14+12 m))+5 (y^(16+12 m))+2 (y^(18+12 m))+(y^(20+12 m))-2 (y^(4+14 m))-4 (y^(6+14 m))-5 (y^(8+14 m))-2 (y^(10+14 m))+(y^(12+14 m))+5 (y^(16+14 m))+2 (y^(18+14 m))+(y^(20+14 m))+2 (y^(2+16 m))-3 (y^(4+16 m))-6 (y^(6+16 m))-12 (y^(8+16 m))-10 (y^(10+16 m))-8 (y^(12+16 m))-2 (y^(14+16 m))+4 (y^(2+18 m))+3 (y^(4+18 m))-3 (y^(8+18 m))-2 (y^(10+18 m))-(y^(12+18 m))+4 (y^(2+20 m))+3 (y^(4+20 m))+2 (y^(2+22 m))))/((((-1+(y^2)))^7) (((1+(y^2)))^4) (((1+(y^4)))^2) (((1+(y^2)+(y^4)))^2));

Evaluate the Poincare polynomial

Nvec={2,2};
Cvec=-AttractorFI[Mat,Nvec];

QCoulomb=Simplify[CoulombBranchFormula[Mat,Cvec,Nvec],$QuiverMultiplierAssumption]

QHiggs=Simplify[HiggsBranchFormula[Mat,Cvec,Nvec],$QuiverMultiplierAssumption]

Simplify[QCoulomb-OmR[Nvec,y],{m∈Integers,y>0}]

Simplify[QHiggs-OmR[Nvec,y],{m∈Integers,y>0}]

Simplify[Table[Series[QCoulomb-OmR[Nvec,y],{y,0,10}],{m,1,5}]]