(* ::Package:: *)

(************************************************************************)
(* This file was generated automatically by the Mathematica front end.  *)
(* It contains Initialization cells from a Notebook file, which         *)
(* typically will have the same name as this file except ending in      *)
(* ".nb" instead of ".m".                                               *)
(*                                                                      *)
(* This file is intended to be loaded into the Mathematica kernel using *)
(* the package loading commands Get or Needs.  Doing so is equivalent   *)
(* to using the Evaluate Initialization Cells menu command in the front *)
(* end.                                                                 *)
(*                                                                      *)
(* DO NOT EDIT THIS FILE.  This entire file is regenerated              *)
(* automatically each time the parent Notebook file is saved in the     *)
(* Mathematica front end.  Any changes you make to this file will be    *)
(* overwritten.                                                         *)
(************************************************************************)



(* ::Input::Initialization:: *)
 \[Epsilon] /: \[Epsilon]^d_. /;d>1 :=0;


(* ::Input::Initialization:: *)
CF[\[ScriptCapitalE]_] := ExpandDenominator@ExpandNumerator@Together[Expand[\[ScriptCapitalE]] //. E^x_ E^y_:>E^(x+y) /. E^x_:>E^CF[x]];
K\[Delta]/:Subscript[K\[Delta], i_,j_]:=If[i===j,1,0];


(* ::Input::Initialization:: *)
\[DoubleStruckCapitalE]/:\[DoubleStruckCapitalE][L1_,Q1_,P1_]\[Congruent]\[DoubleStruckCapitalE][L2_,Q2_,P2_]:=CF[L1==L2]\[And]CF[Q1==Q2]\[And]CF[Normal[P1-P2]==0];
\[DoubleStruckCapitalE]/:\[DoubleStruckCapitalE][L1_,Q1_,P1_]\[DoubleStruckCapitalE][L2_,Q2_,P2_]:=\[DoubleStruckCapitalE][L1+L2,Q1+Q2,P1*P2];


(* ::Input::Initialization:: *)
{SuperStar[t],SuperStar[b],SuperStar[y],SuperStar[a],SuperStar[x],SuperStar[z]}={\[Tau],\[Beta],\[Eta],\[Alpha],\[Xi],\[Zeta]}; {SuperStar[\[Tau]],SuperStar[\[Beta]],SuperStar[\[Eta]],SuperStar[\[Alpha]],SuperStar[\[Xi]],SuperStar[\[Zeta]]}={t,b,y,a,x,z}; SuperStar[(Subscript[u_, i_])]:=Subscript[(SuperStar[u]), i];


(* ::Input::Initialization:: *)
\!\(\*SubscriptBox[\(Zip\), \({}\)]\)[P_]:=P; \!\(\*SubscriptBox[\(Zip\), \({\[Zeta]_, \[Zeta]s___}\)]\)[P_] := (Expand[P//\!\(\*SubscriptBox[\(Zip\), \({\[Zeta]s}\)]\)] /. f_. \[Zeta]^d_.:>\!\(
\*SubscriptBox[\(\[PartialD]\), \({
\*SuperscriptBox[\(\[Zeta]\), \(*\)], d}\)]f\)) /. SuperStar[\[Zeta]]->0


(* ::Input::Initialization:: *)
Subscript[QZip, \[Zeta]s_List,simp_]@\[DoubleStruckCapitalE][L_,Q_,P_] := Module[{\[Zeta],z,zs,c,ys,\[Eta]s,qt,zrule,Q1,Q2},
zs=Table[SuperStar[\[Zeta]],{\[Zeta],\[Zeta]s}];
c=Q/.Alternatives@@(\[Zeta]s\[Union]zs)->0;
ys=Table[\!\(
\*SubscriptBox[\(\[PartialD]\), \(\[Zeta]\)]\((Q /. Alternatives @@ zs -> 0)\)\),{\[Zeta],\[Zeta]s}];
\[Eta]s=Table[\!\(
\*SubscriptBox[\(\[PartialD]\), \(z\)]\((Q /. Alternatives @@ \[Zeta]s -> 0)\)\),{z,zs}];
qt=Inverse@Table[Subscript[K\[Delta], z,SuperStar[\[Zeta]]]-\!\(
\*SubscriptBox[\(\[PartialD]\), \(z, \[Zeta]\)]Q\),{\[Zeta],\[Zeta]s},{z,zs}];
zrule=Thread[zs->qt.(zs+ys)];
Q2=(Q1=c+\[Eta]s.zs/.zrule)/.Alternatives@@zs->0;
simp /@ \[DoubleStruckCapitalE][L,Q2,Det[qt]E^-Q2 Subscript[Zip, \[Zeta]s][E^Q1 (P/.zrule)]] ];
Subscript[QZip, \[Zeta]s_List]:=Subscript[QZip, \[Zeta]s,CF];


(* ::Input::Initialization:: *)
Subscript[LZip, \[Zeta]s_List,simp_]@\[DoubleStruckCapitalE][L_,Q_,P_] := Module[{\[Zeta],z,zs,c,ys,\[Eta]s,\[Eta]srule,lt,zrule,L1,L2,Q1,Q2},
(*Print["LZipping"];*)
(*\[Zeta]s//Echo;*)
zs=Table[SuperStar[\[Zeta]],{\[Zeta],\[Zeta]s}];
c=L/.Alternatives@@(\[Zeta]s\[Union]zs)->0;
ys=Table[\!\(
\*SubscriptBox[\(\[PartialD]\), \(\[Zeta]\)]\((L /. Alternatives @@ zs -> 0)\)\),{\[Zeta],\[Zeta]s}];
\[Eta]s=Table[\!\(
\*SubscriptBox[\(\[PartialD]\), \(z\)]\((L /. Alternatives @@ \[Zeta]s -> 0)\)\),{z,zs}];
\[Eta]srule= Table[SuperStar[z]->\!\(
\*SubscriptBox[\(\[PartialD]\), \(z\)]\((L /. Alternatives @@ \[Zeta]s -> 0)\)\),{z,zs}];(*NEW*)
lt=Inverse@Table[Subscript[K\[Delta], z,SuperStar[\[Zeta]]]-\!\(
\*SubscriptBox[\(\[PartialD]\), \(z, \[Zeta]\)]L\),{\[Zeta],\[Zeta]s},{z,zs}];
zrule=Thread[zs->lt.(zs+ys)];
L1=c+\[Eta]s.zs/.zrule;
L2=L1/.Alternatives@@zs->0;
Q1=Q/.Join[zrule,\[Eta]srule];
Q2=Q1/.Join[{Alternatives@@zs->0},\[Eta]srule];
(*{Det[lt]\[ExponentialE]^(-L2-Q2),Subscript[Zip, \[Zeta]s][\[ExponentialE]^(L1+Q1)(P/.zrule)]}//Echo;*)
simp /@ \[DoubleStruckCapitalE][L2,Q2,Det[lt]E^(-L2-Q2) Subscript[Zip, \[Zeta]s][E^(L1+Q1) (P/.zrule)]] ];
Subscript[LZip, \[Zeta]s_List]:=Subscript[LZip, \[Zeta]s,CF];


(* ::Input::Initialization:: *)
\!\(\*SubscriptBox[\(Bind\), \({}\)]\)[L_,R_]:=L R;
\!\(\*SubscriptBox[\(Bind\), \({is__}\)]\)[L_\[DoubleStruckCapitalE],R_\[DoubleStruckCapitalE]] := Module[{n},
Times[
L/.Table[Subscript[(v:t|b|a|x|y), i]->Subscript[v, n@i],{i,{is}}],
R/.Table[Subscript[(v:\[Tau]|\[Beta]|\[Alpha]|\[Xi]|\[Eta]), i]->Subscript[v, n@i],{i,{is}}]
] // Subscript[LZip, Flatten@Table[{Subscript[\[Tau], n@i],Subscript[\[Beta], n@i],Subscript[a, n@i]},{i,{is}}]] // Subscript[QZip, Flatten@Table[{Subscript[\[Xi], n@i],Subscript[y, n@i]},{i,{is}}]] ];
Subscript[B, l_List]:=Subscript[Bind, l]; Subscript[B, is___]:=\!\(\*SubscriptBox[\(Bind\), \({is}\)]\);


(* ::Input::Initialization:: *)
(*Hopf algebra on the a,x side*)
Subscript[tam, i_,j_->k_]:=\[DoubleStruckCapitalE][(Subscript[\[Alpha], i]+Subscript[\[Alpha], j])Subscript[a, k],(\[DoubleStruckE]^-Subscript[\[Alpha], j] Subscript[\[Xi], i]+Subscript[\[Xi], j])Subscript[x, k],1]
Subscript[ta\[CapitalDelta], i_->j_,k_]:=\[DoubleStruckCapitalE][Subscript[\[Alpha], i](Subscript[a, j]+Subscript[a, k]),Subscript[\[Xi], i](Subscript[x, j]+Subscript[x, k]),1+\[Epsilon] Subscript[\[Xi], i] Subscript[x, k]( -Subscript[a, j]+1/2 Subscript[\[Xi], i] Subscript[x, j])]
Subscript[taS, i_] :=\[DoubleStruckCapitalE][-Subscript[\[Alpha], i] Subscript[a, i],- \[DoubleStruckE]^ Subscript[\[Alpha], i] Subscript[\[Xi], i] Subscript[x, i],1- \[Epsilon] \[DoubleStruckE]^ Subscript[\[Alpha], i] Subscript[\[Xi], i] Subscript[x, i](Subscript[a, i]+ 1/2 \[DoubleStruckE]^ Subscript[\[Alpha], i] Subscript[\[Xi], i] Subscript[x, i])]
Subscript[taSi, i_]:=\[DoubleStruckCapitalE][-Subscript[\[Alpha], i] Subscript[a, i],-\[DoubleStruckE]^ Subscript[\[Alpha], i] Subscript[\[Xi], i] Subscript[x, i],1- \[Epsilon] \[DoubleStruckE]^ Subscript[\[Alpha], i] Subscript[\[Xi], i] Subscript[x, i](Subscript[a, i]-1+ 1/2 \[DoubleStruckE]^ Subscript[\[Alpha], i] Subscript[\[Xi], i] Subscript[x, i])]
(*Hopf algebra on the y,b side*)
Subscript[tbm, i_,j_->k_]:=\[DoubleStruckCapitalE][(Subscript[\[Beta], i]+Subscript[\[Beta], j])Subscript[b, k],(Subscript[\[Eta], i]+Subscript[\[Eta], j])Subscript[y, k],1- \[Epsilon]  Subscript[\[Eta], j] Subscript[y, k] Subscript[\[Beta], i]]
Subscript[tb\[CapitalDelta], i_->j_,k_]:=\[DoubleStruckCapitalE][Subscript[\[Beta], i](Subscript[b, j]+Subscript[b, k]),Subscript[\[Eta], i](\[DoubleStruckE]^-Subscript[b, k] Subscript[y, j]+Subscript[y, k]),1+1/2 \[Epsilon] Subscript[\[Eta], i]^2 Subscript[y, j] Subscript[y, k] \[DoubleStruckE]^-Subscript[b, k]]
Subscript[tbS, i_]:=\[DoubleStruckCapitalE][-Subscript[\[Beta], i] Subscript[b, i],-\[DoubleStruckE]^ Subscript[b, i] Subscript[\[Eta], i] Subscript[y, i],1-\[Epsilon] \[DoubleStruckE]^ Subscript[b, i] Subscript[\[Eta], i] Subscript[y, i]( Subscript[\[Beta], i]+ 1/2 \[DoubleStruckE]^Subscript[b, i] Subscript[\[Eta], i] Subscript[y, i])]
Subscript[tbSi, i_]:=\[DoubleStruckCapitalE][-Subscript[\[Beta], i] Subscript[b, i],-\[DoubleStruckE]^Subscript[b, i] Subscript[\[Eta], i] Subscript[y, i],1-\[Epsilon] \[DoubleStruckE]^ Subscript[b, i] Subscript[\[Eta], i] Subscript[y, i]( Subscript[\[Beta], i]-1+ 1/2 \[DoubleStruckE]^Subscript[b, i] Subscript[\[Eta], i] Subscript[y, i])]


(* ::Input::Initialization:: *)
Subscript[tP, i_,j_]:=\[DoubleStruckCapitalE][Subscript[\[Beta], i] Subscript[\[Alpha], j],Subscript[\[Eta], i] Subscript[\[Xi], j],1+1/4 \[Epsilon] Subscript[\[Eta], i]^2 Subscript[\[Xi], j]^2]


(* ::Input::Initialization:: *)
Subscript[tdm, i_,j_->k_]:= Subscript[tdm, i,j->k]=Simplify/@Expand/@((\[DoubleStruckCapitalE][Subscript[\[Beta], i] Subscript[b, i] + Subscript[\[Alpha], j] Subscript[a, j],Subscript[\[Eta], i] Subscript[y, i]+Subscript[\[Xi], j] Subscript[x, j],1](Subscript[ta\[CapitalDelta], i->h1,h2]~Subscript[B, h2]~Subscript[ta\[CapitalDelta], h2->h2,h3])(Subscript[tb\[CapitalDelta], j->t1,t2]~Subscript[B, t2]~Subscript[tb\[CapitalDelta], t2->t2,t3]))~Subscript[B, h3]~Subscript[taSi, h3]~Subscript[B, t1,h3]~(Subscript[tP, t1,h3])~Subscript[B, t3,h1]~(Subscript[tP, t3,h1])~Subscript[B, h2,j,i,t2]~(Subscript[tam, h2,j->k] Subscript[tbm, i,t2->-k])/.{\[DoubleStruckE]->E,Subscript[u_, -k]:>Subscript[u, k]})


(* ::Input::Initialization:: *)
(*Deriving tdS using tdm*)
Subscript[tdS, i_]:=Subscript[tdS, i]=((Subscript[tbSi, 1] Subscript[taS, 2])~Subscript[B, 1,2]~Subscript[tdm, 2,1->i])/.{\[DoubleStruckE]->E,Subscript[z_, 1|2]->Subscript[z, i]}


(* ::Input::Initialization:: *)
Subscript[tdm, i_,j_->k_]:=\[DoubleStruckCapitalE][Subscript[a, k] (Subscript[\[Alpha], i]+Subscript[\[Alpha], j])+Subscript[b, k] (Subscript[\[Beta], i]+Subscript[\[Beta], j]),Subscript[y, k] (Subscript[\[Eta], i]+E^-Subscript[\[Alpha], i] Subscript[\[Eta], j])+E^-Subscript[b, k] (-1+E^Subscript[b, k]) Subscript[\[Eta], j] Subscript[\[Xi], i]+Subscript[x, k] (E^-Subscript[\[Alpha], j] Subscript[\[Xi], i]+Subscript[\[Xi], j]),1/4 E^(-2 Subscript[b, k]-Subscript[\[Alpha], i]-Subscript[\[Alpha], j]) (2 E^Subscript[b, k] \[Epsilon] Subscript[y, k] Subscript[\[Eta], j] (-2 E^(Subscript[b, k]+Subscript[\[Alpha], j]) Subscript[\[Beta], i]+(2 E^Subscript[b, k] Subscript[x, k]+E^Subscript[\[Alpha], j] (-3+E^Subscript[b, k]) Subscript[\[Eta], j]) Subscript[\[Xi], i])+E^Subscript[\[Alpha], i] (2 E^Subscript[b, k] \[Epsilon] Subscript[x, k] Subscript[\[Xi], i] (-2 E^Subscript[b, k] Subscript[\[Beta], j]+(-3+E^Subscript[b, k]) Subscript[\[Eta], j] Subscript[\[Xi], i])+E^Subscript[\[Alpha], j] (4 E^(2 Subscript[b, k])+4 E^Subscript[b, k] \[Epsilon] Subscript[a, k] Subscript[\[Eta], j] Subscript[\[Xi], i]+(3-4 E^Subscript[b, k]+E^(2 Subscript[b, k])) \[Epsilon] \!\(\*SubsuperscriptBox[\(\[Eta]\), \(j\), \(2\)]\) \!\(\*SubsuperscriptBox[\(\[Xi]\), \(i\), \(2\)]\))))]
Subscript[td\[CapitalDelta], i_->j_,k_]:=\[DoubleStruckCapitalE][Subscript[a, j] Subscript[\[Alpha], i]+Subscript[a, k] Subscript[\[Alpha], i]+Subscript[b, j] Subscript[\[Beta], i]+Subscript[b, k] Subscript[\[Beta], i],\[DoubleStruckE]^-Subscript[b, j] (\[DoubleStruckE]^Subscript[b, j] Subscript[y, j] Subscript[\[Eta], i]+Subscript[y, k] Subscript[\[Eta], i]+\[DoubleStruckE]^Subscript[b, j] Subscript[x, j] Subscript[\[Xi], i]+\[DoubleStruckE]^Subscript[b, j] Subscript[x, k] Subscript[\[Xi], i]),1/2 \[DoubleStruckE]^-Subscript[b, j] (2 \[DoubleStruckE]^Subscript[b, j]+\[Epsilon] Subscript[y, j] Subscript[y, k] \!\(\*SubsuperscriptBox[\(\[Eta]\), \(i\), \(2\)]\)-2 \[DoubleStruckE]^Subscript[b, j] \[Epsilon] Subscript[a, j] Subscript[x, k] Subscript[\[Xi], i]+\[DoubleStruckE]^Subscript[b, j] \[Epsilon] Subscript[x, j] Subscript[x, k] \!\(\*SubsuperscriptBox[\(\[Xi]\), \(i\), \(2\)]\))]
Subscript[tdS, i_]:=\[DoubleStruckCapitalE][-Subscript[a, i] Subscript[\[Alpha], i]-Subscript[b, i] Subscript[\[Beta], i],E^-Subscript[b, i] (-E^(2 Subscript[b, i]+Subscript[\[Alpha], i]) Subscript[y, i] Subscript[\[Eta], i]-E^(Subscript[b, i]+Subscript[\[Alpha], i]) Subscript[x, i] Subscript[\[Xi], i]-E^(Subscript[b, i]+Subscript[\[Alpha], i]) Subscript[\[Eta], i] Subscript[\[Xi], i]+E^(2 Subscript[b, i]+Subscript[\[Alpha], i]) Subscript[\[Eta], i] Subscript[\[Xi], i]),1/4 E^(-2 Subscript[b, i]) (4 E^(2 Subscript[b, i])+4 E^(3 Subscript[b, i]+Subscript[\[Alpha], i]) \[Epsilon] Subscript[y, i] Subscript[\[Eta], i]-4 E^(3 Subscript[b, i]+Subscript[\[Alpha], i]) \[Epsilon] Subscript[y, i] Subscript[\[Beta], i] Subscript[\[Eta], i]-2 E^(4 Subscript[b, i]+2 Subscript[\[Alpha], i]) \[Epsilon] \!\(\*SubsuperscriptBox[\(y\), \(i\), \(2\)]\) \!\(\*SubsuperscriptBox[\(\[Eta]\), \(i\), \(2\)]\)-4 E^(2 Subscript[b, i]+Subscript[\[Alpha], i]) \[Epsilon] Subscript[a, i] Subscript[x, i] Subscript[\[Xi], i]-4 E^(2 Subscript[b, i]+Subscript[\[Alpha], i]) \[Epsilon] Subscript[x, i] Subscript[\[Beta], i] Subscript[\[Xi], i]+4 E^(2 Subscript[b, i]+Subscript[\[Alpha], i]) \[Epsilon] Subscript[\[Eta], i] Subscript[\[Xi], i]-4 E^(3 Subscript[b, i]+Subscript[\[Alpha], i]) \[Epsilon] Subscript[\[Eta], i] Subscript[\[Xi], i]+4 E^(3 Subscript[b, i]+Subscript[\[Alpha], i]) \[Epsilon] Subscript[a, i] Subscript[\[Eta], i] Subscript[\[Xi], i]-4 E^(3 Subscript[b, i]+2 Subscript[\[Alpha], i]) \[Epsilon] Subscript[x, i] Subscript[y, i] Subscript[\[Eta], i] Subscript[\[Xi], i]-4 E^(2 Subscript[b, i]+Subscript[\[Alpha], i]) \[Epsilon] Subscript[\[Beta], i] Subscript[\[Eta], i] Subscript[\[Xi], i]+4 E^(3 Subscript[b, i]+Subscript[\[Alpha], i]) \[Epsilon] Subscript[\[Beta], i] Subscript[\[Eta], i] Subscript[\[Xi], i]-2 E^(3 Subscript[b, i]+2 Subscript[\[Alpha], i]) \[Epsilon] Subscript[y, i] \!\(\*SubsuperscriptBox[\(\[Eta]\), \(i\), \(2\)]\) Subscript[\[Xi], i]+6 E^(4 Subscript[b, i]+2 Subscript[\[Alpha], i]) \[Epsilon] Subscript[y, i] \!\(\*SubsuperscriptBox[\(\[Eta]\), \(i\), \(2\)]\) Subscript[\[Xi], i]-2 E^(2 Subscript[b, i]+2 Subscript[\[Alpha], i]) \[Epsilon] \!\(\*SubsuperscriptBox[\(x\), \(i\), \(2\)]\) \!\(\*SubsuperscriptBox[\(\[Xi]\), \(i\), \(2\)]\)-2 E^(2 Subscript[b, i]+2 Subscript[\[Alpha], i]) \[Epsilon] Subscript[x, i] Subscript[\[Eta], i] \!\(\*SubsuperscriptBox[\(\[Xi]\), \(i\), \(2\)]\)+6 E^(3 Subscript[b, i]+2 Subscript[\[Alpha], i]) \[Epsilon] Subscript[x, i] Subscript[\[Eta], i] \!\(\*SubsuperscriptBox[\(\[Xi]\), \(i\), \(2\)]\)-E^(2 Subscript[b, i]+2 Subscript[\[Alpha], i]) \[Epsilon] \!\(\*SubsuperscriptBox[\(\[Eta]\), \(i\), \(2\)]\) \!\(\*SubsuperscriptBox[\(\[Xi]\), \(i\), \(2\)]\)+4 E^(3 Subscript[b, i]+2 Subscript[\[Alpha], i]) \[Epsilon] \!\(\*SubsuperscriptBox[\(\[Eta]\), \(i\), \(2\)]\) \!\(\*SubsuperscriptBox[\(\[Xi]\), \(i\), \(2\)]\)-3 E^(4 Subscript[b, i]+2 Subscript[\[Alpha], i]) \[Epsilon] \!\(\*SubsuperscriptBox[\(\[Eta]\), \(i\), \(2\)]\) \!\(\*SubsuperscriptBox[\(\[Xi]\), \(i\), \(2\)]\))]


(* ::Input::Initialization:: *)
Subscript[\[DoubleStruckE], q_,k_][x_]:=\[DoubleStruckE]^(\!\(
\*UnderoverscriptBox[\(\[Sum]\), \(j = 1\), \(k + 1\)]\ 
\*FractionBox[\(
\*SuperscriptBox[\((1 - q)\), \(j\)] 
\*SuperscriptBox[\(x\), \(j\)]\), \(j \((1 - 
\*SuperscriptBox[\(q\), \(j\)])\)\)]\))
Subscript[tR, i_,j_]:=\[DoubleStruckCapitalE][Subscript[b, i] Subscript[a, j],Subscript[y, i] Subscript[x, j],1- \[Epsilon] 1/4 \!\(\*SubsuperscriptBox[\(y\), \(i\), \(2\)]\) \!\(\*SubsuperscriptBox[\(x\), \(j\), \(2\)]\) ](*First two terms in Faddeev-Quesne formula*)


(* ::Input::Initialization:: *)
Subscript[\!\(\*OverscriptBox[\(tR\), \(_\)]\), i_,j_]:=\[DoubleStruckCapitalE][-Subscript[a, j] Subscript[b, i],-E^Subscript[b, i] Subscript[x, j] Subscript[y, i],1-E^Subscript[b, i] \[Epsilon] Subscript[a, j] Subscript[x, j] Subscript[y, i]-3/4 E^(2 Subscript[b, i]) \[Epsilon] \!\(\*SubsuperscriptBox[\(x\), \(j\), \(2\)]\) \!\(\*SubsuperscriptBox[\(y\), \(i\), \(2\)]\)]


(* ::Input::Initialization:: *)
Subscript[b2t, i_]:=\[DoubleStruckCapitalE][Subscript[\[Alpha], i] Subscript[a, i]-Subscript[\[Beta], i] Subscript[t, i],Subscript[\[Xi], i] Subscript[x, i]+Subscript[\[Eta], i] Subscript[y, i],1+\[Epsilon] Subscript[\[Beta], i] Subscript[a, i]]
Subscript[t2b, i_]:=\[DoubleStruckCapitalE][Subscript[\[Alpha], i] Subscript[a, i]-Subscript[\[Tau], i] Subscript[b, i],Subscript[\[Xi], i] Subscript[x, i]+Subscript[\[Eta], i] Subscript[y, i],1+\[Epsilon] Subscript[\[Tau], i] Subscript[a, i]]


(* ::Input::Initialization:: *)
Subscript[dm, i_,j_->k_]:=\[DoubleStruckCapitalE][Subscript[a, k] Subscript[\[Alpha], i]+Subscript[a, k] Subscript[\[Alpha], j]+Subscript[t, k] Subscript[\[Tau], i]+Subscript[t, k] Subscript[\[Tau], j],E^(-Subscript[\[Alpha], i]-Subscript[\[Alpha], j]) (E^(Subscript[\[Alpha], i]+Subscript[\[Alpha], j]) Subscript[y, k] Subscript[\[Eta], i]+E^Subscript[\[Alpha], j] Subscript[y, k] Subscript[\[Eta], j]+E^Subscript[\[Alpha], i] Subscript[x, k] Subscript[\[Xi], i]+E^(Subscript[\[Alpha], i]+Subscript[\[Alpha], j]) Subscript[\[Eta], j] Subscript[\[Xi], i]-E^(Subscript[t, k]+Subscript[\[Alpha], i]+Subscript[\[Alpha], j]) Subscript[\[Eta], j] Subscript[\[Xi], i]+E^(Subscript[\[Alpha], i]+Subscript[\[Alpha], j]) Subscript[x, k] Subscript[\[Xi], j]),1/4 E^(-Subscript[\[Alpha], i]-Subscript[\[Alpha], j]) (4 E^(Subscript[\[Alpha], i]+Subscript[\[Alpha], j])+8 E^(Subscript[t, k]+Subscript[\[Alpha], i]+Subscript[\[Alpha], j]) \[Epsilon] Subscript[a, k] Subscript[\[Eta], j] Subscript[\[Xi], i]+4 \[Epsilon] Subscript[x, k] Subscript[y, k] Subscript[\[Eta], j] Subscript[\[Xi], i]+2 E^Subscript[\[Alpha], j] \[Epsilon] Subscript[y, k] \!\(\*SubsuperscriptBox[\(\[Eta]\), \(j\), \(2\)]\) Subscript[\[Xi], i]-6 E^(Subscript[t, k]+Subscript[\[Alpha], j]) \[Epsilon] Subscript[y, k] \!\(\*SubsuperscriptBox[\(\[Eta]\), \(j\), \(2\)]\) Subscript[\[Xi], i]+2 E^Subscript[\[Alpha], i] \[Epsilon] Subscript[x, k] Subscript[\[Eta], j] \!\(\*SubsuperscriptBox[\(\[Xi]\), \(i\), \(2\)]\)-6 E^(Subscript[t, k]+Subscript[\[Alpha], i]) \[Epsilon] Subscript[x, k] Subscript[\[Eta], j] \!\(\*SubsuperscriptBox[\(\[Xi]\), \(i\), \(2\)]\)+E^(Subscript[\[Alpha], i]+Subscript[\[Alpha], j]) \[Epsilon] \!\(\*SubsuperscriptBox[\(\[Eta]\), \(j\), \(2\)]\) \!\(\*SubsuperscriptBox[\(\[Xi]\), \(i\), \(2\)]\)-4 E^(Subscript[t, k]+Subscript[\[Alpha], i]+Subscript[\[Alpha], j]) \[Epsilon] \!\(\*SubsuperscriptBox[\(\[Eta]\), \(j\), \(2\)]\) \!\(\*SubsuperscriptBox[\(\[Xi]\), \(i\), \(2\)]\)+3 E^(2 Subscript[t, k]+Subscript[\[Alpha], i]+Subscript[\[Alpha], j]) \[Epsilon] \!\(\*SubsuperscriptBox[\(\[Eta]\), \(j\), \(2\)]\) \!\(\*SubsuperscriptBox[\(\[Xi]\), \(i\), \(2\)]\))]
Subscript[d\[CapitalDelta], i_->j_,k_]:=\[DoubleStruckCapitalE][Subscript[a, j] Subscript[\[Alpha], i]+Subscript[a, k] Subscript[\[Alpha], i]+Subscript[t, j] Subscript[\[Tau], i]+Subscript[t, k] Subscript[\[Tau], i],Subscript[y, j] Subscript[\[Eta], i]+E^Subscript[t, j] Subscript[y, k] Subscript[\[Eta], i]+Subscript[x, j] Subscript[\[Xi], i]+Subscript[x, k] Subscript[\[Xi], i],1/2 (2-2 E^Subscript[t, j] \[Epsilon] Subscript[a, j] Subscript[y, k] Subscript[\[Eta], i]+E^Subscript[t, j] \[Epsilon] Subscript[y, j] Subscript[y, k] \!\(\*SubsuperscriptBox[\(\[Eta]\), \(i\), \(2\)]\)-2 \[Epsilon] Subscript[a, j] Subscript[x, k] Subscript[\[Xi], i]+\[Epsilon] Subscript[x, j] Subscript[x, k] \!\(\*SubsuperscriptBox[\(\[Xi]\), \(i\), \(2\)]\))]
Subscript[dS, i_]:=\[DoubleStruckCapitalE][-Subscript[a, i] Subscript[\[Alpha], i]-Subscript[t, i] Subscript[\[Tau], i],E^-Subscript[t, i] (-E^Subscript[\[Alpha], i] Subscript[y, i] Subscript[\[Eta], i]-E^(Subscript[t, i]+Subscript[\[Alpha], i]) Subscript[x, i] Subscript[\[Xi], i]+E^Subscript[\[Alpha], i] Subscript[\[Eta], i] Subscript[\[Xi], i]-E^(Subscript[t, i]+Subscript[\[Alpha], i]) Subscript[\[Eta], i] Subscript[\[Xi], i]),1/4 E^(-2 Subscript[t, i]) (4 E^(2 Subscript[t, i])+4 E^(Subscript[t, i]+Subscript[\[Alpha], i]) \[Epsilon] Subscript[y, i] Subscript[\[Eta], i]-4 E^(Subscript[t, i]+Subscript[\[Alpha], i]) \[Epsilon] Subscript[a, i] Subscript[y, i] Subscript[\[Eta], i]-2 E^(2 Subscript[\[Alpha], i]) \[Epsilon] \!\(\*SubsuperscriptBox[\(y\), \(i\), \(2\)]\) \!\(\*SubsuperscriptBox[\(\[Eta]\), \(i\), \(2\)]\)-4 E^(2 Subscript[t, i]+Subscript[\[Alpha], i]) \[Epsilon] Subscript[a, i] Subscript[x, i] Subscript[\[Xi], i]-4 E^(Subscript[t, i]+Subscript[\[Alpha], i]) \[Epsilon] Subscript[\[Eta], i] Subscript[\[Xi], i]+4 E^(2 Subscript[t, i]+Subscript[\[Alpha], i]) \[Epsilon] Subscript[\[Eta], i] Subscript[\[Xi], i]+8 E^(Subscript[t, i]+Subscript[\[Alpha], i]) \[Epsilon] Subscript[a, i] Subscript[\[Eta], i] Subscript[\[Xi], i]-4 E^(Subscript[t, i]+2 Subscript[\[Alpha], i]) \[Epsilon] Subscript[x, i] Subscript[y, i] Subscript[\[Eta], i] Subscript[\[Xi], i]+6 E^(2 Subscript[\[Alpha], i]) \[Epsilon] Subscript[y, i] \!\(\*SubsuperscriptBox[\(\[Eta]\), \(i\), \(2\)]\) Subscript[\[Xi], i]-2 E^(Subscript[t, i]+2 Subscript[\[Alpha], i]) \[Epsilon] Subscript[y, i] \!\(\*SubsuperscriptBox[\(\[Eta]\), \(i\), \(2\)]\) Subscript[\[Xi], i]-2 E^(2 Subscript[t, i]+2 Subscript[\[Alpha], i]) \[Epsilon] \!\(\*SubsuperscriptBox[\(x\), \(i\), \(2\)]\) \!\(\*SubsuperscriptBox[\(\[Xi]\), \(i\), \(2\)]\)+6 E^(Subscript[t, i]+2 Subscript[\[Alpha], i]) \[Epsilon] Subscript[x, i] Subscript[\[Eta], i] \!\(\*SubsuperscriptBox[\(\[Xi]\), \(i\), \(2\)]\)-2 E^(2 Subscript[t, i]+2 Subscript[\[Alpha], i]) \[Epsilon] Subscript[x, i] Subscript[\[Eta], i] \!\(\*SubsuperscriptBox[\(\[Xi]\), \(i\), \(2\)]\)-3 E^(2 Subscript[\[Alpha], i]) \[Epsilon] \!\(\*SubsuperscriptBox[\(\[Eta]\), \(i\), \(2\)]\) \!\(\*SubsuperscriptBox[\(\[Xi]\), \(i\), \(2\)]\)+4 E^(Subscript[t, i]+2 Subscript[\[Alpha], i]) \[Epsilon] \!\(\*SubsuperscriptBox[\(\[Eta]\), \(i\), \(2\)]\) \!\(\*SubsuperscriptBox[\(\[Xi]\), \(i\), \(2\)]\)-E^(2 Subscript[t, i]+2 Subscript[\[Alpha], i]) \[Epsilon] \!\(\*SubsuperscriptBox[\(\[Eta]\), \(i\), \(2\)]\) \!\(\*SubsuperscriptBox[\(\[Xi]\), \(i\), \(2\)]\))]


(* ::Input::Initialization:: *)
Subscript[R, i_,j_]:=\[DoubleStruckCapitalE][-Subscript[a, j] Subscript[t, i],Subscript[x, j] Subscript[y, i],1/4 (4+4 \[Epsilon] Subscript[a, i] Subscript[a, j]-\[Epsilon] \!\(\*SubsuperscriptBox[\(x\), \(j\), \(2\)]\) \!\(\*SubsuperscriptBox[\(y\), \(i\), \(2\)]\))]
Subscript[\!\(\*OverscriptBox[\(R\), \(_\)]\), i_,j_]:=\[DoubleStruckCapitalE][Subscript[a, j] Subscript[t, i],-E^-Subscript[t, i] Subscript[x, j] Subscript[y, i],1/4 E^(-2 Subscript[t, i]) (4 E^(2 Subscript[t, i])-4 E^(2 Subscript[t, i]) \[Epsilon] Subscript[a, i] Subscript[a, j]-4 E^Subscript[t, i] \[Epsilon] Subscript[a, i] Subscript[x, j] Subscript[y, i]-4 E^Subscript[t, i] \[Epsilon] Subscript[a, j] Subscript[x, j] Subscript[y, i]-3 \[Epsilon] \!\(\*SubsuperscriptBox[\(x\), \(j\), \(2\)]\) \!\(\*SubsuperscriptBox[\(y\), \(i\), \(2\)]\))]


(* ::Input::Initialization:: *)
Subscript[u, i_]:=Subscript[R, 1,2]~Subscript[B, 1]~Subscript[dS, 1]~Subscript[B, 1,2]~Subscript[dm, 2,1->i]
Subscript[ui, i_]:=Subscript[R, 1,2]~Subscript[B, 2]~Subscript[dS, 2]~Subscript[B, 2]~Subscript[dS, 2]~Subscript[B, 1,2]~Subscript[dm, 2,1->i]


(* ::Input::Initialization:: *)
Subscript[CC, i_]:=\[DoubleStruckCapitalE][0,0,\[DoubleStruckE]^( 1/2 Subscript[t, i]) (1-\[Epsilon] Subscript[a, i])]
Subscript[\!\(\*OverscriptBox[\(CC\), \(_\)]\), i_]:=\[DoubleStruckCapitalE][0,0,\[DoubleStruckE]^( -1/2 Subscript[t, i]) (1+\[Epsilon] Subscript[a, i])]


(* ::Input::Initialization:: *)
Subscript[Kink, i_]:=\[DoubleStruckCapitalE][-Subscript[a, i] Subscript[t, i],Subscript[x, i] Subscript[y, i],1/4 E^(-(Subscript[t, i]/2)) (4+4 \[Epsilon] Subscript[a, i]+4 \[Epsilon] \!\(\*SubsuperscriptBox[\(a\), \(i\), \(2\)]\)-\[Epsilon] \!\(\*SubsuperscriptBox[\(x\), \(i\), \(2\)]\) \!\(\*SubsuperscriptBox[\(y\), \(i\), \(2\)]\))]
Subscript[\!\(\*OverscriptBox[\(Kink\), \(_\)]\), j_]:=\[DoubleStruckCapitalE][Subscript[a, j] Subscript[t, j],-E^-Subscript[t, j] Subscript[x, j] Subscript[y, j],1/4 E^(-2 Subscript[t, j]) (4 E^((5 Subscript[t, j])/2)-4 E^((5 Subscript[t, j])/2) \[Epsilon] Subscript[a, j]-4 E^((5 Subscript[t, j])/2) \[Epsilon] \!\(\*SubsuperscriptBox[\(a\), \(j\), \(2\)]\)-8 E^((3 Subscript[t, j])/2) \[Epsilon] Subscript[a, j] Subscript[x, j] Subscript[y, j]-3 E^(Subscript[t, j]/2) \[Epsilon] \!\(\*SubsuperscriptBox[\(x\), \(j\), \(2\)]\) \!\(\*SubsuperscriptBox[\(y\), \(j\), \(2\)]\))]



