Pensieve Header: An attempt on an explicit formula for the Alexander blobs vertex.

SetDirectory["C:\\drorbn\\AcademicPensieve\\2011-08\\w-Computations"];
<< "AlexanderBlobs-Program.m"

(b[r[1,3],#]& /@ Table[r[j,0], {j,3}]) /. Diag[hs_, ars___] :> hs Diag[1, ars]

B[n_, expr_] := Module[
{bra},
Transpose[Table[
bra=b[expr,r[i,0]] /. Diag[hs_, ars___] :> hs Diag[1, ars];
Coefficient[bra, r[#,0]]& /@ Range[n],
{i,n}
]]
];
B[3, r[1,3]] // MatrixForm

B3R[i_,j_] := MatrixExp[B[3, r[i,j]]];
B3R[1,3] // MatrixForm

MatrixForm /@ Simplify[{B3R[1,2].B3R[1,3], B3R[1,3].B3R[1,2]}]

MatrixForm /@ Simplify[{B3R[1,2].B3R[1,3].B3R[2,3], B3R[2,3].B3R[1,3].B3R[1,2]}]

v=Diag[f12, ar[1,2]]+Diag[f21, ar[2,1]];
B[3,v] // MatrixForm

MatrixExp[a{{-1,1},{1,-1}}] // MatrixForm

(B3V=MatrixExp[B[3,v]]) // MatrixForm

Simplify[B3V.MatrixExp[B[3,r[1,3]+r[2,3]]]] // MatrixForm

Simplify[B3R[1,3].B3R[2,3].B3V] // MatrixForm

Simplify[B3V.MatrixExp[B[3,r[1,3]+r[2,3]]] == B3R[1,3].B3R[2,3].B3V] // MatrixForm

Solve[{(((E^(-(1+f12) h[1]-(1+f21) h[2])) (-1+(E^(h[1]+h[2]))) ((-f12+f21) h[2]+(E^(f12 h[1]+f21 h[2])) f12 (h[1]+h[2])) h[3])/((h[1]+h[2]) (f12 h[1]+f21 h[2]))) == (((E^(-h[1]-h[2])) (-1+(E^(h[1]))) h[3])/(h[1])), f21==0}, {f12, f21}]

Simplify[((Log[(((E^(-h[1]-f21 h[2])) (-1+(E^(h[1]+h[2]))) h[2])/((-1+(E^(h[2]))) (h[1]+h[2])))])/(h[1])) /. {f21 -> 0, h[1] -> x, h[2] -> y}]

(((Log[(((E^(-h[1]-f21 h[2])) (-1+(E^(h[1]+h[2]))) h[2])/((-1+(E^(h[2]))) (h[1]+h[2])))])/(h[1])) /. {f21-> 0, h[1]-> h x, h[2] -> h y})+O[h]^8

A result from Recycling/AlexanderBlobs-Results-110819.nb:

((-(1/2) Diag[1,ar[1,2]]+(1/24) Diag[h[1],ar[1,2]]-((Diag[((h[1])^3),ar[1,2]])/2880)+((Diag[((h[1])^5),ar[1,2]])/181440)-((Diag[((h[1])^7),ar[1,2]])/9676800)+(1/12) Diag[h[2],ar[1,2]]-(1/720) Diag[((h[1])^2) h[2],ar[1,2]]+((Diag[((h[1])^4) h[2],ar[1,2]])/30240)-((Diag[((h[1])^6) h[2],ar[1,2]])/1209600)-(1/480) Diag[h[1] ((h[2])^2),ar[1,2]]+((Diag[((h[1])^3) ((h[2])^2),ar[1,2]])/12096)-((Diag[((h[1])^5) ((h[2])^2),ar[1,2]])/345600)-(1/720) Diag[((h[2])^3),ar[1,2]]+((Diag[((h[1])^2) ((h[2])^3),ar[1,2]])/9072)-((Diag[((h[1])^4) ((h[2])^3),ar[1,2]])/172800)+((Diag[h[1] ((h[2])^4),ar[1,2]])/12096)-((Diag[((h[1])^3) ((h[2])^4),ar[1,2]])/138240)+((Diag[((h[2])^5),ar[1,2]])/30240)-((Diag[((h[1])^2) ((h[2])^5),ar[1,2]])/172800)-((Diag[h[1] ((h[2])^6),ar[1,2]])/345600)-((Diag[((h[2])^7),ar[1,2]])/1209600)) /. Diag[hs_,ar[1,2]] :> hs /. {h[1]-> h x, h[2] -> h y}) +O[h]^8

MatrixForm @ Simplify[
MatrixExp[B[2,r[1,2]]].MatrixExp[B[2,v]].MatrixExp[B[2, -1/2(r[1,2]+r[2,1])]] /. {h[1]->x, h[2]->y}
]

Simplify[MatrixExp[B[2,v /. {f12 -> g12, f21->g21}]] /. {h[1]->x, h[2]->y}] // MatrixForm

A result from AlexanderBlobs-Results.nb:

pertres=(-(1/4) Diag[1,ar[1,2]]+(1/4) Diag[1,ar[2,1]]+(1/32) Diag[h[1],ar[1,2]]+(5/96) Diag[h[1],ar[2,1]]-((19 Diag[((h[1])^3),ar[1,2]])/46080)-((11 Diag[((h[1])^3),ar[2,1]])/15360)+((37 Diag[((h[1])^5),ar[1,2]])/4644864)+((13 Diag[((h[1])^5),ar[2,1]])/1161216)-((10663 Diag[((h[1])^7),ar[1,2]])/66886041600)-((457 Diag[((h[1])^7),ar[2,1]])/2675441664)+((243091 Diag[((h[1])^9),ar[1,2]])/75676778496000)+((1368247 Diag[((h[1])^9),ar[2,1]])/529737449472000)+(5/96) Diag[h[2],ar[1,2]]+(1/32) Diag[h[2],ar[2,1]]-((169 Diag[((h[1])^2) h[2],ar[1,2]])/138240)-((187 Diag[((h[1])^2) h[2],ar[2,1]])/138240)+((1279 Diag[((h[1])^4) h[2],ar[1,2]])/34836480)+((2669 Diag[((h[1])^4) h[2],ar[2,1]])/69672960)-((205357 Diag[((h[1])^6) h[2],ar[1,2]])/200658124800)-((186139 Diag[((h[1])^6) h[2],ar[2,1]])/200658124800)+((42750073 Diag[((h[1])^8) h[2],ar[1,2]])/1589212348416000)+((33851563 Diag[((h[1])^8) h[2],ar[2,1]])/1589212348416000)-((187 Diag[h[1] ((h[2])^2),ar[1,2]])/138240)-((169 Diag[h[1] ((h[2])^2),ar[2,1]])/138240)+((125 Diag[((h[1])^3) ((h[2])^2),ar[1,2]])/1741824)+((2539 Diag[((h[1])^3) ((h[2])^2),ar[2,1]])/34836480)-((592097 Diag[((h[1])^5) ((h[2])^2),ar[1,2]])/200658124800)-((80849 Diag[((h[1])^5) ((h[2])^2),ar[2,1]])/28665446400)+((20640551 Diag[((h[1])^7) ((h[2])^2),ar[1,2]])/198651543552000)+((18273881 Diag[((h[1])^7) ((h[2])^2),ar[2,1]])/198651543552000)-((11 Diag[((h[2])^3),ar[1,2]])/15360)-((19 Diag[((h[2])^3),ar[2,1]])/46080)+((2539 Diag[((h[1])^2) ((h[2])^3),ar[1,2]])/34836480)+((125 Diag[((h[1])^2) ((h[2])^3),ar[2,1]])/1741824)-((27571 Diag[((h[1])^4) ((h[2])^3),ar[1,2]])/5733089280)-((949967 Diag[((h[1])^4) ((h[2])^3),ar[2,1]])/200658124800)+((1305949 Diag[((h[1])^6) ((h[2])^3),ar[1,2]])/5518098432000)+((306281 Diag[((h[1])^6) ((h[2])^3),ar[2,1]])/1379524608000)+((2669 Diag[h[1] ((h[2])^4),ar[1,2]])/69672960)+((1279 Diag[h[1] ((h[2])^4),ar[2,1]])/34836480)-((949967 Diag[((h[1])^3) ((h[2])^4),ar[1,2]])/200658124800)-((27571 Diag[((h[1])^3) ((h[2])^4),ar[2,1]])/5733089280)+((4387321 Diag[((h[1])^5) ((h[2])^4),ar[1,2]])/12612796416000)+((90369791 Diag[((h[1])^5) ((h[2])^4),ar[2,1]])/264868724736000)+((13 Diag[((h[2])^5),ar[1,2]])/1161216)+((37 Diag[((h[2])^5),ar[2,1]])/4644864)-((80849 Diag[((h[1])^2) ((h[2])^5),ar[1,2]])/28665446400)-((592097 Diag[((h[1])^2) ((h[2])^5),ar[2,1]])/200658124800)+((90369791 Diag[((h[1])^4) ((h[2])^5),ar[1,2]])/264868724736000)+((4387321 Diag[((h[1])^4) ((h[2])^5),ar[2,1]])/12612796416000)-((186139 Diag[h[1] ((h[2])^6),ar[1,2]])/200658124800)-((205357 Diag[h[1] ((h[2])^6),ar[2,1]])/200658124800)+((306281 Diag[((h[1])^3) ((h[2])^6),ar[1,2]])/1379524608000)+((1305949 Diag[((h[1])^3) ((h[2])^6),ar[2,1]])/5518098432000)-((457 Diag[((h[2])^7),ar[1,2]])/2675441664)-((10663 Diag[((h[2])^7),ar[2,1]])/66886041600)+((18273881 Diag[((h[1])^2) ((h[2])^7),ar[1,2]])/198651543552000)+((20640551 Diag[((h[1])^2) ((h[2])^7),ar[2,1]])/198651543552000)+((33851563 Diag[h[1] ((h[2])^8),ar[1,2]])/1589212348416000)+((42750073 Diag[h[1] ((h[2])^8),ar[2,1]])/1589212348416000)+((1368247 Diag[((h[2])^9),ar[1,2]])/529737449472000)+((243091 Diag[((h[2])^9),ar[2,1]])/75676778496000));

pf12=(pertres/. {Diag[hs_,ar[1,2]] :> hs, Diag[hs_,ar[2,1]] :> 0} /. {h[1]-> h x, h[2] -> h y}) + O[h]^11

pf21=(pertres/. {Diag[hs_,ar[1,2]] :> 0, Diag[hs_,ar[2,1]] :> hs} /. {h[1]-> h x, h[2] -> h y}) + O[h]^11

Simplify[(((E^(-(1+f12) h[1]-(1+f21) h[2])) (-1+(E^(h[1]+h[2]))) ((-f12+f21) h[2]+(E^(f12 h[1]+f21 h[2])) f12 (h[1]+h[2])) h[3])/((h[1]+h[2]) (f12 h[1]+f21 h[2]))) == (((E^(-h[1]-h[2])) (-1+(E^(h[1]))) h[3])/(h[1])) /. {h[1]-> h x, h[2] -> h y} /. {f12->pf12, f21->pf21}]

Simplify[((((E^(-(1+f12) x-f21 y)) ((f12-f21) x y+(E^(f12 x+f21 y)) f21 y (x+y)+(E^((x+y)/2)) x (f12 x+f21 y)))/((x+y) (f12 x+f21 y)))==(((E^(-g12 x-g21 y)) g12 x+g21 y)/(g12 x+g21 y))) /. {x->h x, y->h y} /.{
f12->pf12, f21->pf21, g12 -> (pf21 /. {x->y, y->x}), g21 -> (pf12 /. {x->y, y->x})
}]