Bootstrapping 𝒲-algebras

Chiral Algebras for the 21st Century
UC Davis Spring School
April 2019

Load K. Thielemans’ OPEdefs package

Quit[]

SetDirectory[NotebookDirectory[]];
<<OPEDefs`
Prettify[expression_]:=DeleteDuplicates[DeleteCases[Flatten[Simplify[expression]],0]]

?OPEdefsHelp

Heisenberg

Bosonic[j]

OPE[j,j]:=MakeOPE[(1/(((z-w))^2))+Ord[z,w,0]]

OPEJacobi[j,j,j]

Family of Virasoro vectors

Tj[λ_]:=(1/2)NO[j,j]+λ j';

OPEToSeries[OPE[Tj[λ],Tj[λ]]]

Expand[2*OPEPole[4][Tj[λ],Tj[λ]]/.One->1]

Virasoro

Bosonic[T]

OPE[T,T]:=MakeOPE[((cc/2)/(((z-w))^4))+((2T[w])/(((z-w))^2))+((T'[w])/(z-w))+Ord[z,w,0]]

OPEJacobi[T,T,T]

Bootstrapping 𝒲[2,3]

Bosonic[W3]

OPE[T,W3]:=MakeOPE[((3W3)/(((z-w))^2))+((W3')/((z-w)))+Ord[z,w,0]];
OPE[W3,W3]:=MakeOPE[(π/(((z-w))^6))+ ((α1 T)/(((z-w))^4))+((α2 T'+β1 W3)/(((z-w))^3))+((α3 T''+β2 W3')/(((z-w))^2))+((α4 T'''+β3 W3'')/((z-w)))+Ord[z,w,0]]

Flatten[OPEJacobi[T,T,W3]]
Flatten[OPEJacobi[T,W3,T]]
Flatten[OPEJacobi[W3,T,T]]

Prettify[Flatten[

OPEJacobi[T,W3,W3]]

]

Solve[

CoefficientList[

Flatten[OPEJacobi[T,W3,W3]],

{T,T',T'',W3,W3',One}

]==0]

Try again!

Bosonic[W3]

OPE[T,W3]:=MakeOPE[((3W3)/(((z-w))^2))+((W3')/((z-w)))+Ord[z,w,0]];
OPE[W3,W3]:=MakeOPE[(π/(((z-w))^6))+ ((α1 T)/(((z-w))^4))+((α2 T'+β1 W3)/(((z-w))^3))+((α3 T''+β2 W3'+γ1 NO[T,T])/(((z-w))^2))+((α4 T'''+β3 W3''+γ2 NO[T',T]+γ3 NO[T,W])/((z-w)))+Ord[z,w,0]]

Flatten[OPEJacobi[T,T,W3]]
Flatten[OPEJacobi[T,W3,T]]
Flatten[OPEJacobi[W3,T,T]]

Prettify[CoefficientList[

(OPEJacobi[T,W3,W3]~Join~OPEJacobi[W3,T,W3]~Join~OPEJacobi[W3,W3,T]),

{T,T',T'',Derivative[3][T],NO[T,T],NO[T,W],NO[T,W'],NO[W,T],NO[W',T],NO[T',T],NO[W,T'],W3,W3',W3'',One}
]]//TableForm

Solve[%==0]

Simplify[

OPEJacobi[W3,W3,W3]/.

{α1->((6 π)/cc),α2->((3 π)/cc),α3->((9 (-2+cc) π)/(2 cc (22+5 cc))),α4->(((-2+cc) π)/(cc (22+5 cc))),β1->0,β2->0,β3->0,γ1->((96 π)/(cc (22+5 cc))),γ2->((96 π)/(cc (22+5 cc))),γ3->0}
]