Example sheet 2 : Anti-de Sitter Virasoro algebra in Einstein gravity and Topologically Massive Gravity

(* This example uses RG&TC-Code and SurfaceCharges-Code *)

Virasoro algebras of AdS in Einstein gravity

(* ads3 in Poincare patch *)

metric = {{-r^2,0,0},{0,1/(r^ 2),0},{0,0,r^ 2}};

(* BTZ *)(* Based on hep-th/9204099 with a factor 1/8 more in the Lagrangian, and so, in the definition of the charges *)

metric = {{-(r^2/l^2 -8 M ),0,- 8 J/2},{0,1/(r^2/l^2 -8  M +64 J^2/4/r^2),0},{- 8  J/2,0,r^ 2}};

l=1

coord = {t,r,θ};

RGtensors[metric,coord,{0,0}];

(* Virasoro generators *)

Lm =  I{T[t,θ],-r D[Th[t,θ],θ],Th[t,θ]} /. {T[t,θ] -> 1/2/I Exp[I m (t+θ)],Th[t,θ] -> 1/2/I Exp[I m (t+θ)],D[Th[t,θ],θ] ->D[ 1/2/I Exp[I m (t+θ)],θ]}
Ln = Lm /.{m-> n};
Lmn = Lm /.{m-> (m+n)};

Factor[Simplify[ I  Commutator[Lm,Ln] -   (m-n)  Lmn]]

cLgLn = Simplify[Liediffdd[metric,coord,Ln]]

(* Poisson bracket *)

PreCentralCharge =Series[Simplify[Series[ ChargesEinstein[Lm,cLgLn,-1,{1,2}]/GN,{r,∞,0}]],{r,∞,0}]

(* Check that the Iyer-Wald and Barnich-Brandt charges are the same *)

Series[ChargeSuppl,{r,∞,0}]

(* Value of the central charge *)

CentralCharge =12 Coefficient[  Factor[Normal[2π I PreCentralCharge/.{n-> -m}]],m^3]

(* Computation of the BTZ charges *)

dmetric = D[metric,M]δM + D[metric,J]δJ

Simplify[2 Pi ChargesEinstein[{1,0,0},dmetric,-1,{1,2}],{r>0}]

Simplify[2 Pi ChargesEinstein[{0,0,-1},dmetric,-1,{1,2}],{r>0}]

(* Virasoro charges *)

Integrate[Normal[Series[Simplify[ ChargesEinstein[Ln,dmetric,-1,{1,2}],{r>0}],{r,∞,0}]],{θ,0,2 Pi}]

Limit[% ,{n -> 0}]

(* So L0 > 0 for all regular black holes *)

(* Second Virasoro generators *)

Lm =  I{T[t,θ],-r D[Th[t,θ],θ],Th[t,θ]} /. {T[t,θ] -> 1/2/I Exp[I m (t-θ)],Th[t,θ] ->- 1/2/I Exp[I m (t-θ)],D[Th[t,θ],θ] ->D[ -1/2/I Exp[I m (t-θ)],θ]}
Ln = Lm /.{m-> n};
Lmn = Lm /.{m-> (m+n)};

Factor[Simplify[ I  Commutator[Lm,Ln] -   (m-n)  Lmn]]

cLgLn = Simplify[Liediffdd[metric,coord,Ln]]

PreCentralCharge =Series[Simplify[Series[ ChargesEinstein[Lm,cLgLn,-1,{1,2}]/GN,{r,∞,0}]],{r,∞,0}]

(* Check that the Iyer-Wald and Barnich-Brandt charges are the same *)

Series[ChargeSuppl,{r,∞,0}]

(* Value of the central charge *)

CentralCharge = Collect[12/(m^3)  Factor[Normal[2π I PreCentralCharge/.{n-> -m,J -> 0, M -> 0}]],m]

(* Virasoro charges *)

Integrate[Normal[Series[Simplify[ ChargesEinstein[Ln,dmetric,-1,{1,2}],{r>0}],{r,∞,0}]],{θ,0,2 Pi}]

Limit[% ,{n -> 0}]

Virasoro algebras of AdS in Topologically Massive gravity

(* ads3 in Poincare coordinates *)

metric = {{-r^2,0,0},{0,1/(r^ 2),0},{0,0,r^ 2}};

(* BTZ *)(* Based on hep-th/9204099 with a factor 1/8 more in the Lagrangian, and so, in the definition of the charges *)

metric = {{-(r^2/l^2 -8 M ),0,- 8 J/2},{0,1/(r^2/l^2 -8  M +64 J^2/4/r^2),0},{- 8  J/2,0,r^ 2}};

l=1

coord = {t,r,θ};

RGtensors[metric,coord,{0,0}];

Cotton[-1]

EOM = Simplify[EUd-1/l^2 {{1,0,0},{0,1,0},{0,0,1}} ]

(* Virasoro generators *)

Lm =  I{T[t,θ],-r D[Th[t,θ],θ],Th[t,θ]} /. {T[t,θ] -> 1/2/I Exp[I m (t+θ)],Th[t,θ] -> 1/2/I Exp[I m (t+θ)],D[Th[t,θ],θ] ->D[ 1/2/I Exp[I m (t+θ)],θ]}
Ln = Lm /.{m-> n};
Lmn = Lm /.{m-> (m+n)};

Factor[Simplify[ I  Commutator[Lm,Ln] -   (m-n)  Lmn]]

cLgLn = Simplify[Liediffdd[metric,coord,Ln]]

PreCentralCharge =Series[Simplify[Series[ ChargesTMG[Lm,cLgLn,μ,{-1,1},{1,2}]/GN,{r,∞,0}]],{r,∞,0}]

(* Check that the Iyer-Wald and Barnich-Brandt charges are the same *)

Series[ChargeSuppl,{r,∞,0}]

(* Value of the central charge *)

CentralCharge = Expand[Collect[12/(m^3)  Factor[Normal[2π I PreCentralCharge/.{n-> -m,J -> 0,M-> 0}]],m]]

(* Second Virasoro generators *)

L2m =  I{T[t,θ],-r D[Th[t,θ],θ],Th[t,θ]} /. {T[t,θ] -> 1/2/I Exp[I m (t-θ)],Th[t,θ] ->- 1/2/I Exp[I m (t-θ)],D[Th[t,θ],θ] ->D[ -1/2/I Exp[I m (t-θ)],θ]}
L2n = L2m /.{m-> n};
L2mn = L2m /.{m-> (m+n)};

Factor[Simplify[ I  Commutator[L2m,L2n] -   (m-n)  L2mn]]

cLgL2n = Simplify[Liediffdd[metric,coord,L2n]]

PreCentralCharge2 =Series[Simplify[Series[ ChargesTMG[L2m,cLgL2n,μ,{-1,1},{1,2}]/GN,{r,∞,0}]],{r,∞,0}]

(* Check that the Iyer-Wald and Barnich-Brandt charges are the same *)

Series[ChargeSuppl,{r,∞,0}]

(* Value of the central charge *)

CentralCharge2 = Expand[Collect[12/(m^3)  Factor[Normal[2π I PreCentralCharge2/.{n-> -m,J -> 0, M -> 0}]],m]]

(* Check charges *)

dmetric = D[metric,M]δM + D[metric,J]δJ

Simplify[2 Pi ChargesTMG[{1,0,0},dmetric,μ,{-1,1},{1,2}],{r>0}]

Simplify[2 Pi ChargesTMG[{0,0,-1},dmetric,μ,{-1,1},{1,2}],{r>0}]

Integrate[Normal[Series[Simplify[ ChargesTMG[Ln,dmetric,μ,{-1,1},{1,2}],{r>0}],{r,∞,0}]],{θ,0,2 Pi}]

Limit[% ,{n -> 0}]

Expand[Integrate[Normal[Series[Simplify[ ChargesTMG[L2n,dmetric,μ,{-1,1},{1,2}],{r>0}],{r,∞,0}]]/.{n -> 0},{θ,0,2 Pi}] ]

(* So L0 > 0 and bar L0  > 0 for BH for μ > 1 *)