Generalized*Gauge*Theories* in*arbitrary*dimensions N.K. Watabiki gauge*field gauge*parameter derivative curvature gauge*trans. Chern;Simons Topological* Yang;Mills Yang;Mills
= Quantization of Abelian 2-dim. BF theory d 2 x[ϵ µν φ µ ω ν + b µ ω µ i c µ µ c] M 2 d!! = dv BF = B(d! +! 2 ) (4-dim.) φ A sφ A s µ φ A sφ A φ 0 ϵ µν ν c 0 ω ν ν c 0 ϵ νρ ρ c c 0 iω µ 0 c ib 0 iφ b 0 µ c 0 on shell N=2 twisted SUSY invariance s 2 = {s, s} = s 2 = {s µ,s ν } =0, {s, s µ } = i µ, { s, s µ } = iϵ µν ν. S off-shell AQBF = d 2 x[ϵ µν φ µ ω ν + b µ ω µ i c µ µ c iλρ] M 2 = d 2 xs s 1 M 2 2 ϵµν s µ s ν ( i cc). off shel N=2 twisted SUSY invariance Kato, N.K. Uchida 03 φ A sφ A s µ φ A sφ A φ iρ ϵ µν ν c 0 ω ν ν c iϵ µν λ ϵ νρ ρ c c 0 iω µ 0 c ib 0 iφ b 0 µ c iρ λ ϵ µν µ ω ν 0 µ ω µ ρ 0 µ φ ϵ µν ν b 0
Gauge Theory on the Random Lattice Form Simplex Gauge Theory + Gravity Boson Fermion? SUSY? U l =(e! ) l A = dv +[A, V] ( l = dual link) N.K. Watabiki 91
3-dim. Chern- Simon Gravity Ponzano-Regge gravity N.K., H.B.Nielsen & N.Sato (1999)
4-dim. BF Gravity (N.K., K.Sato & Uchida (2000)) S LBF (F a = e d!+!2 + )
S = Tr 1 2 AQA + 1 3 A3 = 1S 1 + is i + js j + ks k 2 A = QV +[A, V] =1 1 + i ˆ0 + j A 0 + k Â1 1. Q 2 =0 2. {! Q, } = Q, [! Q, +] =Q + 3. Tr( + 0 + )=Tr( 0 + +), Tr( +) =Tr( + ), Tr( 0 )= Tr( 0 )
i 2 = j 2 = k 2 = 1, ij = ji = k, jk = kj = i, ki = ik = j S GCS = Tr 1 2 AQA + 1 3 A3 A = QV +[A, V] (Q = jd) a + b + c = odd or even [q(a, b, c), (a 0,b 0,c 0 )]=0 A = q(1, 1, 1) (1,1,1) + q(1, 0, 0) (1,0,0) + q(0, 1, 0) (0,1,0) + q(0, 0, 1) (0,0,1) = 1 (1,1,1) + i (1,0,0) + j (0,1,0) + k (0,0,1) 2 V = q(0, 0, 0) (0,0,0) + q(0, 1, 1) (0,1,1) + q(1, 0, 1) (1,0,1) + q(1, 1, 0) (1,1,0) = 1 (0,0,0) + i (0,1,1) + j (1,0,1) + k (1,1,0) 2 + AA 0 = A 0 A 2 + AV = VA 2 VV 0 = V 0 V 2 + =
(a,b,c)(x) c =0, 1 (0,0,c)(x) Âc(x) ( = direct sum of bosonic even forms, (0,1,c)(x) A c (x) = direct sum of bosonic odd forms, (1,0,c)(x) ˆc(x) = direct sum of fermionic even forms, (1,1,c)(x) c (x) = direct sum of fermionic odd forms d = dx µ µ A Q = q(0, 1, 0)d = jd A = A B T B =(1 B 1 + i ˆB 0 + ja B 0 + kâb 1 )T B V = V B T B =(1â B 0 + ia B 1 + jˆ B 1 + k B 0 )T B
S 1 = S i = S j = Tr Tr Tr apple apple 1(dA 0 + A 2 0 + Â2 1 + ˆ2 0 )+Â1(d ˆ0 +[A 0, ˆ0]) + 1 3 ˆ0(dA 0 + A 2 0 + Â2 1 2 1)  1 (d 1 + {A 0, 1 }) apple 1 2 A 1 0dA 0 3 A3 0 + 1 +[A 0, Â1]) 2Â1(dÂ1 3 1, 1 3 ˆ3 0, + 1 2 ˆ0(d ˆ0 +[A 0, ˆ0]) + 1 2 1 (d 1 + {A 0, 1 }) ˆ0{ 1, Â1} apple S k = Tr  1 (da + A 2 + ˆ2 2 1 1( 0 1) 3Â3 1 1 (d ˆ0 +[A 0, ˆ0]) 0 0 A = dâ +[A 0, â 0 ]+{Â1,a 1 } +[ 1, ˆ 1 ] { ˆ0, 0 0 0, 0] 0 },  = da {A +[Â1, â 0 ]+[ 1, 0 ]+{ ˆ0, 1 1 0,a 1 } ˆ 1 }, 1 = dˆ 1 [A 0, ˆ 1 ] [Â1, 0 ]+[ 1, â 0 ] [ ˆ0,a 1 ], ˆ0 = d 0 + {A 0, 0 } {Â1, ˆ 1 } +[ 1,a 1 ]+[ˆ0, â 0 ]
c {A 0, 0 } = A B 0 C 0 [T B,T C ], { ˆ0, ˆ 1 } = ˆB 0 ˆ C 1 [T B,T C ], {Â1,a 1 } = ÂB 1 a C 1 [T B,T C ] A = 1 1 + i ˆ0 + ja 0 + kâ1 = 1( (1) 1 + (3) 1 + )+i( (0) 0 + (2) 0 + (4) 0 + ) + j(! (1) 0 + (3) 0 + )+k( (0) 1 + B (2) 1 + H (4) 1 + ), V = 1â 0 + ia 1 + jˆ 1 + k 0 = 1(v (0) 0 + b (2) 0 + h (4) 0 + )+i(u (1) 1 + U (3) 1 + ) + j( (0) 1 + (2) 1 + (4) 1 )+k( (1) 0 + (3) 0 + ) (i) c i : i-form (0) 1,!(1) 0,B(2) 1, (3) 0,H(4) 1,.. (i) c i : i-form v (0) 0,u(1) 1,b(2) 0,U(3) 1,h(4) 0,..
=0, =0 S2 k ( = 0) = S j 3 ( = 0) = S4 k ( = 0) = Tr 1(d! 0 +! 2 2 0) 1B 1 apple 1 Tr 2! 1 0d! 0 3!3 2 0 + 1 (db 1 +[! 0,B 1 ]) 0 1, apple Tr B 1 (d! 0 +! 0) 2 1(d 0 + {! 0, 0 } + B1 2 2 H 1 1 1 =[ 1,v 0 ],! 0 = dv 0 +[! 0,v 0 ]+{ 1,u 1 }, B 1 = du 1 {! 0,u 1 } +[ 1,b 0 ]+[B 1,v 0 ], 0 = db 0 +[! 0,b 0 ]+{ 1,U 1 } + {B 1,u 1 } +[ 0,v 0 ] H 1 = du 1 {! 0,U 1 } +[ 1,h 0 ]+[B 1,b 0 ] { 0,u 1 } +[H 1,v 0 ]
S0 k = S j 1 = S2 k = S j 3 = apple Tr apple 1 Tr apple Tr apple Tr 1( ˆ(0) 0 )2 1 3 2 1 (d 1 +[! 0, 1]) ˆ(0) 0 { (1) 3 1 1(d! 0 +! 2 0 + { ˆ(0) 0, ˆ(2) 0 } ( (1) 1 2! 1 0d! 0 3!3 0 + ˆ(0) 0 1, 1} + 1 ˆ(0) 0 2 1 )2 ) (1) 1 (d ˆ(0) 0 +[! 0, ˆ(0) 0 ]) (d ˆ(0) 0 +[! 0, ˆ(0) 0 ]) 2 1 B 1 (d ˆ(2) 0 +[! 0, ˆ(2) 0 ] { (1) 1,B 1} { (3) 1, 1}) S k 4 = + 1 (db 1 +[! 0,B 1 ]) 0 ( 2 +(ˆ(0) 1 0 )2 ) apple Tr B 1 (d! 0 +! 0 2 + { ˆ(0) 0, ˆ(2) 0 } ( (1) 1 )2 ) 1(d 0 + {! 0, 0 } + B1 2 + { ˆ(0) 0, ˆ(4) 0 } { (1) 1, (3) 1 }) H 1(( ˆ(0) 0 )2 + 2 1) (1) 1 (d ˆ(2) 0 +[! 0, ˆ(2) 0 ]+[ 0, ˆ(0) 0 ]) (3) 1 [! 0, ˆ(0) 0 ]
1 =[ 1,v 0 ]+{ ˆ(0) 0, ˆ (0) 1 },! 0 = dv 0 +[! 0,v 0 ]+{ 1,u 1 } +[ (1) 1, ˆ (0) 1 ] { ˆ(0) 0, (1) 0 }, B 1 = du 1 {! 0,u 1 } +[ 1,b 0 ]+[B 1,v 0 ]+{ ˆ(0) 1 1 0, ˆ (2) 1 0 = db 0 +[! 0,b 0 ]+{ 1,U 1 } + {B 1,u 1 } +[ 0,v 0 ] } +[ (1) { ˆ(0) 0, (3) (1) 0 } +[ 1, ˆ (2) 1 ] { ˆ(2) 0, (1) (3) 0 } +[ 1, ˆ (0) 1 ], H 1 = du 1 {! 0,U 1 } +[ 1,h 0 ]+[B 1,b 0 ] { 0,u 1 } +[H 1,v 0 ] + { ˆ(0) 0, (4) 1 } +[ (1) 1, (3) 0 ]+{ ˆ(2) 0, ˆ (2) 1 } +[ (3) 1, (1) 0 1, (1) 0 ]+{ ˆ(4) 0, ˆ (0) 1 } ]+{ ˆ(2) 0, ˆ (0) 1 }, ˆ(0) 0 =[ˆ(0) 0,v 0] { 1, ˆ (0) 1 }, (1) 1 = dˆ (0) 1 [! 0, ˆ (0) 1 ] [ 1, (1) 0 ] [ ˆ(0) 0,u 1]+[ (1) 1,v 0], ˆ(2) 0 = d (1) 0 + {! 0, (1) 0 } { 1, ˆ (2) 1 } {B 1, ˆ (0) 1 } +[ˆ(0) (3) 1 = dˆ (2) 1 [! 0, ˆ (2) 1 ] [ 1, (3) 0 ] [B 1, (1) 0 ] [ 0, ˆ (0) 1 ] [ ˆ(0) 0,U 1]+[ (1) 1,b 0] [ ˆ(2) 0,u 1]+[ (3) 1,v 0], 0,b 0]+[ (1) 1 ˆ(4) 0 = d (3) 0 + {! 0, (3) 0 } { 1, (4) 1 } {B 1, ˆ (2) 1 } + { 0, (1) 0 } {H 1, ˆ (0) 1 } +[ˆ(0) 0,h 0]+[ (1) 1,U1]+[ˆ(2) 0,b 0]+[ (3) 1,u1]+[ˆ(4) 0,v 0],u 1]+[ˆ(2) 0,v 0],
[Q B,A a µ]=d µ c a, [Q B,c a ]= 1 2 f a bcc b c c Q B [Q B,D µ c a ]=0, [Q B, 1 2 f a bcc b c c ]=0 S[J, K] = d d x L + J a µ A a µ + J a c a + K a µ D µ c a 1 2 K afbcc a b c c
0=h[Q B,e is[j,k] ]i = i d d xh J a µ D µ c a 1 2 J afbcc a b c c e is[j,k] i e i [J,K] = he is[j,k] i a µ = J a µ [J, K] =ha a µe is i, a = J a [J, K] =hc a e is i W [,K]= [J, K] + J J a µ = W a µ, J a = W a 0= J a µ + J a K a K a µ [J, K] = W a µ W K K µ a + W W a K a =(W, W ) BRST =(,W)
{F, G} = F G A A F A G A W [, ]= A : A = A 1 d d x(l + J µ a A a µ + J a c a + A µ a D µ c a c a F A =( 1) A ( F + A ) F A 1 2 f a bcc b c c + c ab a ) {W, W } =2 W A a µ W A µ a 2 W c a W c a =2hJ µ a D µ c a J a 1 2 f a bcc b c c i =0 2 W c a W c a {W, W } =0 B A = {W, A }
A A = D A e i ~ W (, ( + ) A ) / A (deg. = 1) D A e i ~ W (, A ) D A i~ BV W 1 2 {W, W } e i ~ W W = c a µ A µ a A a µ = = c a µ c a = c a = A µ a c a =0 c a = µa µ a i~ BV W 1 {W, W } =0 2 ~! 0 : calssical master equation, {W, W } =0
H(q, p) =p i q i L(q, q), p i = L q i H p i = q i, H q i = L q i = d dt L q i = ṗ i F (q, p) ={F, H} = F H q i p i F p i H q i = F q i qi + F p i ṗ i H t = {H, H} =0 L(q, q) =p i q i H(q, p) S = dt(p q H)
(M,{, }, ), {, } =0 Q {, } Q = Q i x i = q i p i p i q i, (xi )=(q i,p i ) Q 2 = Q{, } = {, {, }} = 1 {{, }, } =0 2 = 0
d! =0,! =! ab dx a ^ dx b {F, G} =(! 1 ab F G ) x a x b L Q! =(i Q d + di Q )! = di Q! =0 i Q = Q a (dx a ) i Q! = d {, } =0 S = d d xd d (pdq + ) (d = µ µ, µ as odd coordinate)
v i (t) = dx i (t) dt = X i (x(t)) X = X i x i x i (t) =x i + tv i deg{t} = 1, t 2 =0 X i (x i (t)) = X i (x + vt) =X i (x)+tv j Xi x j = tx j Xi x j dx i (x(t)) dt = X j Xi x j =0 X i = Q i Q 2 i = Q j Qi =0$ {, } =0 xj
(q a,p a ) grading (1,n 1) : (x i (q a,p a )) {F, G} = F q a G p a +( 1) n F p a G q a F x a =( a 1) x ( F! x a ) F x a (q, p) = 1 2 f a bcp a q b q c {, } =0 (M,{, }, ) =0 Q = {, } = q a Q 2 =0 + p a p a =( 1)n qa f c abq b p c p a + 1 2 f a bcq b q c q a
! M :(x, ) 7! (q a,p a )=( a (x, ), a (x, )) a(x, ) = a (x, ) =c a (x)+ µ A a µ + 1 2 µ a µ (x), a (x)+ µ A a,µ(x)+ 1 2 µ c a,µ (x) (' = anti-field of ') ( ' = ' 1) d = µ µ d 2 xd 2 µ µ = µ dx µ ^ dx W = d 2 x = d 2 x d 2 = d 2 x d 2 ( a d a + (, )) a d a + 1 2 f bc a b 1 2 a µ Fµ a µ A a, ( µ c a + fbca a b µc c )+ 1 4 µ c a,µ fbcc a b c c + 1 b µ 2 µ fbc a c ac c
M (q, p) = 1 2 f a bcp a q b q c {, } = { 1 + 3, 1 + 3 } =0
a (x, ) = a 1(x)+ µ a 1,µ(x)+ 1 2 µ B a 1,µ (x) a (x, ) = ˆa 0 (x)+ µ! a 0,µ(x)+ 1 2 µ ˆa 0,µ (x) W AKS = = d 2 xd 2 ( a d a + 1 (, )+ 3 (, )) d 2 xd 2 a d a + 1 2 f bc a b a c + 1 3! f abc a b c = Tr{ 1 (d! 0 +! 0 2 + { ˆ(0) 0, ˆ(2) 0 } + 1) 2 ) + 1 (d ˆ(0) 0 +[! 0, ˆ(0) 0 ]) + B 1( ( 2 +(ˆ(0) 1 0 )2 )}
a (x, ) = a 1(x)+ µ a 1,µ(x)+ 1 2 µ B a 1,µ (x) a (x, ) = ˆa 0 (x)+ µ! a 0,µ(x)+ 1 2 µ ˆa 0,µ (x) (F, G) = d 2 x F 1a a (x, ) = a 1(x) µ µ! 1a(x)+ 2 c a 1 (x) a (x, ) =c a 0(x)+ µ! 0µ(x)+ a 1 2 G a 0 F a 0 G + F 1a! 0µ a G! µ 1a F! µ 1a a 0 (x) G! a 0µ + F c a 0 ( ' = ' 1) G c 1a F c 1a! G c a 0 F =( 1) ( + F ) [ ]([ ]+[F ]) F ( 1)! W AKS = W BF + W 0
W AKS = W BF + W 0 (W AKS,W AKS )=0 W BF = d 2 x 1 2 1a µ Fµ a d 2 x! µ 1a ( µc a 0 + fbc! a 0µc b c 0)+ 1 2 f bcc a 1ac b 0c c 0 a 0 fabc c b 0 1c W 0 = 1 2 d 2 x f a bc 1a µ! bµ 1! 1 c a b + f abc 1 1 c c 1 W AKS W 0 ghost number of B1 a = c a 1 = 2 1 2 c a = 1 (x) a (x, ) = a 1(x)+ µ a 1,µ(x)+ 1 2 µ B a 1,µ (x) a (x, ) = ˆa 0 (x)+ µ! a 0,µ(x)+ 1 2 µ ˆa 0,µ (x) = c a 0(x)
S k 2 ( =0)= Tr 1(d! 0 +! 0) 2 2 1BB 1 1 =[ 1,v 0 ],! 0 = dv 0 +[! 0,v 0 ]+{ 1,u 1 }, B 1 = du 1 {! 0,u 1 } +[ 1,b 0 ]+[B 1,v 0 ], 0 = db 0 +[! 0,b 0 ]+{ 1,U 1 } + {B 1,u 1 } +[ 0,v 0 ] H 1 = du 1 {! 0,U 1 } +[ 1,h 0 ]+[B 1,b 0 ] { 0,u 1 } +[H 1,v 0 ] X a, (0, 1), Y a, (1, 0), a, (2, 1), X a = a 1 + Y a = + µ! 0µ + a = + 1 2 µ B 1µ (ghost number, hidden grading) Q 1,! 0,B 1
(X(2m) a,ya (2m+1)), (m = 1,, 1) Q = X m Q m, Q 2 =0 Q 2m+1 = A 2 f a bc X k Q 2m = Af a bc Y b (2k+1) Y c ( 2k+2m+1) X k! Y b (2k+1) Xc ( Y a (2m+1) 2k+2m)! + B 2 f a bc X a (2m) X k X b (2k) Xc ( 2k+2m+2)! Y a (2m+1) (A, B = const.) Q 2 = {, {, }} = 1 {{, }, } =0 2
= A 2 X X k m f abc Y b (2k+1) Y c ( 2k+2m+1) Xa ( 2m) + B 3! S = d 2 xd 2 X m X ( X X f abc X(2k) a Xb ( 2k+2m+2) Xc ( 2m) k m 2m)a dy a (2m+1) +! S = 1X 1X d 2 xtr {Cn B ( µ µ Cn B + ( µ CmµC B B (m+n) + {CF m, C F (m+n) } µ CmµC F F (m+n) )) + C 1X n B m= 1 n= 1 C F n µ ( µ C F nµ + m=1 (C F mc F (m+n) + CB mc B (m+n) ) 1X [Cmµ,C B F (m+n) ])} m= 1 V 2 = 1 A =[Q + A, V 1 ] F =0 2 1A =[Q + A, V 2 ]={Q + A, [Q + A, V]} = 1 2 [{Q + A,Q+ A}, V] =1 [F, V] =0 2
Y a (2m+1) = CF,a 2m+1 + µ C B,a 2mµ + 1 2 CF,a 2m 1µ Y a X a (2m) = CB,a 2m + µ C F,a 2m 1µ + 1 2 CB,a 2m 2µ (2m+1) = a (2m+1) + µ! a (2m)µ + 1 2 µ ˆa (2m 1)µ X a (2m) = a (2m) + µ a (2m 1)µ + 1 2 µ B a (2m 2)µ a = X m Y a (2m+1) = X m a (2m+1) + µ X m! a (2m)µ + 1 2 µ X m ˆa (2m 1)µ = a + µ! µ a + µ ˆa µ a = X m X a (2m) = X m a (2m) + µ X m a (2m 1)µ + 1 2 µ X m B a (2m 2)µ = a + µ µ a + µ B µ a a,! µ,b a µ a
(x µ ( ),p µ ( )) ( µ ( ), µ ( )) Bx µ ( ) = µ ( ) B µ ( ) =0 Bp µ ( ) =0 B µ ( ) = p µ ( ) S = d B ( µ (ẋ µ p µ )) = d ( B µ (ẋ µ p µ ) µ B ẋ µ ) = d (p µ ẋ µ p µ p µ + µ µ ) 2p µ =ẋ µ S on-shell = d 1 4ẋµ ẋ µ + µ µ
x,µ = L ẋ µ = p µ,µ = L µ = µ [x µ,p ]=i µ { µ, } = i µ Q B = i µ p µ = µ µ = d x, > (x, ) =<x, >= (x)+ µ µ(x)+ 1 2 µ B µ (x, ) =<x, >= ˆ(x)+ µ! µ (x)+ 1 2 µ ˆµ
<x, Q B >= µ µ (x, ) =0 <x, Q B >= µ µ (x, ) =0 >= Q B B >, >= Q B F > S =< Q B >= d 2 xd 2 (x, )Q B (x, ) S = d 2 xd 2 a d a + 1 2 f a bc a b c + 1 3! f abc a b c
deg. of (q a,q a )=(1, 0) {F, G} = F q a G q a S = F q a G q a = 1 3! f abcq a q b q c, {, } =0 1 2 q adq a + 1 3! f abcq a q b q c q a! A a = 1 1 a + i ˆa 0 + ja a 0 + kâa 1 Q = jd V a = 1â a 0 + ia a 1 + jˆ a 1 + k a 0 S GCS = Tr 1 2 AQA + 1 3 A3 (A = A a T a, V = V a T a ) A = QV +[A, V]
A a = 1 a 1 + i ˆa 0 + ja a 0 + kâa 1 = 1 a 1 + kâa 1 + i( ˆa 0 ka a 0)! ( a 1 + Âa 1)+(ˆa 0 A a 0) a 1 + a 0 = q a = S = = 1 2 q adq a + 1 3! f abcq a q b q c apple 1 2 ( 1a + 0a)d( a a 1 + 0)+ 1 3! f abc( a 1 + a 0)( b 1 + b 0)( c 1 + c 0) = S E + S O a S E = 1ad 0 + 1 2 f a b c abc 1 0 0 + 1 3! f a b abc 1 1 c 1 S O = 1 2 ( 1ad a 1 + 0ad a 0)+ 1 2 f abc a 0 b 1 c 1 + 1 3! f abc a 0 b 0 c 0 S O