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High-order Virasoro gravity
16 May 1996
PHYSICS
ELSEVIER
LETTERS 6
Physics Letters B 375 (1996) 65-68
High-order Virasoro gravity * Chao-Zheng Zha, Wei-Zhong Zhao CCAST (World Laboratory), PO. Box 8730, BeQing 100080, China and Center for Theoretical Physics, Xinjiang University. Vrumqi, Xinjiang 830046, China
’
Received 18 December 1995 Editor: H. Georgi
Abstract We construct
a gauge theory of high-order
Virasoro algebra and give the explicit results.
1. Introduction
spin generators. The W, algebra which admits central terms is given by Ref. [ 41:
Recently W algebra is one of central interest in two-dimensional conformal field theory. W gravity is a higher-spin generalization of gravity and can be thought of as the gauge theory of local W algebra symmetries. A field-theoretic realization of the W3 algebra was given in Ref. [ 11, where a free action realizing a chiral W3 symmetry was Noether-coupled to spin-2 and spin-3 background gauge fields. This work was extended in Ref. [2] to a non-chiral gauged theory of W3. In terms of the methods of Refs. [ 1,2], chiral and nonchiral W, gravity were constructed by E. Bergshoeff et al. [ 31, where the W, algebra is:
[UjJ,]
= [(Zf
l)m-
(j+
l)n]u$!fl.
+ q2jCjT712i+3Sijti~~+n,~
where the structure constants #(m, #(m,n)
I 3 -;I i,
I-4
4F3
;, + 5 ;1
_~-;,-j-~,~+j-~+z
[
1
(4)
and NY(m, n) is the “Clebsh-Gordan” part, ifl Ny(m,n)=x(-l)k
0370-2693/96/$12.00 @ 1996 Elsevier Science B.V. All rights reserved 0026
-
-37
“This work is supported by the NSFC. ’ Mailing address.
(96)
(3)
# are functions given in terms of a saalschutzian 4F3( 1) generalized hyper-geometric function:
4;’ =
PII SO370-2693
n) are given by:
= 2(1_: l),d:Nf’(m,n),
(I)
Here ufr,corresponds to generators with conformal spin 1 + 2. w, algebra is linear. However, only the Virasoro subsector of the w, algebra admits a central extension. From the point of view of representation theory it is thus not very interesting, since physical states would have to transform trivially under all the higher-
(2)
(
k=O x
[2j
x
[j
+ 2 +
If1 k
k][+l-k[i
1 $_ n]k.
(2i+2 - I)/( )
+
1 + m]l+l-_k (5)
C.-Z. Zha, W.-Z. Zhao / Physics Leriers B 375 (1996) 65-68
66
The gauge theory of the W, algebra was constructed by E. Bergshoeff et al. [ 51, but the method is different from those used in Refs. [l-3]. In Ref. [ 51, the currents that generate the algebra are built from higher-derivative bilinears in field, whereas those in Refs. [ l-31 are built from higher powers of first derivative of fields. The form of the W, algebra is more complicated. However, in investigating the higher-order differential neighborhoods of holomorphic mapping of St, the formalism the of high-order Virasoro algebra (HOVA) can be obtained [ 61. The form of HOVA is more simple than that of the W, algebra. In this paper, we will construct a gauge theory of HOVA which seems to be more easily tackled in further explorations for gravity of high conformal spins.
In fact, the commutator of the W, defined by that of the HOVA [ 71:
(10)
B(s)
s
4,a = W"+'s_lK&_, B(s) =
a
(11)
2J-3(S - l)!
(12)
(2s - 3)!!
(s+a-
KS=
s_a_,
l)!
a!(a-
l)!
(13)
.
Now we consider the following 2. High-order Virasoro algebra (HOVA) The High-Order by Ref. [6]:
[L;,,
X(z)
Virasoro Algebra (HOVA) is given
L;+"] =-&-l,i'(C1'B;+k+r- Ck”_t,BG’+k) p=I
dz>cp*(w)- -
(14)
log( z - w) .
(15)
Then we have the operator products expansion as follows:
(OPE)
2 = I c + (z _ w),K(w) 2 (z - w)4
1 + (z _W)avl(w)
1 (k+ t-)!k! 2(2k+r+l)!
(m + k)!c
’
currents:
where q is a free complex scalar, with 2-point function:
vl(z)vl(w) k+,.+,
bilinear
= (-l)iai4p(z)a(p*(z)
ktr
tt-1)
algebra can be
(m-k_r_
~)!8n’+n~o
where LL, are the generators given as follows:
(6)
vl(z)h(w)
=-
c
3 + (z _ w)2 h(w)
of the HOVA and can be
2
(z - w)5 -
(z -
w)3h(w)
1 + (z _ w) d&(w)
(17)
2
2c
v2(z)K(w) = (z - w)6 + (z _ w),ab(w) 1 + ----a%(w) (z -w) 2 + ---av3(w) z-w
k!
c; =
(k-p)!p!
=o B;:
p>n,
(18)
... (9)
I
i = (nn’p)!
4 + (z _ w)2h(w)
pLn.
The general formulisms
of the OPE are:
C.-Z. Zha, W.-Z.Zhao/Physics LettersB 375 (1996) 65-68
y(z)r/;+,(w)
=
(-l)“+‘tl);,;y;;fcz
67
where the ki are functions of X+ only. In fact, the transformations (25) close to form the algebra
i+r
-c (-l)PC” p=i+l
p![C,”
+k
C
/‘=I I’
P! r+r (z _ w)p+l b-p(W) - (-l)“CC,]
(z - W)Pfl
max(p,4)
- C,qk,a;kp)a~+Y-‘p
V2i+r-y ( w)
max(a4)
(26)
(p - q) ply-”
+c q=,
cz _ W)/:-41:*
@~i+r-_p(W)
>
(19) where
They are consistent with the forms of OPE cited in Ref. [6]. Under the symmetry generated by the current Vi, the transformation of any field f is given by the standard OPE expression ‘fjk,_f =
(20)
$ki(z)V’(Z)f.
Taking f = q( z ) we have Sk,+)(Z)
=
(-l)‘ki(Z)&‘(Z>.
dz
f’Giz
.
(27)
J
d2x(a+qo*a_q
-
c
(28)
Ai+,Vi+‘).
i+r>l
The action St is invariant under (25) together the transformation of the gauge fields Ai+r,
00 ki(?.) = c kk,Z”‘+’ . n,=-cm Using (9), (20),
- C,Yk,a:k,)
Using (21), (22) and (26) we recover the HOVA at once. To gauge this chiral HOVA symmetry, we introduce gauge fields Ai and add to the action (24) the Noether coupling, St =
Let
L; =
Kpfy+- = (-l)‘(C,Pk,a$k,
SAi+r= a_ Ki+,
+ c
C(
/>I
(21) and (22) we have x
“‘f’E(z)
-
Using the OPE ( 19), we can obtain that the commutator of Li, and Lt+r is just Eq. (6).
with
- 1 )i+r+wq!;;;,f-~
.>=I
Ai+r+sa~r’2S-‘ki+r+s_~ i+s [C”l+S - ( -l)PC;+p_S]A cc 320 p=l ifs
r+p_,.a”k. ( + r+s
p
+ CCC(-i)Ytlc~,c;-“a:
s>o
3. Chiral and non-chiral HOVA gravity x
In order to construct chiral and non-chiral gravity, we consider the following action: So =
J
HOVA
(24)
d2xa+cp*a_(o
where the two-dimensional space-time has null coordinates xfi = (x+, X-) which are related to the usual Cartesian coordinates by X* = x0 f x1. It is easy to find that this action is invariant under the following global transformations: Sk;+7= ( -1 )ikia;p
i = 1,2...
(25)
p=l
q=l
(Ar+p-.d-Yki+r)
(29)
.
To gauge the non-chiral HOVA, we introduce gauge fields Ah, corresponding to the left- and right-handed HOVA symmetries, and auxiliary fields J+. Then the complete action can be written as dX2{-a+p*a_p
s2 =
- JZJ-
+ a_pJT
J +
a+p*J-
-
C [A+(i+r)V’f’( i+r>l
- J+ J” + a-p* J+ f a+pJ+ J-> + A-(i+rjVF’(
J+> I} (30)
C.-Z. Zha, W.-Z. Zhm/Physics
68
Letters B 375 (1996) 65-68
S-f&= &(&,d
where
Vi(J*)
= (-l)‘&J*,*J$.
The equations give Ji = &p
-
(31)
of motion
x[
for the auxiliary
fields J*
(-l)‘A_,d;-‘.I+
ill
+ (-l)‘A+&‘J_]
.
(32)
The action S2 is invariant under the following transformations:
640= ~w)iK-ic$9+
+ (_l)‘K+&pJ_]
+I
(33)
(34)
.
(35)
In conclusion, we have constructed a gauge theory of HOVA. Because the forms of HOVA and the bilinear currents realizing HOVA are more simple than those of the W, algebra, we give the explicit results of the gauge theory of HOVA. It is much more convenient in discussing the quantisation and anomalies of the theory which will be explored in a forthcoming publication.
References [ 1I C.M. Hull, Gauging the Zamolodchikov W-algebra, preprint, QMC/PH/89/18 (1989). [21 K. Schoutens, A. Sew-in and P Van Nienwenhuizen, A new gauge theory of W-type algebra, Stony Brook preprint, ITPSB-9-19 (1990). 131 E. Bergshoeff, C.N. Pope et al., Phys. Lett. B 243 (1990) 350. [4] C.N. Pope, L.J. Romans and X. Shen, Phys. Lett. B 236 (1990) 173. [S] E. Bergshoeff, C.N. Pope, L.J. Romans, E. Sezgin and X. Shen, Mod. Phys. Lett A 5 (1990) 1957. [6] C.Z. Zha, Phys. Lett. B 288 (1992) 269. 171 C.Z. Zha, J. Math. Phys. 35 (1994) 517.
PHYSICS
ELSEVIER
LETTERS 6
Physics Letters B 375 (1996) 65-68
High-order Virasoro gravity * Chao-Zheng Zha, Wei-Zhong Zhao CCAST (World Laboratory), PO. Box 8730, BeQing 100080, China and Center for Theoretical Physics, Xinjiang University. Vrumqi, Xinjiang 830046, China
’
Received 18 December 1995 Editor: H. Georgi
Abstract We construct
a gauge theory of high-order
Virasoro algebra and give the explicit results.
1. Introduction
spin generators. The W, algebra which admits central terms is given by Ref. [ 41:
Recently W algebra is one of central interest in two-dimensional conformal field theory. W gravity is a higher-spin generalization of gravity and can be thought of as the gauge theory of local W algebra symmetries. A field-theoretic realization of the W3 algebra was given in Ref. [ 11, where a free action realizing a chiral W3 symmetry was Noether-coupled to spin-2 and spin-3 background gauge fields. This work was extended in Ref. [2] to a non-chiral gauged theory of W3. In terms of the methods of Refs. [ 1,2], chiral and nonchiral W, gravity were constructed by E. Bergshoeff et al. [ 31, where the W, algebra is:
[UjJ,]
= [(Zf
l)m-
(j+
l)n]u$!fl.
+ q2jCjT712i+3Sijti~~+n,~
where the structure constants #(m, #(m,n)
I 3 -;I i,
I-4
4F3
;, + 5 ;1
_~-;,-j-~,~+j-~+z
[
1
(4)
and NY(m, n) is the “Clebsh-Gordan” part, ifl Ny(m,n)=x(-l)k
0370-2693/96/$12.00 @ 1996 Elsevier Science B.V. All rights reserved 0026
-
-37
“This work is supported by the NSFC. ’ Mailing address.
(96)
(3)
# are functions given in terms of a saalschutzian 4F3( 1) generalized hyper-geometric function:
4;’ =
PII SO370-2693
n) are given by:
= 2(1_: l),d:Nf’(m,n),
(I)
Here ufr,corresponds to generators with conformal spin 1 + 2. w, algebra is linear. However, only the Virasoro subsector of the w, algebra admits a central extension. From the point of view of representation theory it is thus not very interesting, since physical states would have to transform trivially under all the higher-
(2)
(
k=O x
[2j
x
[j
+ 2 +
If1 k
k][+l-k[i
1 $_ n]k.
(2i+2 - I)/( )
+
1 + m]l+l-_k (5)
C.-Z. Zha, W.-Z. Zhao / Physics Leriers B 375 (1996) 65-68
66
The gauge theory of the W, algebra was constructed by E. Bergshoeff et al. [ 51, but the method is different from those used in Refs. [l-3]. In Ref. [ 51, the currents that generate the algebra are built from higher-derivative bilinears in field, whereas those in Refs. [ l-31 are built from higher powers of first derivative of fields. The form of the W, algebra is more complicated. However, in investigating the higher-order differential neighborhoods of holomorphic mapping of St, the formalism the of high-order Virasoro algebra (HOVA) can be obtained [ 61. The form of HOVA is more simple than that of the W, algebra. In this paper, we will construct a gauge theory of HOVA which seems to be more easily tackled in further explorations for gravity of high conformal spins.
In fact, the commutator of the W, defined by that of the HOVA [ 71:
(10)
B(s)
s
4,a = W"+'s_lK&_, B(s) =
a
(11)
2J-3(S - l)!
(12)
(2s - 3)!!
(s+a-
KS=
s_a_,
l)!
a!(a-
l)!
(13)
.
Now we consider the following 2. High-order Virasoro algebra (HOVA) The High-Order by Ref. [6]:
[L;,,
X(z)
Virasoro Algebra (HOVA) is given
L;+"] =-&-l,i'(C1'B;+k+r- Ck”_t,BG’+k) p=I
dz>cp*(w)- -
(14)
log( z - w) .
(15)
Then we have the operator products expansion as follows:
(OPE)
2 = I c + (z _ w),K(w) 2 (z - w)4
1 + (z _W)avl(w)
1 (k+ t-)!k! 2(2k+r+l)!
(m + k)!c
’
currents:
where q is a free complex scalar, with 2-point function:
vl(z)vl(w) k+,.+,
bilinear
= (-l)iai4p(z)a(p*(z)
ktr
tt-1)
algebra can be
(m-k_r_
~)!8n’+n~o
where LL, are the generators given as follows:
(6)
vl(z)h(w)
=-
c
3 + (z _ w)2 h(w)
of the HOVA and can be
2
(z - w)5 -
(z -
w)3h(w)
1 + (z _ w) d&(w)
(17)
2
2c
v2(z)K(w) = (z - w)6 + (z _ w),ab(w) 1 + ----a%(w) (z -w) 2 + ---av3(w) z-w
k!
c; =
(k-p)!p!
=o B;:
p>n,
(18)
... (9)
I
i = (nn’p)!
4 + (z _ w)2h(w)
pLn.
The general formulisms
of the OPE are:
C.-Z. Zha, W.-Z.Zhao/Physics LettersB 375 (1996) 65-68
y(z)r/;+,(w)
=
(-l)“+‘tl);,;y;;fcz
67
where the ki are functions of X+ only. In fact, the transformations (25) close to form the algebra
i+r
-c (-l)PC” p=i+l
p![C,”
+k
C
/‘=I I’
P! r+r (z _ w)p+l b-p(W) - (-l)“CC,]
(z - W)Pfl
max(p,4)
- C,qk,a;kp)a~+Y-‘p
V2i+r-y ( w)
max(a4)
(26)
(p - q) ply-”
+c q=,
cz _ W)/:-41:*
@~i+r-_p(W)
>
(19) where
They are consistent with the forms of OPE cited in Ref. [6]. Under the symmetry generated by the current Vi, the transformation of any field f is given by the standard OPE expression ‘fjk,_f =
(20)
$ki(z)V’(Z)f.
Taking f = q( z ) we have Sk,+)(Z)
=
(-l)‘ki(Z)&‘(Z>.
dz
f’Giz
.
(27)
J
d2x(a+qo*a_q
-
c
(28)
Ai+,Vi+‘).
i+r>l
The action St is invariant under (25) together the transformation of the gauge fields Ai+r,
00 ki(?.) = c kk,Z”‘+’ . n,=-cm Using (9), (20),
- C,Yk,a:k,)
Using (21), (22) and (26) we recover the HOVA at once. To gauge this chiral HOVA symmetry, we introduce gauge fields Ai and add to the action (24) the Noether coupling, St =
Let
L; =
Kpfy+- = (-l)‘(C,Pk,a$k,
SAi+r= a_ Ki+,
+ c
C(
/>I
(21) and (22) we have x
“‘f’E(z)
-
Using the OPE ( 19), we can obtain that the commutator of Li, and Lt+r is just Eq. (6).
with
- 1 )i+r+wq!;;;,f-~
.>=I
Ai+r+sa~r’2S-‘ki+r+s_~ i+s [C”l+S - ( -l)PC;+p_S]A cc 320 p=l ifs
r+p_,.a”k. ( + r+s
p
+ CCC(-i)Ytlc~,c;-“a:
s>o
3. Chiral and non-chiral HOVA gravity x
In order to construct chiral and non-chiral gravity, we consider the following action: So =
J
HOVA
(24)
d2xa+cp*a_(o
where the two-dimensional space-time has null coordinates xfi = (x+, X-) which are related to the usual Cartesian coordinates by X* = x0 f x1. It is easy to find that this action is invariant under the following global transformations: Sk;+7= ( -1 )ikia;p
i = 1,2...
(25)
p=l
q=l
(Ar+p-.d-Yki+r)
(29)
.
To gauge the non-chiral HOVA, we introduce gauge fields Ah, corresponding to the left- and right-handed HOVA symmetries, and auxiliary fields J+. Then the complete action can be written as dX2{-a+p*a_p
s2 =
- JZJ-
+ a_pJT
J +
a+p*J-
-
C [A+(i+r)V’f’( i+r>l
- J+ J” + a-p* J+ f a+pJ+ J-> + A-(i+rjVF’(
J+> I} (30)
C.-Z. Zha, W.-Z. Zhm/Physics
68
Letters B 375 (1996) 65-68
S-f&= &(&,d
where
Vi(J*)
= (-l)‘&J*,*J$.
The equations give Ji = &p
-
(31)
of motion
x[
for the auxiliary
fields J*
(-l)‘A_,d;-‘.I+
ill
+ (-l)‘A+&‘J_]
.
(32)
The action S2 is invariant under the following transformations:
640= ~w)iK-ic$9+
+ (_l)‘K+&pJ_]
+I
(33)
(34)
.
(35)
In conclusion, we have constructed a gauge theory of HOVA. Because the forms of HOVA and the bilinear currents realizing HOVA are more simple than those of the W, algebra, we give the explicit results of the gauge theory of HOVA. It is much more convenient in discussing the quantisation and anomalies of the theory which will be explored in a forthcoming publication.
References [ 1I C.M. Hull, Gauging the Zamolodchikov W-algebra, preprint, QMC/PH/89/18 (1989). [21 K. Schoutens, A. Sew-in and P Van Nienwenhuizen, A new gauge theory of W-type algebra, Stony Brook preprint, ITPSB-9-19 (1990). 131 E. Bergshoeff, C.N. Pope et al., Phys. Lett. B 243 (1990) 350. [4] C.N. Pope, L.J. Romans and X. Shen, Phys. Lett. B 236 (1990) 173. [S] E. Bergshoeff, C.N. Pope, L.J. Romans, E. Sezgin and X. Shen, Mod. Phys. Lett A 5 (1990) 1957. [6] C.Z. Zha, Phys. Lett. B 288 (1992) 269. 171 C.Z. Zha, J. Math. Phys. 35 (1994) 517.