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A note on the connection between the SL(2,R ) current algebra and general covariance
Volume 234. number 3
PHYSICS LETTERS B
11 January 1990
A N O T E ON T I l E C O N N E C T I O N B E T W E E N T H E SL(2, ~) C U R R E N T ALGEBRA AND G E N E R A L COVARIANCE E.N. ARGYRES. C.G. P A P A D O P O U L O S Institute o['NuclearPhysics.NR(" "'Dcmocrttos", GR-153 10Athens. Greece and E.G. FLORATOS
t'h),swsDepartment. Universityo.f('rete attd 1"0R771. lraklion. Crete. Greece Received 16 October 1989
We propose a partial gauge fixing,which enables one to derive all the symmetriesof two-dimensionalinduced quantum gravity in the light-conegauge, including the SL( 2, P ) current algebra, as the result of general covariance.
in order to study two-dimensional gravity induced by string theory, Polyakov [ 1 ] has proposed thc special gauge (so-called light-cone ( LC ) gauge ) g__=0,
g._=l,
g÷+=2h(x+,x-).
(I)
where h (x "..v - ) is the remaining degree of freedom of the metric. Then by requiring e n e r g y - m o m e n t u m conservation and using the known form of the trace anomaly in two dimensions, he derived differential equations for correlation functions of the field h. Furthermore, using the classical equations of motion. arising from the induced action
S= f d 2 - x x / - g R - I R ,
(2)
M
where A-g'aJV.VB, he showed the existence of an SL(2, ~ ) current algebra. From another point of view. the SL(2. ~ ) current algebra can be derived from the W Z N W model [ 1,2 ]. As has been shown, one can establish an equivalence (at lcast on-shell ) between the W Z N W model with group SL(2, ~ ) and two-dimensional induced gravity. This can be achieved in the LC gauge as well as in the usual contbrmal gauge. In this note we propose an alternative approach, in
304
order to investigate the connection between two-dimensional induced gravity and the SL(2, D) current algebra. Wc prove that the SL(2, >) symmetry resuits from a general coordinate transformation ( G C T ) invariancc of the action, preserving the form, not of the LC gauge, but of partial gauge fixing. Before analysing our approach, it is worthwhile to consider what kind of transformation this SL(2, .'4) current algebra induces on h, and whether it maintains the form of the LC gauge. Wc dcfine
dh-d~) h = ~ d l- -'+ - t " ~ ( y + , l i ( a ) ( i, + ) h ( x ' . x - ) . . . . + 2zti
(3)
where a takes the values + , - . 0. Using the operator product expansion j(a)(x
+ )j(')( V '
)
j"~d) (y + ) =2~"%1'a ~ : T - v ;
t]ah c 12 ( x + - y + )
2'
(4)
where ~ +o= 1. q~ _ = ~ and qoo= l , a n d t h c c x p a n s i o n o f h in powers o f x - . h ( x +. x - ) =./~+ ' ( x ' ) - 2 / " " ( x ' ).v+i<
'(.~" ) ( x ) -~.
(5)
0370-2693/90/$ 03.50 53 ElsevierScience Publishers B.V. ( North-Ilolland )
('O. - hO _ +
~_ tl ),5.\-- ,
(6)
X c x p ( - S I + S ~ "v )
with
= f [d.W'] [de_ l tdc+ ] [ d h - - ]
dr- =~->(x
' )-2~(°~(x
+(~+~(x+)(.\.
' ).\-
g__=0,
(8)
We notice that the parametrization (8) is an interpolation between L(" formal ( h = 0 ) gauges. Starting with over the string coordinates, fixingg
of the metric in (0 = 0) and conthe path integral _ = 0 , wc get
] [db--]
× c x p ( -S~ + S v P ) .
(9)
where "/'[ / ..... ,S'I = ~_ , d2-v~/' ( - g ) 8,*X*'O/~'k'"g"/51t,,,
A'~'0_ A'")tl~ ....
,)
(10a) and
Tf 5
whcrc
-
"l'[d2x(h--0 2 J
d~-xcxp(~/2)h--(~_c.-~..Oc_). (10b)
The induced action in the LC gauge can easily be obtained by integrating over 0, using the gauge fixing 0 = O,
c +h+~0+c+)
(12)
which contributes - 26 to the trace anomaly. The GCT translbrmations which preserve the gauge (8) are given by ~5.\+ = f ( x + ). 6 x - arbitrary, under which 0 and h transform as follows:
60=2(0+dx++0
6\-)+0,Odx'+3_Od.v-.
(13a) ~5t1=(0+-h0_
g++ = 2 e x p ( o / 2 ) h .
f. = "/" | d2x(O+ .¥~'0_ X " - h ~ _
(ll)
(7)
g+_=cxp(0/2),
Z = f [dh] [dO] [d.W'] [de
[dh++ l
X cxp ( - .S'~ + .S'~,L~' ) .
)2.
These transformations can be thought of as a combination o f a G C T plus a Weyl transformation, with parameters ,iv + = 0 , ~t\-- given by (7) and 2=-0 dv . One can easily verify that these transformations preserve the form of the LC gauge. What we wish to do. is to consider the symmetries of induced gra~.ity in the I.C gauge as the result of the general covariancc of the theory. We define the partial gauge fixing
,,-] l, - -
r J [do] [dX"] [de_] [ d h - - ]
exp[-St(h)]=
wc find ,~h=
11 January 1900
PHYSICS LETTERS B
Volume 234, number 3
+0_h)dx-+0+
(h,J.v +) .
(13b)
What is the interesting point of (13), is that the transtbrmations tbr O and 11 arc dccoupled, which means that ( 13b ) is still present after the integration over 0 and its F a d d c c v - P o p o v partners, and therefore is an invariance of the action S t ( h ) . the invariance is automatically guarantccd, sincc (13b) is nothing but a (iCT transformation. All the known symmetries of the action St,(t1 ) are included in the gcncral tbrm (13b). These symmetries manifest themselves, in the L(: gauge, as a combination o f a G C T and a Wcyl transformation: (1) d \ arbitrary, d x ~ = J ~ x +) and 2 = - ( 0 , 6 x + + 0_ ~,t-- ) which has bccn studicd in rcf. [31. (II) dx* =((.x -+ ), 6,\- = - . \ - ~3+ ~+11(.\'+ ). ).= 0. which is thc residual G C T symmctry of thc LC gaugc [4], gcncrating the Virasoro and the 0 ( I ) K a e - M o o d y algcbra. Following the reasoning of refs. [ 1,5 ], we derive perturbatively the equations of motion, which in the limit O = 0 take the standard from 0~ h = 0 . I f n o w wc requirc that the transformations (13b) preserve the form of the equations of motion, the independent variations 6x arc restricted at the classical level to bc of thc form (7) inducing thus the SL(2. ~) current algebra of the h-transformations. Furthermore. one can obtain the correct central charge dependence by replacing tl with ~ch. Thus. SL(2, ~) comcs into 305
Volume 234, number 3
PHYSICS LETTERS B
play without any information concerning the corrclation functions ofh and in this sense is a purely classical result. Wc conclude that using the more general metric parametrization (8), one can study all the symmetries of SL(h). In fact these symmetries manifest themsclvcs as GCT transformations, which guarantee automatically the invariance of the action. SL(2, P) seems to be an inherent characteristic of induced gravity, related not to the special form of the LC gauge, but to the "'spin" or scale dimensions of the metric [ 6 ].
306
11 January 1990
References [ 1 ] A.M. Polyakov, Mod. Phys. Lett. A 2 (1987) 893. [2] P. Forgacs et al., Phys. Lelt. B 227 (1989) 214; M. Bershadsky and H. Ooguri, preprint IASSN-HEP-89/ 09(1989). [3] E. Abdalla, M.C.B. Abdalla and A. Zadra, preprint ICTP/ 89/56. [41 A.M. Polyakov and A.B. Zamolodchikov, Mod. Phys. Lett. A3 (1988)819. [5] AI.B. Zamolodchikov, preprint ITEP 89-84 ( 1989 ). [6] Y. Matsuo, Phys. Lctt. B 227 (1989) 209.
PHYSICS LETTERS B
11 January 1990
A N O T E ON T I l E C O N N E C T I O N B E T W E E N T H E SL(2, ~) C U R R E N T ALGEBRA AND G E N E R A L COVARIANCE E.N. ARGYRES. C.G. P A P A D O P O U L O S Institute o['NuclearPhysics.NR(" "'Dcmocrttos", GR-153 10Athens. Greece and E.G. FLORATOS
t'h),swsDepartment. Universityo.f('rete attd 1"0R771. lraklion. Crete. Greece Received 16 October 1989
We propose a partial gauge fixing,which enables one to derive all the symmetriesof two-dimensionalinduced quantum gravity in the light-conegauge, including the SL( 2, P ) current algebra, as the result of general covariance.
in order to study two-dimensional gravity induced by string theory, Polyakov [ 1 ] has proposed thc special gauge (so-called light-cone ( LC ) gauge ) g__=0,
g._=l,
g÷+=2h(x+,x-).
(I)
where h (x "..v - ) is the remaining degree of freedom of the metric. Then by requiring e n e r g y - m o m e n t u m conservation and using the known form of the trace anomaly in two dimensions, he derived differential equations for correlation functions of the field h. Furthermore, using the classical equations of motion. arising from the induced action
S= f d 2 - x x / - g R - I R ,
(2)
M
where A-g'aJV.VB, he showed the existence of an SL(2, ~ ) current algebra. From another point of view. the SL(2. ~ ) current algebra can be derived from the W Z N W model [ 1,2 ]. As has been shown, one can establish an equivalence (at lcast on-shell ) between the W Z N W model with group SL(2, ~ ) and two-dimensional induced gravity. This can be achieved in the LC gauge as well as in the usual contbrmal gauge. In this note we propose an alternative approach, in
304
order to investigate the connection between two-dimensional induced gravity and the SL(2, D) current algebra. Wc prove that the SL(2, >) symmetry resuits from a general coordinate transformation ( G C T ) invariancc of the action, preserving the form, not of the LC gauge, but of partial gauge fixing. Before analysing our approach, it is worthwhile to consider what kind of transformation this SL(2, .'4) current algebra induces on h, and whether it maintains the form of the LC gauge. Wc dcfine
dh-d~) h = ~ d l- -'+ - t " ~ ( y + , l i ( a ) ( i, + ) h ( x ' . x - ) . . . . + 2zti
(3)
where a takes the values + , - . 0. Using the operator product expansion j(a)(x
+ )j(')( V '
)
j"~d) (y + ) =2~"%1'a ~ : T - v ;
t]ah c 12 ( x + - y + )
2'
(4)
where ~ +o= 1. q~ _ = ~ and qoo= l , a n d t h c c x p a n s i o n o f h in powers o f x - . h ( x +. x - ) =./~+ ' ( x ' ) - 2 / " " ( x ' ).v+i<
'(.~" ) ( x ) -~.
(5)
0370-2693/90/$ 03.50 53 ElsevierScience Publishers B.V. ( North-Ilolland )
('O. - hO _ +
~_ tl ),5.\-- ,
(6)
X c x p ( - S I + S ~ "v )
with
= f [d.W'] [de_ l tdc+ ] [ d h - - ]
dr- =~->(x
' )-2~(°~(x
+(~+~(x+)(.\.
' ).\-
g__=0,
(8)
We notice that the parametrization (8) is an interpolation between L(" formal ( h = 0 ) gauges. Starting with over the string coordinates, fixingg
of the metric in (0 = 0) and conthe path integral _ = 0 , wc get
] [db--]
× c x p ( -S~ + S v P ) .
(9)
where "/'[ / ..... ,S'I = ~_ , d2-v~/' ( - g ) 8,*X*'O/~'k'"g"/51t,,,
A'~'0_ A'")tl~ ....
,)
(10a) and
Tf 5
whcrc
-
"l'[d2x(h--0 2 J
d~-xcxp(~/2)h--(~_c.-~..Oc_). (10b)
The induced action in the LC gauge can easily be obtained by integrating over 0, using the gauge fixing 0 = O,
c +h+~0+c+)
(12)
which contributes - 26 to the trace anomaly. The GCT translbrmations which preserve the gauge (8) are given by ~5.\+ = f ( x + ). 6 x - arbitrary, under which 0 and h transform as follows:
60=2(0+dx++0
6\-)+0,Odx'+3_Od.v-.
(13a) ~5t1=(0+-h0_
g++ = 2 e x p ( o / 2 ) h .
f. = "/" | d2x(O+ .¥~'0_ X " - h ~ _
(ll)
(7)
g+_=cxp(0/2),
Z = f [dh] [dO] [d.W'] [de
[dh++ l
X cxp ( - .S'~ + .S'~,L~' ) .
)2.
These transformations can be thought of as a combination o f a G C T plus a Weyl transformation, with parameters ,iv + = 0 , ~t\-- given by (7) and 2=-0 dv . One can easily verify that these transformations preserve the form of the LC gauge. What we wish to do. is to consider the symmetries of induced gra~.ity in the I.C gauge as the result of the general covariancc of the theory. We define the partial gauge fixing
,,-] l, - -
r J [do] [dX"] [de_] [ d h - - ]
exp[-St(h)]=
wc find ,~h=
11 January 1900
PHYSICS LETTERS B
Volume 234, number 3
+0_h)dx-+0+
(h,J.v +) .
(13b)
What is the interesting point of (13), is that the transtbrmations tbr O and 11 arc dccoupled, which means that ( 13b ) is still present after the integration over 0 and its F a d d c c v - P o p o v partners, and therefore is an invariance of the action S t ( h ) . the invariance is automatically guarantccd, sincc (13b) is nothing but a (iCT transformation. All the known symmetries of the action St,(t1 ) are included in the gcncral tbrm (13b). These symmetries manifest themselves, in the L(: gauge, as a combination o f a G C T and a Wcyl transformation: (1) d \ arbitrary, d x ~ = J ~ x +) and 2 = - ( 0 , 6 x + + 0_ ~,t-- ) which has bccn studicd in rcf. [31. (II) dx* =((.x -+ ), 6,\- = - . \ - ~3+ ~+11(.\'+ ). ).= 0. which is thc residual G C T symmctry of thc LC gaugc [4], gcncrating the Virasoro and the 0 ( I ) K a e - M o o d y algcbra. Following the reasoning of refs. [ 1,5 ], we derive perturbatively the equations of motion, which in the limit O = 0 take the standard from 0~ h = 0 . I f n o w wc requirc that the transformations (13b) preserve the form of the equations of motion, the independent variations 6x arc restricted at the classical level to bc of thc form (7) inducing thus the SL(2. ~) current algebra of the h-transformations. Furthermore. one can obtain the correct central charge dependence by replacing tl with ~ch. Thus. SL(2, ~) comcs into 305
Volume 234, number 3
PHYSICS LETTERS B
play without any information concerning the corrclation functions ofh and in this sense is a purely classical result. Wc conclude that using the more general metric parametrization (8), one can study all the symmetries of SL(h). In fact these symmetries manifest themsclvcs as GCT transformations, which guarantee automatically the invariance of the action. SL(2, P) seems to be an inherent characteristic of induced gravity, related not to the special form of the LC gauge, but to the "'spin" or scale dimensions of the metric [ 6 ].
306
11 January 1990
References [ 1 ] A.M. Polyakov, Mod. Phys. Lett. A 2 (1987) 893. [2] P. Forgacs et al., Phys. Lelt. B 227 (1989) 214; M. Bershadsky and H. Ooguri, preprint IASSN-HEP-89/ 09(1989). [3] E. Abdalla, M.C.B. Abdalla and A. Zadra, preprint ICTP/ 89/56. [41 A.M. Polyakov and A.B. Zamolodchikov, Mod. Phys. Lett. A3 (1988)819. [5] AI.B. Zamolodchikov, preprint ITEP 89-84 ( 1989 ). [6] Y. Matsuo, Phys. Lctt. B 227 (1989) 209.