- Email: [email protected]
Two-dimensional quantum gravity and current algebra on a torus
Volume 240, number 3,4
PHYSICS LETTERS B
26 April 1990
T W O - D I M E N S I O N A L Q U A N T U M GRAVITY A N D C U R R E N T ALGEBRA O N A T O R U S Jung-Jai LEE, Won-Sang C H U N G , Sin-Kyu K A N G and Jae-Kwan KIM
Department of Physics, KoreaAdvancedInstitute of Science and Technology, P.O. Box 150, Chungryang,Seoul, Korea Received 24 January 1990
The euclidean generating functional of the correlation functions of the energy-momentum tensor in conformal field theory on a torus is derived. The quantization of the effective action for induced two-dimensional gravity on the torus is still related to the SU(2) XSU(2) current algebra.
The quantization of 2D gravity plays a very important role in string theory and 2D gravity-matter coupling theory. In refs. [ 1,2] Polyakov et al. have developed the quantization of 2D gravity and its relation to the SL(2, ) current algebra. The supersymmetric extension and more detailed investigations have been developed. In this stage it is very interesting to consider the quantization of 2D gravity on a different topology from the complex plane which plays the role o f the background field vacuum in quantum gravity. The local property of correlation in 2D quantum gravity theory on the torus is the same as that of the complex plane, but its global property will be given as nontrivial topological information of the gravitational correlation functions on the torus. In this paper, we will derive the OPE form of the gravitational correlation function from the Ward identity given in 2D conformal field theory on the torus [ 3 ] since it is known that the generating functional for correlation functions of the stress-energy tensor in conformal field theory is equal to the effective action for 2D induced gravity in the light-cone gauge recently given by Polyakov [ 1 ]. The correlation functions of the holomorphic components of the stress-energy m o m e n t u m tensor T ( Z ) on a general Riemann surface [ 3 ] is
( T ( z ) T ( w i ) ... T(w~.)) =
C
N
3
~
24nj~l (V~j) (Tz~(wj, z ) ( T ( w l ) ... T(w,v))
N
3g- 3
- ~ [2Vw,(7(wj, z ) + G ( w j , z ) V w , ] ( T ( w l ) . . . T ( w u ) ) + j=l
r
~ ITzz ~ d v 6 ~ ( T ( v ) T ( w , ) . . . T ( w u ) ) , i=1
cycles
(1) where the H~-z are holomorphic quadratic two-forms, 6~,~ the discontinuities on the Riemann surface and (7 is a modified Green's function which obeys V'(7(z, w ) = -
I
~gg6(Z-W) .
(2)
From eq. ( 1 ), using the generating functional method [4] or anomaly relations [ 1 ], we can find the Ward identity
c (Vz)3h=vzSW(h) -2Vzh 8W
24n
5~
~-
-
hv6W ~
,
0370-2693/90/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland )
(3)
341
Volume 240, number 3,4
PHYSICS LETTERS B
26 April 1990
where W and h are, respectively, the generating functional and the source function of the quantum fields on the Riemann surface. After the procedure of renormalization of the central charge c-~c' = c - 2 6 [1,4 ], we obtain the quantum Ward identity of the quantum field h(z, 2):
/ -VzJ+V.Jh+2JVzh-
<
~(Vz)3h
) =0,
(4)
where we introduce a source function J to quantize the field h(z, if) by using the generating functional method [4]. On the other hand, by choosing the moduli parameter r in the standard euclidean metric on the torus, ds 2-- Idzl 2 z-- o~ + roe, 0 < al, a2 < 1, we can find the Ward identities for the individual correlation functions: C' N 24n0-3 ( h ( z , £)h(w,, if1) ... h(wx, if~,,) ) = ~ O t ~ ( z - w f l ( h ( w , , ift) ... h(wN, ifu) ) j=l
N -
(5)
Y~ [ ~ ( z - w , ) O w , + O z 6 ( z - w j ) ] ( h ( w l , if1) ...h(WN, ifu)) • j=l
In order to find the OPE ofeq. (5), we introduce the f u n c t i o n f ( z , w) which obeys the following relation on a torus:
03f(z, W) = ~ ( Z - - W )
.
(6)
After the rescaling h--.h/k, where the relation between k and c was discussed in ref. [2], the result is N
( h(z, ~.)h(wl, if1 ) ... h(WN, if:u) ) =k ~. Off(z, %) ( h(wl, if1 ) ... h(wN, ifN) ) j=l
- ~. Lf(z, wj)Ow,+O,f(z, w j ) ] ( h ( z l , Z , ) . . . h ( z N , ~ u ) > + ~ d v ( h ( v , O ) h ( w , , i f l ) . . . h ( w N , j=l
ifx)),
(7)
where the integral is along a-cycles on the torus. The leading singular term o f f ( z , w) is the first one given in appendix A:
f(z, w) ~ ( z - w)20~ In [O1 ( g - ifl f) ] •
(8)
To analyze eq. (7), let us consider the correlation function for arbitrary fields with the transformation laws under the change of coordinate z: t~¢= ~O:¢+,;t(OzE)O.
(9)
In this case, we have ( h(z, ff')Ol(Wl, i f 1 ) " " ¢ N ( W N , ifN) ) = E If(z, W])Ow~-l-2,~ Ozf(z, Wj) ](~)l(Wl, if1)""~)N(WN, iflV) ) J
+ ~ dv(h(v, 0)01(wl, if1) ... ON(Wx, ifu)) ,
(10)
which shows that with the locally expressed form of eq. (10) on the torus, the operator h(z, ~) is antihoiomorphic, except for the quadratic dependence on z. From this fact we can write
h(z, ~) = J ( - 1)(£) _ 2zj(O)(2) _ z 2 j ( +) ( ~ ) ,
( I 1)
introducing the antiholomorphic operators J~ ($) (a = + 1, 0). We find, after substituting eq. ( 11 ) into eq. ( ! 0 ) ( J a ( Z ) ¢ l ( Wl, ifl ) "'" ON( WN, if N) )
= ~ t~G(w~,z)(fbI(Wl,If,)...dpN(WN, IfN))+ 3= I
342
do(J"(O)fbl(W,,ifl)...fgu(WN, ifN)),
(12)
Volume 240, number 3,4
PHYSICS LETTERSB
26 April 1990
where the function G on the right-hand side is given by the elliptic theta function:
OzOl(z-wlr) Ol(z-wlr)
G(w,z)-
We see that the t] are generators of SU(2) and the j a are the corresponding currents. Looking at the short distance behavior implied by eq. (7), we obtain the usual current algebra for the group SU (2) on the torus, as discovered in ref. [5]. This interpretation is further confirmed by the use of eq. (7): k
(ja (~,)jb,(~,) ... jbu(I~N) ) = 2 ~, rlab,OzG( wj, Z) (jb,( ~, ) ... jb,(w2) "'" ja,,(wu) ) + y" ffb'~q¢'e,G( wj, z) (J~'( ~ ) ... jc,(v~j) ... jh~(¢vN) ) + ~ d 0 ( J " ( 0 ) J b ' ( ~ ) ... jb,,(WN) ) •
( 13 )
J
The short distance behavior of eq. ( 13 ) is similar to that of the auxiliary operator J(z~, z2) introduced in ref. [ 5 ]. This result allows us to check the fact that t h e f ~t~ are the structure constants of SU (2) in the Cartan basis (I ±, I °), while the qab are the inverse of the Killing form coefficients for the same group, qoo= 2r/± = 1. The local form ofeq. ( 13 ) has non-abelian structure discovered by Polyakov [ 1 ], who identified it as the SL (2, ~ ) algebra while working on 2D gravity in Minkowski space. So, we derived the Ward identity for 2D induced quantum gravity on a torus. We think that this theory can be extended to 2D supergravity on a super torus [6 ]. The tasks remaining for this theory are as follows. One is that this theory should be generalized to the general (super) Riemann surface, which is supposed needing further careful investigation. Another is that the conformal version of this theory should be given manifestly, the fiat case of which is given in ref. [ 7 ]. We hope that these problems will be cleared in the near future. This research is supported by the Korea Science and Engineering Foundation (KOSEF).
Appendix In this appendix we calculate explicitly the function f ( z , w) given in eq. (8). f ( z , w) satisfies the following relation on a torus:
03f(z, w) = tt6(z, w) .
• (14)
We choose the standard euclidean metric on the torus, ds 2= Idzl 2 ,
(15)
where
z=a~+ra2,
Z=a~+faz
(O
(16, 17)
and r is a modular parameter on the torus and f is its complex conjugate. Using the mode expansion on the toms, we get f o r f ( z , w)
f(z,w)_
(Im r)3
~2 --
~,'
tl,rtlEl
exp{(n/Imr)[(m-n~)(z-w)-(m-nr)(2-vO)]} (rn_nf)3
r)________ (Im7[2 3 ~ , exp I--mr ( - f a + r a )
~' K(m),
(18)
m
where Ol~Z--W
,
c~=g-~,
K ( m ) = Z',,
exp(nm/Im r) ( a - a ) (m_nf)3
(19,20,21) 343
Volume 240, number 3,4
PHYSICS LETTERS B
26 April 1990
and the prime denotes the exclusion o f the zero m o d e ( r n = n = 0 ) . Since K(m) has a pole at rn=n'~, we can convert the sum o f K ( m ) into a contour integral by using the S o m m e r f e l d - W a t s o n transformation [ 8 ]: 1 lim I q~ e x p ( i n z ) exp(nz/Im z ) ( a - 6 t ) a-o 2i o s i ~ n z (z_n.~)3+2 3 - 2--3,
K(m)= ,~
(22)
¢
where 2 is a regularization p a r a m e t e r arising due to the zero m o d e insertion (the n = m = 0 above transformation we can obtain
f(z,w)=
(Imr) 3
re2
case). Using the
~ exp(2nina)B(n, ot, a , t ) ,
(23)
tl
where
B(n,a,a,f)=
+ \Imr,/
\n]
--
ln[l-exp(2nine)]+
In[ 1 - e x p ( 2 n i n , )
~mr(a-a)
ln[l-exp(2nin~)l
(24)
] .
Using the relation which originates from the well-known free boson propagator in string theory, 4n2i exp(2~rinr) 0z In l o t (zl z) ] = I m r ~ exp(2ninz) 1 - e x p ( 2 z t i n z ) '
(25)
we obtain /i__
_\ 3
f(z, 0 ) = (z--g)2Ozln[tgl(g,,)]-b(~--~)(Z--Z.)OI(O,)
-2
ln[Ot (g,'~) ]
4 ( I m r)~ (Of)2(O.. ) _ 3 l n [ O , ( g [ ~ ) ] . 7t
(26)
Using the relation O¢O~ = ( 1/4hi )03 0~, the second and third term o f eq. ( 2 6 ) have the following local forms: 0¢ (O~) - 2 In [ O~ (g[ f ) ] ~ in g + higher o r d e r t e r m s , (Of)2
(Oz-) -
3 In [ O~(zl z) ] ~ - - z + g In g + higher o r d e r t e r m s .
(27)
Therefore, the leading term o f f ( z , 0 ) can be written as
f(z, O) ~ z20. ln[Ot (el e) ] +a(g)z+ b(g), where a(:?) and b(~) are analytic functions o f z?.
References [ I ] A.M. Polyakov, Mod. Phys. Left. A 2 ( 1987 ) 893. [2] V.G. Knizhnik, A.M. Polyakov and A.B. Zamolodchikov, Mod. Phys. Lett. A 3 (1988) 819. [3] T. Eguchi and H. Ooguri, Nucl. Phys. B 282 (1987) 308. [4] K. Yoshida, Mod. Phys. Lett. A 4 (1989) 71. [5] V.G. Knizhnik and A.B. Zamolodchikov, Nucl. Phys. B 247 (1984) 83. [6] J. Polchinski, Nucl. Phys. B 324 (1989) 123. [7] J.J. Lee,W.S. Chung, S.K. Kang and J.K. Kim, in preparation. [8] J. Polchinski, Commun. Math. Phys. 104 (1986) 37. [9] J. Distler and H. Kawai, Nucl. Phys. B 321 (1989) 509. 344
(28)
PHYSICS LETTERS B
26 April 1990
T W O - D I M E N S I O N A L Q U A N T U M GRAVITY A N D C U R R E N T ALGEBRA O N A T O R U S Jung-Jai LEE, Won-Sang C H U N G , Sin-Kyu K A N G and Jae-Kwan KIM
Department of Physics, KoreaAdvancedInstitute of Science and Technology, P.O. Box 150, Chungryang,Seoul, Korea Received 24 January 1990
The euclidean generating functional of the correlation functions of the energy-momentum tensor in conformal field theory on a torus is derived. The quantization of the effective action for induced two-dimensional gravity on the torus is still related to the SU(2) XSU(2) current algebra.
The quantization of 2D gravity plays a very important role in string theory and 2D gravity-matter coupling theory. In refs. [ 1,2] Polyakov et al. have developed the quantization of 2D gravity and its relation to the SL(2, ) current algebra. The supersymmetric extension and more detailed investigations have been developed. In this stage it is very interesting to consider the quantization of 2D gravity on a different topology from the complex plane which plays the role o f the background field vacuum in quantum gravity. The local property of correlation in 2D quantum gravity theory on the torus is the same as that of the complex plane, but its global property will be given as nontrivial topological information of the gravitational correlation functions on the torus. In this paper, we will derive the OPE form of the gravitational correlation function from the Ward identity given in 2D conformal field theory on the torus [ 3 ] since it is known that the generating functional for correlation functions of the stress-energy tensor in conformal field theory is equal to the effective action for 2D induced gravity in the light-cone gauge recently given by Polyakov [ 1 ]. The correlation functions of the holomorphic components of the stress-energy m o m e n t u m tensor T ( Z ) on a general Riemann surface [ 3 ] is
( T ( z ) T ( w i ) ... T(w~.)) =
C
N
3
~
24nj~l (V~j) (Tz~(wj, z ) ( T ( w l ) ... T(w,v))
N
3g- 3
- ~ [2Vw,(7(wj, z ) + G ( w j , z ) V w , ] ( T ( w l ) . . . T ( w u ) ) + j=l
r
~ ITzz ~ d v 6 ~ ( T ( v ) T ( w , ) . . . T ( w u ) ) , i=1
cycles
(1) where the H~-z are holomorphic quadratic two-forms, 6~,~ the discontinuities on the Riemann surface and (7 is a modified Green's function which obeys V'(7(z, w ) = -
I
~gg6(Z-W) .
(2)
From eq. ( 1 ), using the generating functional method [4] or anomaly relations [ 1 ], we can find the Ward identity
c (Vz)3h=vzSW(h) -2Vzh 8W
24n
5~
~-
-
hv6W ~
,
0370-2693/90/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland )
(3)
341
Volume 240, number 3,4
PHYSICS LETTERS B
26 April 1990
where W and h are, respectively, the generating functional and the source function of the quantum fields on the Riemann surface. After the procedure of renormalization of the central charge c-~c' = c - 2 6 [1,4 ], we obtain the quantum Ward identity of the quantum field h(z, 2):
/ -VzJ+V.Jh+2JVzh-
<
~(Vz)3h
) =0,
(4)
where we introduce a source function J to quantize the field h(z, if) by using the generating functional method [4]. On the other hand, by choosing the moduli parameter r in the standard euclidean metric on the torus, ds 2-- Idzl 2 z-- o~ + roe, 0 < al, a2 < 1, we can find the Ward identities for the individual correlation functions: C' N 24n0-3 ( h ( z , £)h(w,, if1) ... h(wx, if~,,) ) = ~ O t ~ ( z - w f l ( h ( w , , ift) ... h(wN, ifu) ) j=l
N -
(5)
Y~ [ ~ ( z - w , ) O w , + O z 6 ( z - w j ) ] ( h ( w l , if1) ...h(WN, ifu)) • j=l
In order to find the OPE ofeq. (5), we introduce the f u n c t i o n f ( z , w) which obeys the following relation on a torus:
03f(z, W) = ~ ( Z - - W )
.
(6)
After the rescaling h--.h/k, where the relation between k and c was discussed in ref. [2], the result is N
( h(z, ~.)h(wl, if1 ) ... h(WN, if:u) ) =k ~. Off(z, %) ( h(wl, if1 ) ... h(wN, ifN) ) j=l
- ~. Lf(z, wj)Ow,+O,f(z, w j ) ] ( h ( z l , Z , ) . . . h ( z N , ~ u ) > + ~ d v ( h ( v , O ) h ( w , , i f l ) . . . h ( w N , j=l
ifx)),
(7)
where the integral is along a-cycles on the torus. The leading singular term o f f ( z , w) is the first one given in appendix A:
f(z, w) ~ ( z - w)20~ In [O1 ( g - ifl f) ] •
(8)
To analyze eq. (7), let us consider the correlation function for arbitrary fields with the transformation laws under the change of coordinate z: t~¢= ~O:¢+,;t(OzE)O.
(9)
In this case, we have ( h(z, ff')Ol(Wl, i f 1 ) " " ¢ N ( W N , ifN) ) = E If(z, W])Ow~-l-2,~ Ozf(z, Wj) ](~)l(Wl, if1)""~)N(WN, iflV) ) J
+ ~ dv(h(v, 0)01(wl, if1) ... ON(Wx, ifu)) ,
(10)
which shows that with the locally expressed form of eq. (10) on the torus, the operator h(z, ~) is antihoiomorphic, except for the quadratic dependence on z. From this fact we can write
h(z, ~) = J ( - 1)(£) _ 2zj(O)(2) _ z 2 j ( +) ( ~ ) ,
( I 1)
introducing the antiholomorphic operators J~ ($) (a = + 1, 0). We find, after substituting eq. ( 11 ) into eq. ( ! 0 ) ( J a ( Z ) ¢ l ( Wl, ifl ) "'" ON( WN, if N) )
= ~ t~G(w~,z)(fbI(Wl,If,)...dpN(WN, IfN))+ 3= I
342
do(J"(O)fbl(W,,ifl)...fgu(WN, ifN)),
(12)
Volume 240, number 3,4
PHYSICS LETTERSB
26 April 1990
where the function G on the right-hand side is given by the elliptic theta function:
OzOl(z-wlr) Ol(z-wlr)
G(w,z)-
We see that the t] are generators of SU(2) and the j a are the corresponding currents. Looking at the short distance behavior implied by eq. (7), we obtain the usual current algebra for the group SU (2) on the torus, as discovered in ref. [5]. This interpretation is further confirmed by the use of eq. (7): k
(ja (~,)jb,(~,) ... jbu(I~N) ) = 2 ~, rlab,OzG( wj, Z) (jb,( ~, ) ... jb,(w2) "'" ja,,(wu) ) + y" ffb'~q¢'e,G( wj, z) (J~'( ~ ) ... jc,(v~j) ... jh~(¢vN) ) + ~ d 0 ( J " ( 0 ) J b ' ( ~ ) ... jb,,(WN) ) •
( 13 )
J
The short distance behavior of eq. ( 13 ) is similar to that of the auxiliary operator J(z~, z2) introduced in ref. [ 5 ]. This result allows us to check the fact that t h e f ~t~ are the structure constants of SU (2) in the Cartan basis (I ±, I °), while the qab are the inverse of the Killing form coefficients for the same group, qoo= 2r/± = 1. The local form ofeq. ( 13 ) has non-abelian structure discovered by Polyakov [ 1 ], who identified it as the SL (2, ~ ) algebra while working on 2D gravity in Minkowski space. So, we derived the Ward identity for 2D induced quantum gravity on a torus. We think that this theory can be extended to 2D supergravity on a super torus [6 ]. The tasks remaining for this theory are as follows. One is that this theory should be generalized to the general (super) Riemann surface, which is supposed needing further careful investigation. Another is that the conformal version of this theory should be given manifestly, the fiat case of which is given in ref. [ 7 ]. We hope that these problems will be cleared in the near future. This research is supported by the Korea Science and Engineering Foundation (KOSEF).
Appendix In this appendix we calculate explicitly the function f ( z , w) given in eq. (8). f ( z , w) satisfies the following relation on a torus:
03f(z, w) = tt6(z, w) .
• (14)
We choose the standard euclidean metric on the torus, ds 2= Idzl 2 ,
(15)
where
z=a~+ra2,
Z=a~+faz
(O
(16, 17)
and r is a modular parameter on the torus and f is its complex conjugate. Using the mode expansion on the toms, we get f o r f ( z , w)
f(z,w)_
(Im r)3
~2 --
~,'
tl,rtlEl
exp{(n/Imr)[(m-n~)(z-w)-(m-nr)(2-vO)]} (rn_nf)3
r)________ (Im7[2 3 ~ , exp I--mr ( - f a + r a )
~' K(m),
(18)
m
where Ol~Z--W
,
c~=g-~,
K ( m ) = Z',,
exp(nm/Im r) ( a - a ) (m_nf)3
(19,20,21) 343
Volume 240, number 3,4
PHYSICS LETTERS B
26 April 1990
and the prime denotes the exclusion o f the zero m o d e ( r n = n = 0 ) . Since K(m) has a pole at rn=n'~, we can convert the sum o f K ( m ) into a contour integral by using the S o m m e r f e l d - W a t s o n transformation [ 8 ]: 1 lim I q~ e x p ( i n z ) exp(nz/Im z ) ( a - 6 t ) a-o 2i o s i ~ n z (z_n.~)3+2 3 - 2--3,
K(m)= ,~
(22)
¢
where 2 is a regularization p a r a m e t e r arising due to the zero m o d e insertion (the n = m = 0 above transformation we can obtain
f(z,w)=
(Imr) 3
re2
case). Using the
~ exp(2nina)B(n, ot, a , t ) ,
(23)
tl
where
B(n,a,a,f)=
+ \Imr,/
\n]
--
ln[l-exp(2nine)]+
In[ 1 - e x p ( 2 n i n , )
~mr(a-a)
ln[l-exp(2nin~)l
(24)
] .
Using the relation which originates from the well-known free boson propagator in string theory, 4n2i exp(2~rinr) 0z In l o t (zl z) ] = I m r ~ exp(2ninz) 1 - e x p ( 2 z t i n z ) '
(25)
we obtain /i__
_\ 3
f(z, 0 ) = (z--g)2Ozln[tgl(g,,)]-b(~--~)(Z--Z.)OI(O,)
-2
ln[Ot (g,'~) ]
4 ( I m r)~ (Of)2(O.. ) _ 3 l n [ O , ( g [ ~ ) ] . 7t
(26)
Using the relation O¢O~ = ( 1/4hi )03 0~, the second and third term o f eq. ( 2 6 ) have the following local forms: 0¢ (O~) - 2 In [ O~ (g[ f ) ] ~ in g + higher o r d e r t e r m s , (Of)2
(Oz-) -
3 In [ O~(zl z) ] ~ - - z + g In g + higher o r d e r t e r m s .
(27)
Therefore, the leading term o f f ( z , 0 ) can be written as
f(z, O) ~ z20. ln[Ot (el e) ] +a(g)z+ b(g), where a(:?) and b(~) are analytic functions o f z?.
References [ I ] A.M. Polyakov, Mod. Phys. Left. A 2 ( 1987 ) 893. [2] V.G. Knizhnik, A.M. Polyakov and A.B. Zamolodchikov, Mod. Phys. Lett. A 3 (1988) 819. [3] T. Eguchi and H. Ooguri, Nucl. Phys. B 282 (1987) 308. [4] K. Yoshida, Mod. Phys. Lett. A 4 (1989) 71. [5] V.G. Knizhnik and A.B. Zamolodchikov, Nucl. Phys. B 247 (1984) 83. [6] J. Polchinski, Nucl. Phys. B 324 (1989) 123. [7] J.J. Lee,W.S. Chung, S.K. Kang and J.K. Kim, in preparation. [8] J. Polchinski, Commun. Math. Phys. 104 (1986) 37. [9] J. Distler and H. Kawai, Nucl. Phys. B 321 (1989) 509. 344
(28)