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Two types of matrix models as a coordinate-momentum interchange in a black hole model
Physics Letters B 322 (1994) 188-191 North-Holland
P H YSIC S k ETT ER S B
Two types of matrix models as a coordinate-momentum interchange in a black hole model Kei Ito Department of Physics, Nagoya Institute of Technology, Gokiso, Showa-ku, Nagoya 466, Japan Received 29 June 1993 Editor: M. Dine
We find a simple physical interpretation of the existence of two types of matrix models (Penner's model and the standard c---1 model) which have many similarities. If we look at the continuum theories underlying these matrix models, we find that the momentum representation of one model is the same as the coordinate representation of the other. A possible implication of this fact on the relation between four-dimensional gravity and the 2D string is also discussed.
Although there has been a remarkable progress in our understanding of two-dimensional q u a n t u m gravity, there remain some mysteries. One of them the understanding o f which might play a key role in extending our knowledge from two to four dimensions - is the question why there exist two types o f matrix models (Penner's model [ 1 ] and the standard c = 1 matrix model [ 2 ] ), which have m a n y similarities [ 1,3 ]. This phenomenon is rather universal and not restricted to the c = 1 model. For c < 1 models, for each value o f the central charge c, there exists a so-called Kontsevich type model [4,5 ], besides the standard multi-matrix model. Therefore we have two apparently distinct matrix models for each value o f the central charge of the matter which couples the twodimensional quantum gravity. Here we find a simple and clear physical interpretation of the existence of two types of matrix models (Penner's model and the standard c = 1 model ). If we look at the continuum theories underlying these matrix models, we find that the m o m e n t u m representation of one model is the same as the coordinate representation o f the other one. The underlying continuum theory of Penner's model was found by Witten [6]. It is the S U ( 2 ) / U ( 1 ) gauged W Z W model with the level value k analytically continued to a negative value, which is - 3 . He suggested it would be more natural to start with a gauged W Z W model of SL(2, ~ ) / U ( 1 ), rather than S U ( 2 ) / U ( 1 ) 188
[ 6 ]. We show that the SL (2, R ) / U ( 1 ) gauged W Z W model with the particular level value k = 3 is the continuum theory underlying Penner's matrix model, following Witten's argument [6 ]. Actually, this model is related to the k = 3 black hole model [7], by the target-space duality. (This was also suggested in ref. [ 6 ]. ) On the other hand, the continuum theory underlying the c = 1 matrix model, is c = 2 5 Liouville theory coupled to a c = 1 scalar field. The value of the conformal anomaly of the Liouville theory, which is c = 2 5 , is very important, since this determines the vanishing string susceptibility [ 8 ], which characterizes the scaling properties of the amplitudes of the c = 1 matrix model [ 2 ]. The crucial observation in this paper is that the black hole model, with the particular value k = 3, when represented in m o m e n t u m space, is the same as the coordinate representation of a Liouville theory coupled to a scalar field, with the conformal anomaly o f the Liouville theory being precisely c = 25. In order to derive from the coordinate representation to the m o m e n t u m representation in a black hole model, we follow Martinec and Shatashvili [ 9 ]. First we show that the underlying continuum field theory of Penner's matrix model is the k = 3, SL(2, ) / U (1) gauged W Z W model coupled to topological gravity. This can be done by simply changing Witten's argument [6] by replacing S U ( 2 ) / U ( 1 ) by
0370-2693/94/$ 07.00 © 1994 Elsevier Science B.V. All rights reserved. SSDI 0370-2693 (93)E1562-C
Volume 322, number 3
PHYSICS LETTERS B
17 February 1994
SL (2, g~)/U ( 1 ), k by - k, and ¢ by - ip (with p defined below). The partition function of Penner's matrix model is the Euler characteristic Z of the moduli space of Riemann surfaces [ 1 ]. Therefore it is necessary to show that the partition function of the k = 3 SL (2, R ) / U ( 1 ) gauged WZW model coupled to topological gravity is alsoz [6]. The lagrangian of the level k, SL(2, ~ ) / U ( 1 ) gauged WZW model is given by [ 6 ]
If the theory is coupled to topological gravity, the correlation function
k Lowzw = ~
where CT(V) is the top Chern class of the vector bundle v, whose fiber is
[OzpOzp-L'IzOzp+iAeO~p
+ e ~ ( O~- i & ) u (O~+ iA,) a ] ,
( 1)
where the SL (2, ~ ) group element is parametrized by
of the observables U~---(e°) r is given by (_k+2)g_
CT(V),
V=H°(~"K®[I+I/(k-2)]@ O(zi)-l/(k-2))
[9]
(9) 1
g=(u
0
1
0~(epl]\0 e - V ) ( 0
112) ,
(2)
and the U ( 1 ) symmetry p-*p,
u-*e-i~u,
ti-,ei~ti
(3)
is gauged by the introduction of the gauge fields A~(Az), which transform under the U(1 ) transformation as 8Az= -0~e (SAc= - 0 ~ ) . This model can be made topologically invariant and the gauge fixing of the topological invariance results in the appearance of the ghost field a and the corresponding antighost fl~ [6]. The fermion pair (a, fl~) develops a chiral anomaly, whose effective action is given by [ 6 ]
&u=-- ~1 I p.(2F+½R).
(4)
This effective action together with the p part of the gauged WZW action gives
S'(p,A)= k f x/%hUO~pOjp 1
+
(8)
Mg,.~
2n
This takes a manifestly topological invariant form, if the gauge field is shifted: 1
[
S"(p,A)= -~np. (k--2)F-½R]
(6)
where K is the cotangent space of the Riemann surface Z. In particular, if the level k equals 3, and if we consider the partition function, the fiber V of the vector bundle in question is V = H°(Z, K ® K ) ,
(10)
which is the cotangent bundle T *lf,Is to the moduli space of the Riemann surfaces at Z. Hence the genus g partition function Zg is
Zg = f
cx(T*lVl,) = ( - 1 ) 3g-3X(iglg),
( 11 )
where X is the Euler characteristic of the moduli space of Riemann surfaces [ 6 ]. Now, in order to connect the k = 3, SL (2, R ) / U ( 1 ) gauged WZW model to c= 25 Liouville theory coupled to a scalar, we first perform a target-space duality transformation [ 10 ]. The target-space duality is a generalization of the R ~ 1/R symmetry ofa compactiffed boson on a circle with radius R. In a conformal field theory this corresponds to changing the chirality of the left mover while fixing the chirality of the right mover [ 10 ]. In our case this transformation can be implemented by changing the U ( 1 ) embedding, and now the U( 1 ) symmetry which should be gauged, is p-~p-~,
u-~e'u , a~e~ .
(12)
The Lagrangian of the gauged WZW model which has this U ( 1 ) symmetry is 189
Volume 322, number 3
PHYSICSLETTERSB
k
L~wzw = ~ [ (O~p-A~) (Ozp-Ag)
+eZP(O~+Az)U(G + Az)a] .
-
(13)
This is nothing but the Lagrangian for a two-dimensional black hole with level k [ 7,9 ]. (The fact that these two distinct U ( 1 ) embeddings are related by target-space duality, was suggested in ref. [ 6 ]. ) Following Martinec and Shatashvili [ 9 ], we represent this Lagrangian in terms of conjugate momentum fields: ~L k ePHp- ~(0op) - 2rc (Oop-Ao) , ~SL
k -
5L
1
e2P(
Ha- ~i(0ofi) - 2rc
4rc
Normalizing the fields by fO= ~
1
O
,
0
0"= ~ ,
lIu=otue ~+i°, H~=aue ~'-i°, one obtains the classical Lagrangian 4rc
R~
arc { /2--0) + T # e x p k 4 ~S-~
+ ~00~0+
iR0.
(23)
CL=I+
(17)
L= ~ "2(Oz¢Oz~+O~OOzO)+--~//e 2~ ,
(22)
(16)
Parametrizing the conjugate momentum fields by ~o and 0 as
(18)
where lt=a~fftu is the cosmological constant [9]. Upon quantization of the theory, the effective Lagrangian acquires a term which represents a coupling of ¢ to the scalar curvature:
190
(21)
(15) Upon elimination of u, a, this reads [ 9 ]
and an anomaly term [ 9 ]
00=-00.
This is nothing but a Lagrangian of the Liouville theory (Liouville field ~) coupled to a scalar field 0 [9 ]. The conformal anomaly of the Liouville theory CLcan be read off from the Lagrangian, which is [ 91
t=x/~ (lluOzu+ l-laOza)- -~--/Tj / a ( 1 - u a ) .
k
00= 00,
(14)
After eliminating A1, by use of the equation of motion, substituting the constraint and fixing the unitary gauge condition, we obtain [ 9 ]
k 0~Hu 2--F 4rc L= ~ /Tu -k- I/7ul2
(20)
In order for 0 to couple to the scalar curvature in the standard form, 0 should be transformed to 0, by the target-space duality transformation [ 9 ]
1 k-1 Lqu = ~ 0~ 0 0 - 4 r c ~
1 x~e2p(Oz+Az)U.
- 4 ~ R~o,
2n
r J dZz(2 0z~ 0zq~- ½R~).
one obtains finally the quantum effective Lagrangian, which reads [ 9 ]
,i5
k
1
17 February 1994
(19)
6(k-l) 2 k-~
(24)
Now the crucial observation is that when k= 3 (which is the value of the level of the black hole model underlying Penner's model), then CL= 25. Thus we have shown that the momentum representation of the k = 3 SL(2, ~ ) / U ( 1 ) WZW model is the same as the coordinate representation of the Liouville theory with c=25. To be more precise, to derive the c= 25 Liouville theory from the k = 3 SL(2, R ) / U ( 1 ) WZW model coupled to topological gravity, first we made a target-space duality transformation, then went from the coordinate representation to the momentum representation, and finally made a target-space duality transformation again. Therefore, if we denote the coordinate-momentum interchange operation by V and the target-space duality transformation by U, then the transformation f, which sends one model to the other, is given by
Volume 322, number 3
f=U-I°VoU.
PHYSICS LETTERS B (25)
Since the purpose o f this p a p e r is to show a simple physical i n t e r p r e t a t i o n o f the existence o f two types o f m a t r i x models, rather than to prove the equivalence o f the two, we d i d not work out the ghost terms. Consequently, the scalar field 6~ has a negative conformal a n o m a l y which would split into c = 1 m a t t e r and ghost fields if all the ghost contributions were taken into account. In conclusion, we have found a simple a n d clear physical i n t e r p r e t a t i o n o f the existence o f two types o f matrix models ( P e n n e r ' s m a t r i x m o d e l a n d the s t a n d a r d c = 1 m a t r i x m o d e l ) which have m a n y similarities. I f we look at the c o n t i n u u m theories underlying these m a t r i x models, the m o m e n t u m representation o f one m o d e l is the same as the c o o r d i n a t e representation o f the other one. This fact is very suggestive when one tries to construct a four-dimensional self-dual gravity which is exactly soluble. A deep connection between f o u r - d i m e n s i o n a l self-dual gravity a n d the c = 1 string has been p o i n t e d out by m a n y authors [ 1 1-13 ]. One thing which is c o m m o n to t h e m is that the f o u r - d i m e n s i o n a l space is realized as a space s p a n n e d by a t w o - d i m e n s i o n a l space a n d the two-dimensional conjugate m o m e n t u m space, and a canonical t r a n s f o r m a t i o n which preserves a symplectic form, generates an area-preserving diffeom o r p h i s m , which is then connected to the Woo symm e t r y o f the c = 1 string. F r o m this p o i n t o f view, the fact that the m o m e n t u m representation o f one m o d e l is the same as the c o o r d i n a t e representation o f the other one, a n d the fact that the k = 3 SL(2, ~ ) / U ( 1 ) W Z W m o d e l is exactly soluble, i m p l y that we have a good chance o f constructing exactly soluble four-di-
17 February 1994
mensional self-dual gravity out o f these two-dimensional models. After the m a i n part o f this work was completed, we b e c a m e aware o f ref. [ 14 ], in which the same problem is also discussed, b u t from a quite different point o f view. O u r c o o r d i n a t e - m o m e n t u m interchange interpretation might b e c o m e i m p o r t a n t when one tries to construct a four-dimensional integrable selfdual gravity, for the reasons we mentioned in the text.
References [1]J. Distler and C. Vafa, Mod. Phys. Lett. A 6 (1991) 259, and references therein. [2] D. Gross and M. Miljkovic, Phys. Lett. B 238 (1990) 217; E. Brezin, V. Kazakov and AI.B.Zamolodchikov, Nucl. Phys. B338 (1990) 673; P. Ginsparg and J. Zinn-Justin, Phys. Len. B 240 (1990) 333. [3] R. Dijkgraaf, G. Moore and R. Plesser, Nucl. Phys. B 394 (1993) 356. [ 4 ] E. Witten, IAS preprint IASSNS-HEP-91/24 ( 1991 ), and references therein. [ 5 ] S. Kharchev et al., Phys. Lett. B 275 ( 1992 ) 311. [6] E. Witten, Nucl. Phys. B 371 (1992) 191. [ 7 ] E. Witten, Phys. Rev. D 44 ( 1991 ) 314. [ 8 ] J. Distler and H. Kawai, Nucl. Phys. B 321 (1989) 509. [ 9 ] E.J. Martinec and S.L. Shatashvili, Nucl. Phys. B 368 (1992) 338. [10] R. Dijkgraaf, E. Verlinde and H. Verlinde, Nucl. Phys. B 371 (1992) 269. [ll]Q.H. Park, Phys. Lett. B 236 (1990) 429; B 238 (1990) 287. [12] K. Yamagishi and G. Chapline, UC Livermore preprint (1990). [ 13 ] H. Ooguri and C. Vafa, Nucl. Phys. B 361 ( 1991 ) 469. [14] S. Mukhi and C. Vafa, preprint HUTP-93/A002(TIFR/ TH/93-01 ) (1993).
191
P H YSIC S k ETT ER S B
Two types of matrix models as a coordinate-momentum interchange in a black hole model Kei Ito Department of Physics, Nagoya Institute of Technology, Gokiso, Showa-ku, Nagoya 466, Japan Received 29 June 1993 Editor: M. Dine
We find a simple physical interpretation of the existence of two types of matrix models (Penner's model and the standard c---1 model) which have many similarities. If we look at the continuum theories underlying these matrix models, we find that the momentum representation of one model is the same as the coordinate representation of the other. A possible implication of this fact on the relation between four-dimensional gravity and the 2D string is also discussed.
Although there has been a remarkable progress in our understanding of two-dimensional q u a n t u m gravity, there remain some mysteries. One of them the understanding o f which might play a key role in extending our knowledge from two to four dimensions - is the question why there exist two types o f matrix models (Penner's model [ 1 ] and the standard c = 1 matrix model [ 2 ] ), which have m a n y similarities [ 1,3 ]. This phenomenon is rather universal and not restricted to the c = 1 model. For c < 1 models, for each value o f the central charge c, there exists a so-called Kontsevich type model [4,5 ], besides the standard multi-matrix model. Therefore we have two apparently distinct matrix models for each value o f the central charge of the matter which couples the twodimensional quantum gravity. Here we find a simple and clear physical interpretation of the existence of two types of matrix models (Penner's model and the standard c = 1 model ). If we look at the continuum theories underlying these matrix models, we find that the m o m e n t u m representation of one model is the same as the coordinate representation o f the other one. The underlying continuum theory of Penner's model was found by Witten [6]. It is the S U ( 2 ) / U ( 1 ) gauged W Z W model with the level value k analytically continued to a negative value, which is - 3 . He suggested it would be more natural to start with a gauged W Z W model of SL(2, ~ ) / U ( 1 ), rather than S U ( 2 ) / U ( 1 ) 188
[ 6 ]. We show that the SL (2, R ) / U ( 1 ) gauged W Z W model with the particular level value k = 3 is the continuum theory underlying Penner's matrix model, following Witten's argument [6 ]. Actually, this model is related to the k = 3 black hole model [7], by the target-space duality. (This was also suggested in ref. [ 6 ]. ) On the other hand, the continuum theory underlying the c = 1 matrix model, is c = 2 5 Liouville theory coupled to a c = 1 scalar field. The value of the conformal anomaly of the Liouville theory, which is c = 2 5 , is very important, since this determines the vanishing string susceptibility [ 8 ], which characterizes the scaling properties of the amplitudes of the c = 1 matrix model [ 2 ]. The crucial observation in this paper is that the black hole model, with the particular value k = 3, when represented in m o m e n t u m space, is the same as the coordinate representation of a Liouville theory coupled to a scalar field, with the conformal anomaly o f the Liouville theory being precisely c = 25. In order to derive from the coordinate representation to the m o m e n t u m representation in a black hole model, we follow Martinec and Shatashvili [ 9 ]. First we show that the underlying continuum field theory of Penner's matrix model is the k = 3, SL(2, ) / U (1) gauged W Z W model coupled to topological gravity. This can be done by simply changing Witten's argument [6] by replacing S U ( 2 ) / U ( 1 ) by
0370-2693/94/$ 07.00 © 1994 Elsevier Science B.V. All rights reserved. SSDI 0370-2693 (93)E1562-C
Volume 322, number 3
PHYSICS LETTERS B
17 February 1994
SL (2, g~)/U ( 1 ), k by - k, and ¢ by - ip (with p defined below). The partition function of Penner's matrix model is the Euler characteristic Z of the moduli space of Riemann surfaces [ 1 ]. Therefore it is necessary to show that the partition function of the k = 3 SL (2, R ) / U ( 1 ) gauged WZW model coupled to topological gravity is alsoz [6]. The lagrangian of the level k, SL(2, ~ ) / U ( 1 ) gauged WZW model is given by [ 6 ]
If the theory is coupled to topological gravity, the correlation function
k Lowzw = ~
where CT(V) is the top Chern class of the vector bundle v, whose fiber is
[OzpOzp-L'IzOzp+iAeO~p
+ e ~ ( O~- i & ) u (O~+ iA,) a ] ,
( 1)
where the SL (2, ~ ) group element is parametrized by
of the observables U~---(e°) r is given by (_k+2)g_
CT(V),
V=H°(~"K®[I+I/(k-2)]@ O(zi)-l/(k-2))
[9]
(9) 1
g=(u
0
1
0~(epl]\0 e - V ) ( 0
112) ,
(2)
and the U ( 1 ) symmetry p-*p,
u-*e-i~u,
ti-,ei~ti
(3)
is gauged by the introduction of the gauge fields A~(Az), which transform under the U(1 ) transformation as 8Az= -0~e (SAc= - 0 ~ ) . This model can be made topologically invariant and the gauge fixing of the topological invariance results in the appearance of the ghost field a and the corresponding antighost fl~ [6]. The fermion pair (a, fl~) develops a chiral anomaly, whose effective action is given by [ 6 ]
&u=-- ~1 I p.(2F+½R).
(4)
This effective action together with the p part of the gauged WZW action gives
S'(p,A)= k f x/%hUO~pOjp 1
+
(8)
Mg,.~
2n
This takes a manifestly topological invariant form, if the gauge field is shifted: 1
[
S"(p,A)= -~np. (k--2)F-½R]
(6)
where K is the cotangent space of the Riemann surface Z. In particular, if the level k equals 3, and if we consider the partition function, the fiber V of the vector bundle in question is V = H°(Z, K ® K ) ,
(10)
which is the cotangent bundle T *lf,Is to the moduli space of the Riemann surfaces at Z. Hence the genus g partition function Zg is
Zg = f
cx(T*lVl,) = ( - 1 ) 3g-3X(iglg),
( 11 )
where X is the Euler characteristic of the moduli space of Riemann surfaces [ 6 ]. Now, in order to connect the k = 3, SL (2, R ) / U ( 1 ) gauged WZW model to c= 25 Liouville theory coupled to a scalar, we first perform a target-space duality transformation [ 10 ]. The target-space duality is a generalization of the R ~ 1/R symmetry ofa compactiffed boson on a circle with radius R. In a conformal field theory this corresponds to changing the chirality of the left mover while fixing the chirality of the right mover [ 10 ]. In our case this transformation can be implemented by changing the U ( 1 ) embedding, and now the U( 1 ) symmetry which should be gauged, is p-~p-~,
u-~e'u , a~e~ .
(12)
The Lagrangian of the gauged WZW model which has this U ( 1 ) symmetry is 189
Volume 322, number 3
PHYSICSLETTERSB
k
L~wzw = ~ [ (O~p-A~) (Ozp-Ag)
+eZP(O~+Az)U(G + Az)a] .
-
(13)
This is nothing but the Lagrangian for a two-dimensional black hole with level k [ 7,9 ]. (The fact that these two distinct U ( 1 ) embeddings are related by target-space duality, was suggested in ref. [ 6 ]. ) Following Martinec and Shatashvili [ 9 ], we represent this Lagrangian in terms of conjugate momentum fields: ~L k ePHp- ~(0op) - 2rc (Oop-Ao) , ~SL
k -
5L
1
e2P(
Ha- ~i(0ofi) - 2rc
4rc
Normalizing the fields by fO= ~
1
O
,
0
0"= ~ ,
lIu=otue ~+i°, H~=aue ~'-i°, one obtains the classical Lagrangian 4rc
R~
arc { /2--0) + T # e x p k 4 ~S-~
+ ~00~0+
iR0.
(23)
CL=I+
(17)
L= ~ "2(Oz¢Oz~+O~OOzO)+--~//e 2~ ,
(22)
(16)
Parametrizing the conjugate momentum fields by ~o and 0 as
(18)
where lt=a~fftu is the cosmological constant [9]. Upon quantization of the theory, the effective Lagrangian acquires a term which represents a coupling of ¢ to the scalar curvature:
190
(21)
(15) Upon elimination of u, a, this reads [ 9 ]
and an anomaly term [ 9 ]
00=-00.
This is nothing but a Lagrangian of the Liouville theory (Liouville field ~) coupled to a scalar field 0 [9 ]. The conformal anomaly of the Liouville theory CLcan be read off from the Lagrangian, which is [ 91
t=x/~ (lluOzu+ l-laOza)- -~--/Tj / a ( 1 - u a ) .
k
00= 00,
(14)
After eliminating A1, by use of the equation of motion, substituting the constraint and fixing the unitary gauge condition, we obtain [ 9 ]
k 0~Hu 2--F 4rc L= ~ /Tu -k- I/7ul2
(20)
In order for 0 to couple to the scalar curvature in the standard form, 0 should be transformed to 0, by the target-space duality transformation [ 9 ]
1 k-1 Lqu = ~ 0~ 0 0 - 4 r c ~
1 x~e2p(Oz+Az)U.
- 4 ~ R~o,
2n
r J dZz(2 0z~ 0zq~- ½R~).
one obtains finally the quantum effective Lagrangian, which reads [ 9 ]
,i5
k
1
17 February 1994
(19)
6(k-l) 2 k-~
(24)
Now the crucial observation is that when k= 3 (which is the value of the level of the black hole model underlying Penner's model), then CL= 25. Thus we have shown that the momentum representation of the k = 3 SL(2, ~ ) / U ( 1 ) WZW model is the same as the coordinate representation of the Liouville theory with c=25. To be more precise, to derive the c= 25 Liouville theory from the k = 3 SL(2, R ) / U ( 1 ) WZW model coupled to topological gravity, first we made a target-space duality transformation, then went from the coordinate representation to the momentum representation, and finally made a target-space duality transformation again. Therefore, if we denote the coordinate-momentum interchange operation by V and the target-space duality transformation by U, then the transformation f, which sends one model to the other, is given by
Volume 322, number 3
f=U-I°VoU.
PHYSICS LETTERS B (25)
Since the purpose o f this p a p e r is to show a simple physical i n t e r p r e t a t i o n o f the existence o f two types o f m a t r i x models, rather than to prove the equivalence o f the two, we d i d not work out the ghost terms. Consequently, the scalar field 6~ has a negative conformal a n o m a l y which would split into c = 1 m a t t e r and ghost fields if all the ghost contributions were taken into account. In conclusion, we have found a simple a n d clear physical i n t e r p r e t a t i o n o f the existence o f two types o f matrix models ( P e n n e r ' s m a t r i x m o d e l a n d the s t a n d a r d c = 1 m a t r i x m o d e l ) which have m a n y similarities. I f we look at the c o n t i n u u m theories underlying these m a t r i x models, the m o m e n t u m representation o f one m o d e l is the same as the c o o r d i n a t e representation o f the other one. This fact is very suggestive when one tries to construct a four-dimensional self-dual gravity which is exactly soluble. A deep connection between f o u r - d i m e n s i o n a l self-dual gravity a n d the c = 1 string has been p o i n t e d out by m a n y authors [ 1 1-13 ]. One thing which is c o m m o n to t h e m is that the f o u r - d i m e n s i o n a l space is realized as a space s p a n n e d by a t w o - d i m e n s i o n a l space a n d the two-dimensional conjugate m o m e n t u m space, and a canonical t r a n s f o r m a t i o n which preserves a symplectic form, generates an area-preserving diffeom o r p h i s m , which is then connected to the Woo symm e t r y o f the c = 1 string. F r o m this p o i n t o f view, the fact that the m o m e n t u m representation o f one m o d e l is the same as the c o o r d i n a t e representation o f the other one, a n d the fact that the k = 3 SL(2, ~ ) / U ( 1 ) W Z W m o d e l is exactly soluble, i m p l y that we have a good chance o f constructing exactly soluble four-di-
17 February 1994
mensional self-dual gravity out o f these two-dimensional models. After the m a i n part o f this work was completed, we b e c a m e aware o f ref. [ 14 ], in which the same problem is also discussed, b u t from a quite different point o f view. O u r c o o r d i n a t e - m o m e n t u m interchange interpretation might b e c o m e i m p o r t a n t when one tries to construct a four-dimensional integrable selfdual gravity, for the reasons we mentioned in the text.
References [1]J. Distler and C. Vafa, Mod. Phys. Lett. A 6 (1991) 259, and references therein. [2] D. Gross and M. Miljkovic, Phys. Lett. B 238 (1990) 217; E. Brezin, V. Kazakov and AI.B.Zamolodchikov, Nucl. Phys. B338 (1990) 673; P. Ginsparg and J. Zinn-Justin, Phys. Len. B 240 (1990) 333. [3] R. Dijkgraaf, G. Moore and R. Plesser, Nucl. Phys. B 394 (1993) 356. [ 4 ] E. Witten, IAS preprint IASSNS-HEP-91/24 ( 1991 ), and references therein. [ 5 ] S. Kharchev et al., Phys. Lett. B 275 ( 1992 ) 311. [6] E. Witten, Nucl. Phys. B 371 (1992) 191. [ 7 ] E. Witten, Phys. Rev. D 44 ( 1991 ) 314. [ 8 ] J. Distler and H. Kawai, Nucl. Phys. B 321 (1989) 509. [ 9 ] E.J. Martinec and S.L. Shatashvili, Nucl. Phys. B 368 (1992) 338. [10] R. Dijkgraaf, E. Verlinde and H. Verlinde, Nucl. Phys. B 371 (1992) 269. [ll]Q.H. Park, Phys. Lett. B 236 (1990) 429; B 238 (1990) 287. [12] K. Yamagishi and G. Chapline, UC Livermore preprint (1990). [ 13 ] H. Ooguri and C. Vafa, Nucl. Phys. B 361 ( 1991 ) 469. [14] S. Mukhi and C. Vafa, preprint HUTP-93/A002(TIFR/ TH/93-01 ) (1993).
191