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Gravitation as gauge theory of the diffeomorphism group
Physics Letters B 286 (1992)251-255 North-Holland
PHYSICS kETTER5 13
Gravitation as gauge theory of the diffeomorphism group Y.M. Cho 1, K.S. Soh 2, J.H. Yoon Centerfor Theoretical Physics, Seoul National University, Seou1151- 742, South Korea
and Q-Han Park Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 9EW,, UK Received 15 April 1992
The (m + n)-dimensional Einstein theory of gravitation is identified with an m-dimensional generally invariant gauge theory of DiffN, where N is an n-dimensional manifold. This means that the four-dimensional Einstein gravity can be identified as a lower dimensional gauge theory of an infinite dimensional group of diffeomorphism. We discuss the physical implications of the results.
Recently one o f us ( Y . M . C . ) has f o r m u l a t e d the five-dimensional Einstein theory o f gravitation as a four-dimensional generally invariant gauge theory of D i f f S J, a n d argued that any ( m + n ) - d i m e n s i o n a l generally invariant field theory can be identified as an m-dimensional generally invariant gauge theory of D i f f N , where N is an n-dimensional m a n i f o l d [ 1,2 ]. This implies that one can formulate the four-dimensional Einstein theory as a two-dimensional gauge theory o f D i f f S 2, or a three-dimensional gauge theory o f D i f f S ~. The purpose o f this letter is to prove this claim affirmatively. We start from an ( m + n ) - d i m e n s i o n a l Einstein theory o f gravitation based on a local direct product space P--- M × N, where M and N are arbitrary manifolds o f dimensions m and n, and show explicitly how the theory can be identified as a generally invariant theory o f D i f f N based on M. F o r m a l l y this is quite similar to " t h e d i m e n s i o n a l r e d u c t i o n " in K a l u z a Klein unification [ 3,4 ], where one assumes an n-dimensional isometry G on the ( m + n ) - d i m e n s i o n a l Department of Physics, Seoul National University, Seoul 151742, South Korea. 2 Department of Physics Education, Seoul National University, Seoul 151-742, South Korea.
metric. This dimensional reduction by isometry is possible when the internal space N is i s o m o r p h i c to the group manifold G, in which case the unified space P becomes a principal fibre bundle P ( M , G ) , where G becomes the structure group. W i t h o u t the isometry, however, our results tells that the structure group becomes the infinite d i m e n s i o n a l d i f f e o m o r p h i s m group D i f f N o f the manifold N. Clearly without any isometry the m - d i m e n s i o n a l gauge theory o f the diff e o m o r p h i s m group maintains the full content o f the ( m + n )-dimensional gravitation. Consider an (m + n ) - d i m e n s i o n a l m a n i f o l d P e q u i p p e d with a metric gAB (A, B = 1, 2, ..., m + n ) o f signature ( - + +... + ). Locally, we m a y regard P as a product o f lower dimensional manifolds, P = M × N. Let Ou=O/Ox~' ( # = 1, 2, ..., m ) and Oa--O/Oy a ( a = 1, 2, ..., n) be a coordinate basis o f M and N, and choose 0A= (0~,, 0a) as a coordinate basis o f P. In this basis the most general metric on P can be written as [ 3,4 ]
. __(~,uv q'-e 2 ~al,A/, a Av b ~;AB-- I b \ e~abAu
eAu~q)~b~ Oab } '
( 1)
where e is a gauge coupling constant. In the s t a n d a r d K a l u z a - K l e i n d i m e n s i o n a l reduction one requires a restriction on the metric, an n-dimensional isometry
0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
251
Volume 286, number 3,4
PHYSICS LETTERSB
condition, to make Au ~ a gauge connection associated with the gauge group generated by the isometry G. In this case the ( m + n )-dimensional Einstein theory reduces to an m-dimensional Einstein-YangMills theory interacting with a non-linear sigma field Oab [3,4]. Here, however, we drop the isometry restriction, and allow all the fields to depend arbitrarily on both x u and y". Nevertheless, Au~(x, y) can still be identified as a connection, but now associated with an infinite dimensional gauge group DiffN. To show this, we consider the following diffeomorphism of N:
30 July 1992
Note that the field strength transforms covariantly under the gauge transformation (3), 8F~
~=
- [ ~, F,,.] a •
(8)
Under this transformation, we have
From these one can show that the (m+n)-dimensional Einstein-Hilbert action on P can be interpreted as a generally invariant gaugetheory of DiffN on M, where Aua becomes the gauge potential of the diffeomorphism group. To express the (m + n )-dimensional Einstein theory as an m-dimensional field theory, we must compute the (m + n)-dimensional scalar curvature. To do this we introduce the non-coordinate basis ~A= (~,, ~a) where [1,21
6 A ~ = _ 1 [ 0 ~ ~ _e(A~,8,.~ ~_~,OcA~a) ]
O~,--Ou-eAua(x,Y)Oa,
8xP=O,
8ya_~a(xl
.... , X,n, y l , "", y n ) .
(2)
~ a - a~ •
(9)
e
From the definition we have 1
_-
e ( 0 1 ' ~ - e m.~ )
[~A, ~I~] =fABC(x, Y)~C,
1 (8.~_e[Au ,~]~),
where the structure coefficientsfAsC(x, y) are given by
e
f~.a(x,y)=-eF.~a(x,y),
8Oab = __~c Oc(~ab __ (Oa~C)Ocb -- ( Ob~C)(~ac =
-
~Oa~
By.. = - ~
=
-- [
f.~b(x, y) = - £ . b ( x , y ) = e aaA~(x, y ) ,
~, ¢)] aS,
OcX,,.= -t2~7.. = - [¢, Xa.],
(3)
where t2~ represents the Lie derivative along the vector fields ¢= ~a 0~. Clearly (3) defines a gauge transformation which leaves the line element
fARc = 0,
otherwise.
gAB(X,y)=(y;v
0) ~)ab
(4)
invariant. Now we define the covariant derivative D u of the gauge transformation by
D~ = Or, - eP.Au,
(5)
where the Lie derivative is taken along the vector field A . = A ~ ~ 0~. With this definition, we have '
(12) '
which drastically simplifies the computation of the Einstein-Hilbert action. In the non-coordinate basis the Levi-Civita connections and the Riemann tensors are given by I~
C__ 1 .CD[ OAgBD + ~BgAD -- ~DgAB) "Jr 1 ~(-'DI r
~g,
tlABD --JBDA --f4DB) ,
RABC D = ~4 FBC D
8A,~=_ 1 D~ e
( 11 )
The virtue of this basis is that it brings the metric ( 1 ) into a block diagonal form,
ds 2= (7~. + eZ ~bAu~A~ b) dx u dx"
+ 2e~bA~ b¢Lr ~ dye+Cab dy ~ dy b
( 1O)
-
-
OBFAc D "b FAED F B c 1:-
(6)
_ FBeDFAc E__fAleFeC D,
which clearly indicates that Au a is the gauge potential of the diffeomorphism group. The field strength F~,. corresponding to Au ~ can now be defined as
( 13 )
where fAce= gc~)f4R °. So the scalar curvature is given by
[Du, D . ] = -eFu~ ~ O~
= - e ( O ~ A . ~ - O.Aa~-e[Au, A.] ~) 0~. 252
(7)
=7~'"(R~,..C~+R~,ay)+¢~b(R.,+"+R~,b").
(14)
Volume 286, number 3,4
PHYSICS LETTERSB
After a lengthy but straightforward calculation, we obtain the ( m + n)-dimensional scalar curvature R (re+n) of P,
30 July 1992 1
× {TU"R(u~' + ¢~bR~7,) + ~e2Oa~Fu~F u~b
R ( m + n ) __ v , u v R ( m ) ..{_¢ a b R (ag) + l e 2 f b a K F U v T a ~ F u a ~ F ~ B b --I *'t~V
+ ~yu"O~b(Ua[ (DuO~) (D,Oea) - (D~,O~) (D.Oca) ]
+v~j ~ ,
(15)
where the lower dimensional "Ricci" tensors R(m) ~ktP and Ra(~) are defined by _ o ^ ,,r~ .R. ~( m ) =
o~- ^o a r . . o~+r~a a
L.
B -r~
(16)
V~,ju+ V~j ~ ,
× (Tu~Oab ~O~b-2e~ ~" O~A~~) , V,~ja=(Oa+F~f +F,~a°~)(oabTu" ObTu.) •
(17)
At this point it is important to notice the following. First, DuO~, written as
D uOa~= OuO~t,- eP-auOab =
Ou¢~b
--e[AuC( 0~(9~) + (OaAS)O~ + (0~AS)0~] , (18) indeed transforms covariantly under the gauge transformation (3), 8 (Duq~) = -2E¢(D¢,0a~)= - [~, D,,O]ab •
(19)
Secondly, the derivative ~u, when applied to 7u~, becomes the covariant derivative Ou 7 o ~ = Ou 7 a # - - e ~ A u 7~,e -----Du 7-a,
(20)
so that ~ ~-a transforms covariantly, 8(0u7,~ ~) = - e¢(Du7~) = - [~, DuT,~] •
(23)
N
and the total divergence term is given by vAjA=
where ~= det 7u~ and ~ = det 0ab- From this we obtain the m-dimensional lagrangian S,~ = f ~ d " y .
fl r,,~ a
R~'~)--OaFb~b--ObF=cb+e~ahFb~a--eufe=c b,
+ ~0°bT,'~r~P[ (0a~.~) (0~y~p) -- (aorta) (0b~p) 1}, (22)
(21)
Obviously these observations will play an important role when we discuss the gauge invariance of the ( m + n )-dimensional Einstein theory. With the ( m + n ) - d i m e n s i o n a l scalar curvature at hand, one can easily write down the lagrangian for the Einstein-Hilbert action on P. From ( 15 ) we have
Note that we have neglected the total divergence term (17) in the lagrangian (22), which one can justify with a proper boundary condition. But one has to keep in mind that, with an arbitrary boundary condition, the total divergence term can contribute to the lagrangian (23). Clearly the lagrangian describes an mdimensional generally invariant field theory which is invariant under the gauge transformation (3) of DiffN. This must be so because the gauge potential Au" couples covariantly to both 7u~ and {)ab, Therefore each term in (23) is independently invariant under the infinite dimensional gauge transformation (3). To understand the physical contents of the theory we note the following. First, unlike the ordinary Einstein theory the metric 7~,~of M here is "charged", because it couples to Au a covariantly. Secondly, the metric ¢ab of N can be identified as a non-linear sigma field, whose self-interaction potential ¢abR~g) is uniquely specified by the geometry. So the theory describes a generally invariant gauge theory of DiffN interacting with the "gauged" gravity and the non-linear sigma field based on M. Notice that when N is a homogeneous space G / H or in general a compact space, one can have the harmonic expansion of the fields on N. In this case the lagrangian (23), after the integration over N, reduces to an infinite-component field theory on M [ 5 ]. Furthermore, when N becomes isomorphic to a group space G, the inner automorphism of the group space naturally becomes a subgroup of DiffN. In this case one can consistently reduce the gauge group DiffN down to G, by requiring that G be an isometry of the (m + n)-dimensional metric. This is the dimensional reduction by isometry [ 3,4 ], and the resulting theory becomes the well-known 253
Volume 286, number 3,4
PHYSICS LETTERS B
Einstein-Yang-Mills theory interacting with the nonlinear sigma field. Now we apply the above analysis to the four-dimensional Einstein gravity and reduce the theory to gauge theories of diffeomorphism groups of lower dimensional manifolds. (A) m = 1, n = 3 . In this case we have p _ ~ l XN 3, locally. With m = 1, we can drop the space-time indices in (15), and denote the one-dimensional metric tensor by 7. Then the quadratic terms of 0~y~,~in ( 15 ) become zero, since they cancel each other. Also the one-dimensional covariantized scalar curvature vanishes. Moreover, the field strength vanishes identically for any gauge potential Aft. So ( 15 ) simplifies to R (4 ) ..~ (~abR ~a3 ) .+ ~ ¢al'¢cd(DtOac DtCbd--DtOab Dt¢cd) .
(24)
Note that since no time derivative of Aft appears in (24), A," becomes an "external" gauge potential valued in DiffN 3. In this case (24) becomes precisely the well-known canonical l + 3 decomposition of the four-dimensional gravity, R (4)~ R (3).-k KahK ab- ( Kaa) 2 ,
(25)
Kaa =¢abKab,
where the extrinsic curvature K,b is given by K~b --- ½:e,O~h= ~
1
(0,¢~/, - ~ ¢ ~ b ) •
(26)
Here we used the decomposition of 0t such that 0 , - - x / ~ n + r , where n = n "4Oa is the unit normal to N 3 and r = z ~ Oa is the tangential to N 3. To see this note that the kinetic term of ¢~, is written as Dt(bab = C~,¢ab - e [ A , , ¢]~#, = 0,¢~1, - e~A,¢.t, •
(27)
With the identification ofeA~ '~ as the shift vector r ~, we find that (24) and (25) are exactly the same. This shows that (22) is indeed the generalization of the canonical 1 + 3 decomposition of the four-dimensional Einstein theory. Clearly the result allows us to interpret the four-dimensional Einstein gravity as a theory of"charged" point particles valued in D i f f N 3, minimally coupled to the external gauge field A, ~ moving under the influence of the potential ,~bo 5~' *'~ at3) b • (B) m =2, n = 2 . Here, the four-dimensional grav254
30 July 1992
ity is locally identified with a two-dimensional gauge theory of D i f f N 2. Since our analysis is independent of the signature of the space-time, the gauge theory of D i f f N 2 may be compared with the self-dual gravity extensively studied by one of us (Q.P.) recently, where the self-dual Einstein equation is identified with the equation of the two-dimensional sigma model with the gauge symmetry of SDiffN 2, the area preserving diffeomorphism of N 2 [6,7]. Thus, the present work generalizes the self-dual case to the general case without the self-duality restriction. Unlike the self-dual case, (22) possesses the full two-dimensional diffeomorphism DiffN 2 as the gauge symmetry where the volume ~ = det Gab of N 2 also becomes dynamically independent. When the topology of N 2 becomes a two-sphere S2, the theory describes a gauge theory of DiffS 2, which is especially interesting in connection with the membrane theories. (C) m = 3 , n = 1. Similarly to the 1+3 case, (15) now reduces to R (4)~_ 7,,,,,R~3) + ~eZOFI,,,F..
+ ~0~,'~[ (o,#,,.) (0,#~ e) - (0y~,~) (0yy.~) ]. (28) The theory becomes a gauge theory of DiffS J, when the topology of N becomes S ~. In this case the dimensional reduction becomes almost identical to the Kaluza-Klein dimensional reduction of the five-dimensional Einstein theory to the four-dimensional gauge theory of DiffS ~. This decomposition has been studied rigorously recently [1,2]. The only difference in this case from the Kaluza-Klein unification is that we are viewing the four-dimensional gravity from three space-time dimensions, where the usual three-dimensional Einstein gravity becomes topological. We refer the reader to refs. [ 1,2 ] for the detailed analysis. We conclude with the following remarks. (i) Viewing the four-dimensional gravity as a lower dimensional gauge theory, one may obtain a perturbative expansion of the four-dimensional gravity in terms of the gauge coupling constant e. The zeroth order, namely the case when e = 0, corresponds to the direct product manifold P = M × N, whereas higherorder corrections in perturbation "localize" the product through the interaction with the gauge field m~t a ,
(ii) Various two Killing symmetry reductions of
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PHYSICS LETTERS B
the four-dimensional gravity have been known for some time which led to the discovery o f m a n y exact solutions to the Einstein equation. To a certain extent, (22) provides a vantage point to these reductions from which different reductions arise. Moreover, one could attempt to simplify (22) by imposing restrictions other than the isometry (e.g., the selfduality constraint). On the other hand, it would be interesting to re-examine the constraint algebras o f general relativity themselves in the context o f a twodimensional gauge theory where some of the hamiltonian and the m o m e n t u m constraints could be replaced by the a p p r o p r i a t e gauge conditions. In this way one might reduce the difficulty o f solving the constraint algebras, or at least reformulate the constraint algebras from a different point o f view. (iii) It is well-known that the three-dimensional gravity is exactly solvable [ 8 ]. On the other hand, our analysis tells that the three-dimensional gravity is equivalent to a two-dimensional gauge theory o f D i f f S I. This implies that one could view the two-dimensional gauge theory o f D i f f S 1 as a two-dimensional completely integrable field theory. To find out how the two-dimensional gauge theory o f D i f f S l can actually describe an exactly solvable theory would be a very interesting task. This work is under progress a n d will be reported elsewhere [ 9 ].
30 July 1992
sions and the Center for Theoretical Physics o f Seoul N a t i o n a l U n i v e r s i t y for help during his visit. This work is supported in part by the Ministry o f Education a n d by the K o r e a Science and Engineering Foundation.
References [ 1] Y.M. Cho and S.W. Zoh, preprint SNUTP 90-16, Phys. Rev. D, in press; preprint SNUTP 91-28, submitted to Phys. Rev. D; C.S. Aulakh and D. Sahdev, Phys. Lett. B 164 (1985) 293. [2 ] Y.M. Cho, Phys. Rev. Lett. 67 ( 1991 ) 3469; preprint SNUTP 91-57, submitted to Phys. Rev. Lett. [ 3 ] Y.M. Cho, J. Math. Phys. 16 ( 1975 ) 2029; Y.M. Cho and P.G.O. Freund, Phys. Rev. D 12 (1975) 1711; Y.M. Cho and P.S. Jang, Phys. Rev. D 12 ( 1975 ) 3789. [4] Y.M. Cho, Phys. Lett. B 186 (1987) 38; B 199 (1987) 358; Phys. Rev. D 35 (1987) 2628; Y.M. Cho and D.S. Kimm, J. Math. Phys. 30 (1989) 1590; D. Brill and J.H. Yoon, Class. Quantum Grav. 7 (1990) 1253. [ 5 ] C.S. Aulakh and D. Sahdev, Phys. Lett. B 173 (1986) 241. [6] Q-H. Park, Phys. Len. B 238 (1990) 287;B 257 (1991) 105; Intern. J. Mod. Phys. A 7 (1992) 1415. [7] Q-H. Park, Phys. Lett. B 269 (1991) 271; 4-D instantons from 2-D sigma model, in: Infinite analysis, RIMS91 project (1991). [ 8 ] E. Witten, Nucl. Phys. B 311 (1988) 46; Phys. R ev. Left. 62 (1989) 501. [9] Y.M. Cho, Q-H. Park and J.H. Yoon, in preparation.
J.H.Y. thanks S. N a m for useful discussions. Q.P. thanks G. G i b b o n s and P.K. Townsend for discus-
255
PHYSICS kETTER5 13
Gravitation as gauge theory of the diffeomorphism group Y.M. Cho 1, K.S. Soh 2, J.H. Yoon Centerfor Theoretical Physics, Seoul National University, Seou1151- 742, South Korea
and Q-Han Park Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 9EW,, UK Received 15 April 1992
The (m + n)-dimensional Einstein theory of gravitation is identified with an m-dimensional generally invariant gauge theory of DiffN, where N is an n-dimensional manifold. This means that the four-dimensional Einstein gravity can be identified as a lower dimensional gauge theory of an infinite dimensional group of diffeomorphism. We discuss the physical implications of the results.
Recently one o f us ( Y . M . C . ) has f o r m u l a t e d the five-dimensional Einstein theory o f gravitation as a four-dimensional generally invariant gauge theory of D i f f S J, a n d argued that any ( m + n ) - d i m e n s i o n a l generally invariant field theory can be identified as an m-dimensional generally invariant gauge theory of D i f f N , where N is an n-dimensional m a n i f o l d [ 1,2 ]. This implies that one can formulate the four-dimensional Einstein theory as a two-dimensional gauge theory o f D i f f S 2, or a three-dimensional gauge theory o f D i f f S ~. The purpose o f this letter is to prove this claim affirmatively. We start from an ( m + n ) - d i m e n s i o n a l Einstein theory o f gravitation based on a local direct product space P--- M × N, where M and N are arbitrary manifolds o f dimensions m and n, and show explicitly how the theory can be identified as a generally invariant theory o f D i f f N based on M. F o r m a l l y this is quite similar to " t h e d i m e n s i o n a l r e d u c t i o n " in K a l u z a Klein unification [ 3,4 ], where one assumes an n-dimensional isometry G on the ( m + n ) - d i m e n s i o n a l Department of Physics, Seoul National University, Seoul 151742, South Korea. 2 Department of Physics Education, Seoul National University, Seoul 151-742, South Korea.
metric. This dimensional reduction by isometry is possible when the internal space N is i s o m o r p h i c to the group manifold G, in which case the unified space P becomes a principal fibre bundle P ( M , G ) , where G becomes the structure group. W i t h o u t the isometry, however, our results tells that the structure group becomes the infinite d i m e n s i o n a l d i f f e o m o r p h i s m group D i f f N o f the manifold N. Clearly without any isometry the m - d i m e n s i o n a l gauge theory o f the diff e o m o r p h i s m group maintains the full content o f the ( m + n )-dimensional gravitation. Consider an (m + n ) - d i m e n s i o n a l m a n i f o l d P e q u i p p e d with a metric gAB (A, B = 1, 2, ..., m + n ) o f signature ( - + +... + ). Locally, we m a y regard P as a product o f lower dimensional manifolds, P = M × N. Let Ou=O/Ox~' ( # = 1, 2, ..., m ) and Oa--O/Oy a ( a = 1, 2, ..., n) be a coordinate basis o f M and N, and choose 0A= (0~,, 0a) as a coordinate basis o f P. In this basis the most general metric on P can be written as [ 3,4 ]
. __(~,uv q'-e 2 ~al,A/, a Av b ~;AB-- I b \ e~abAu
eAu~q)~b~ Oab } '
( 1)
where e is a gauge coupling constant. In the s t a n d a r d K a l u z a - K l e i n d i m e n s i o n a l reduction one requires a restriction on the metric, an n-dimensional isometry
0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
251
Volume 286, number 3,4
PHYSICS LETTERSB
condition, to make Au ~ a gauge connection associated with the gauge group generated by the isometry G. In this case the ( m + n )-dimensional Einstein theory reduces to an m-dimensional Einstein-YangMills theory interacting with a non-linear sigma field Oab [3,4]. Here, however, we drop the isometry restriction, and allow all the fields to depend arbitrarily on both x u and y". Nevertheless, Au~(x, y) can still be identified as a connection, but now associated with an infinite dimensional gauge group DiffN. To show this, we consider the following diffeomorphism of N:
30 July 1992
Note that the field strength transforms covariantly under the gauge transformation (3), 8F~
~=
- [ ~, F,,.] a •
(8)
Under this transformation, we have
From these one can show that the (m+n)-dimensional Einstein-Hilbert action on P can be interpreted as a generally invariant gaugetheory of DiffN on M, where Aua becomes the gauge potential of the diffeomorphism group. To express the (m + n )-dimensional Einstein theory as an m-dimensional field theory, we must compute the (m + n)-dimensional scalar curvature. To do this we introduce the non-coordinate basis ~A= (~,, ~a) where [1,21
6 A ~ = _ 1 [ 0 ~ ~ _e(A~,8,.~ ~_~,OcA~a) ]
O~,--Ou-eAua(x,Y)Oa,
8xP=O,
8ya_~a(xl
.... , X,n, y l , "", y n ) .
(2)
~ a - a~ •
(9)
e
From the definition we have 1
_-
e ( 0 1 ' ~ - e m.~ )
[~A, ~I~] =fABC(x, Y)~C,
1 (8.~_e[Au ,~]~),
where the structure coefficientsfAsC(x, y) are given by
e
f~.a(x,y)=-eF.~a(x,y),
8Oab = __~c Oc(~ab __ (Oa~C)Ocb -- ( Ob~C)(~ac =
-
~Oa~
By.. = - ~
=
-- [
f.~b(x, y) = - £ . b ( x , y ) = e aaA~(x, y ) ,
~, ¢)] aS,
OcX,,.= -t2~7.. = - [¢, Xa.],
(3)
where t2~ represents the Lie derivative along the vector fields ¢= ~a 0~. Clearly (3) defines a gauge transformation which leaves the line element
fARc = 0,
otherwise.
gAB(X,y)=(y;v
0) ~)ab
(4)
invariant. Now we define the covariant derivative D u of the gauge transformation by
D~ = Or, - eP.Au,
(5)
where the Lie derivative is taken along the vector field A . = A ~ ~ 0~. With this definition, we have '
(12) '
which drastically simplifies the computation of the Einstein-Hilbert action. In the non-coordinate basis the Levi-Civita connections and the Riemann tensors are given by I~
C__ 1 .CD[ OAgBD + ~BgAD -- ~DgAB) "Jr 1 ~(-'DI r
~g,
tlABD --JBDA --f4DB) ,
RABC D = ~4 FBC D
8A,~=_ 1 D~ e
( 11 )
The virtue of this basis is that it brings the metric ( 1 ) into a block diagonal form,
ds 2= (7~. + eZ ~bAu~A~ b) dx u dx"
+ 2e~bA~ b¢Lr ~ dye+Cab dy ~ dy b
( 1O)
-
-
OBFAc D "b FAED F B c 1:-
(6)
_ FBeDFAc E__fAleFeC D,
which clearly indicates that Au a is the gauge potential of the diffeomorphism group. The field strength F~,. corresponding to Au ~ can now be defined as
( 13 )
where fAce= gc~)f4R °. So the scalar curvature is given by
[Du, D . ] = -eFu~ ~ O~
= - e ( O ~ A . ~ - O.Aa~-e[Au, A.] ~) 0~. 252
(7)
=7~'"(R~,..C~+R~,ay)+¢~b(R.,+"+R~,b").
(14)
Volume 286, number 3,4
PHYSICS LETTERSB
After a lengthy but straightforward calculation, we obtain the ( m + n)-dimensional scalar curvature R (re+n) of P,
30 July 1992 1
× {TU"R(u~' + ¢~bR~7,) + ~e2Oa~Fu~F u~b
R ( m + n ) __ v , u v R ( m ) ..{_¢ a b R (ag) + l e 2 f b a K F U v T a ~ F u a ~ F ~ B b --I *'t~V
+ ~yu"O~b(Ua[ (DuO~) (D,Oea) - (D~,O~) (D.Oca) ]
+v~j ~ ,
(15)
where the lower dimensional "Ricci" tensors R(m) ~ktP and Ra(~) are defined by _ o ^ ,,r~ .R. ~( m ) =
o~- ^o a r . . o~+r~a a
L.
B -r~
(16)
V~,ju+ V~j ~ ,
× (Tu~Oab ~O~b-2e~ ~" O~A~~) , V,~ja=(Oa+F~f +F,~a°~)(oabTu" ObTu.) •
(17)
At this point it is important to notice the following. First, DuO~, written as
D uOa~= OuO~t,- eP-auOab =
Ou¢~b
--e[AuC( 0~(9~) + (OaAS)O~ + (0~AS)0~] , (18) indeed transforms covariantly under the gauge transformation (3), 8 (Duq~) = -2E¢(D¢,0a~)= - [~, D,,O]ab •
(19)
Secondly, the derivative ~u, when applied to 7u~, becomes the covariant derivative Ou 7 o ~ = Ou 7 a # - - e ~ A u 7~,e -----Du 7-a,
(20)
so that ~ ~-a transforms covariantly, 8(0u7,~ ~) = - e¢(Du7~) = - [~, DuT,~] •
(23)
N
and the total divergence term is given by vAjA=
where ~= det 7u~ and ~ = det 0ab- From this we obtain the m-dimensional lagrangian S,~ = f ~ d " y .
fl r,,~ a
R~'~)--OaFb~b--ObF=cb+e~ahFb~a--eufe=c b,
+ ~0°bT,'~r~P[ (0a~.~) (0~y~p) -- (aorta) (0b~p) 1}, (22)
(21)
Obviously these observations will play an important role when we discuss the gauge invariance of the ( m + n )-dimensional Einstein theory. With the ( m + n ) - d i m e n s i o n a l scalar curvature at hand, one can easily write down the lagrangian for the Einstein-Hilbert action on P. From ( 15 ) we have
Note that we have neglected the total divergence term (17) in the lagrangian (22), which one can justify with a proper boundary condition. But one has to keep in mind that, with an arbitrary boundary condition, the total divergence term can contribute to the lagrangian (23). Clearly the lagrangian describes an mdimensional generally invariant field theory which is invariant under the gauge transformation (3) of DiffN. This must be so because the gauge potential Au" couples covariantly to both 7u~ and {)ab, Therefore each term in (23) is independently invariant under the infinite dimensional gauge transformation (3). To understand the physical contents of the theory we note the following. First, unlike the ordinary Einstein theory the metric 7~,~of M here is "charged", because it couples to Au a covariantly. Secondly, the metric ¢ab of N can be identified as a non-linear sigma field, whose self-interaction potential ¢abR~g) is uniquely specified by the geometry. So the theory describes a generally invariant gauge theory of DiffN interacting with the "gauged" gravity and the non-linear sigma field based on M. Notice that when N is a homogeneous space G / H or in general a compact space, one can have the harmonic expansion of the fields on N. In this case the lagrangian (23), after the integration over N, reduces to an infinite-component field theory on M [ 5 ]. Furthermore, when N becomes isomorphic to a group space G, the inner automorphism of the group space naturally becomes a subgroup of DiffN. In this case one can consistently reduce the gauge group DiffN down to G, by requiring that G be an isometry of the (m + n)-dimensional metric. This is the dimensional reduction by isometry [ 3,4 ], and the resulting theory becomes the well-known 253
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Einstein-Yang-Mills theory interacting with the nonlinear sigma field. Now we apply the above analysis to the four-dimensional Einstein gravity and reduce the theory to gauge theories of diffeomorphism groups of lower dimensional manifolds. (A) m = 1, n = 3 . In this case we have p _ ~ l XN 3, locally. With m = 1, we can drop the space-time indices in (15), and denote the one-dimensional metric tensor by 7. Then the quadratic terms of 0~y~,~in ( 15 ) become zero, since they cancel each other. Also the one-dimensional covariantized scalar curvature vanishes. Moreover, the field strength vanishes identically for any gauge potential Aft. So ( 15 ) simplifies to R (4 ) ..~ (~abR ~a3 ) .+ ~ ¢al'¢cd(DtOac DtCbd--DtOab Dt¢cd) .
(24)
Note that since no time derivative of Aft appears in (24), A," becomes an "external" gauge potential valued in DiffN 3. In this case (24) becomes precisely the well-known canonical l + 3 decomposition of the four-dimensional gravity, R (4)~ R (3).-k KahK ab- ( Kaa) 2 ,
(25)
Kaa =¢abKab,
where the extrinsic curvature K,b is given by K~b --- ½:e,O~h= ~
1
(0,¢~/, - ~ ¢ ~ b ) •
(26)
Here we used the decomposition of 0t such that 0 , - - x / ~ n + r , where n = n "4Oa is the unit normal to N 3 and r = z ~ Oa is the tangential to N 3. To see this note that the kinetic term of ¢~, is written as Dt(bab = C~,¢ab - e [ A , , ¢]~#, = 0,¢~1, - e~A,¢.t, •
(27)
With the identification ofeA~ '~ as the shift vector r ~, we find that (24) and (25) are exactly the same. This shows that (22) is indeed the generalization of the canonical 1 + 3 decomposition of the four-dimensional Einstein theory. Clearly the result allows us to interpret the four-dimensional Einstein gravity as a theory of"charged" point particles valued in D i f f N 3, minimally coupled to the external gauge field A, ~ moving under the influence of the potential ,~bo 5~' *'~ at3) b • (B) m =2, n = 2 . Here, the four-dimensional grav254
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ity is locally identified with a two-dimensional gauge theory of D i f f N 2. Since our analysis is independent of the signature of the space-time, the gauge theory of D i f f N 2 may be compared with the self-dual gravity extensively studied by one of us (Q.P.) recently, where the self-dual Einstein equation is identified with the equation of the two-dimensional sigma model with the gauge symmetry of SDiffN 2, the area preserving diffeomorphism of N 2 [6,7]. Thus, the present work generalizes the self-dual case to the general case without the self-duality restriction. Unlike the self-dual case, (22) possesses the full two-dimensional diffeomorphism DiffN 2 as the gauge symmetry where the volume ~ = det Gab of N 2 also becomes dynamically independent. When the topology of N 2 becomes a two-sphere S2, the theory describes a gauge theory of DiffS 2, which is especially interesting in connection with the membrane theories. (C) m = 3 , n = 1. Similarly to the 1+3 case, (15) now reduces to R (4)~_ 7,,,,,R~3) + ~eZOFI,,,F..
+ ~0~,'~[ (o,#,,.) (0,#~ e) - (0y~,~) (0yy.~) ]. (28) The theory becomes a gauge theory of DiffS J, when the topology of N becomes S ~. In this case the dimensional reduction becomes almost identical to the Kaluza-Klein dimensional reduction of the five-dimensional Einstein theory to the four-dimensional gauge theory of DiffS ~. This decomposition has been studied rigorously recently [1,2]. The only difference in this case from the Kaluza-Klein unification is that we are viewing the four-dimensional gravity from three space-time dimensions, where the usual three-dimensional Einstein gravity becomes topological. We refer the reader to refs. [ 1,2 ] for the detailed analysis. We conclude with the following remarks. (i) Viewing the four-dimensional gravity as a lower dimensional gauge theory, one may obtain a perturbative expansion of the four-dimensional gravity in terms of the gauge coupling constant e. The zeroth order, namely the case when e = 0, corresponds to the direct product manifold P = M × N, whereas higherorder corrections in perturbation "localize" the product through the interaction with the gauge field m~t a ,
(ii) Various two Killing symmetry reductions of
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the four-dimensional gravity have been known for some time which led to the discovery o f m a n y exact solutions to the Einstein equation. To a certain extent, (22) provides a vantage point to these reductions from which different reductions arise. Moreover, one could attempt to simplify (22) by imposing restrictions other than the isometry (e.g., the selfduality constraint). On the other hand, it would be interesting to re-examine the constraint algebras o f general relativity themselves in the context o f a twodimensional gauge theory where some of the hamiltonian and the m o m e n t u m constraints could be replaced by the a p p r o p r i a t e gauge conditions. In this way one might reduce the difficulty o f solving the constraint algebras, or at least reformulate the constraint algebras from a different point o f view. (iii) It is well-known that the three-dimensional gravity is exactly solvable [ 8 ]. On the other hand, our analysis tells that the three-dimensional gravity is equivalent to a two-dimensional gauge theory o f D i f f S I. This implies that one could view the two-dimensional gauge theory o f D i f f S 1 as a two-dimensional completely integrable field theory. To find out how the two-dimensional gauge theory o f D i f f S l can actually describe an exactly solvable theory would be a very interesting task. This work is under progress a n d will be reported elsewhere [ 9 ].
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sions and the Center for Theoretical Physics o f Seoul N a t i o n a l U n i v e r s i t y for help during his visit. This work is supported in part by the Ministry o f Education a n d by the K o r e a Science and Engineering Foundation.
References [ 1] Y.M. Cho and S.W. Zoh, preprint SNUTP 90-16, Phys. Rev. D, in press; preprint SNUTP 91-28, submitted to Phys. Rev. D; C.S. Aulakh and D. Sahdev, Phys. Lett. B 164 (1985) 293. [2 ] Y.M. Cho, Phys. Rev. Lett. 67 ( 1991 ) 3469; preprint SNUTP 91-57, submitted to Phys. Rev. Lett. [ 3 ] Y.M. Cho, J. Math. Phys. 16 ( 1975 ) 2029; Y.M. Cho and P.G.O. Freund, Phys. Rev. D 12 (1975) 1711; Y.M. Cho and P.S. Jang, Phys. Rev. D 12 ( 1975 ) 3789. [4] Y.M. Cho, Phys. Lett. B 186 (1987) 38; B 199 (1987) 358; Phys. Rev. D 35 (1987) 2628; Y.M. Cho and D.S. Kimm, J. Math. Phys. 30 (1989) 1590; D. Brill and J.H. Yoon, Class. Quantum Grav. 7 (1990) 1253. [ 5 ] C.S. Aulakh and D. Sahdev, Phys. Lett. B 173 (1986) 241. [6] Q-H. Park, Phys. Len. B 238 (1990) 287;B 257 (1991) 105; Intern. J. Mod. Phys. A 7 (1992) 1415. [7] Q-H. Park, Phys. Lett. B 269 (1991) 271; 4-D instantons from 2-D sigma model, in: Infinite analysis, RIMS91 project (1991). [ 8 ] E. Witten, Nucl. Phys. B 311 (1988) 46; Phys. R ev. Left. 62 (1989) 501. [9] Y.M. Cho, Q-H. Park and J.H. Yoon, in preparation.
J.H.Y. thanks S. N a m for useful discussions. Q.P. thanks G. G i b b o n s and P.K. Townsend for discus-
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