- Email: [email protected]
Super W-symmetries, covariantly constant forms and duality transformations
25 August 1994
PHYSICS LETTERS B t-I XI.:VIt'R
Physics Letters B 335 (1994) 51-55
Super W-symmetries, covariantly constant forms and duality transformations Byungbae Kim 1 Institute fi~r Theoretical Physics. State Universi O' oJ New York at Stony Brook, Stony Brook. NY I 1794. USA Received 28 June 1994 FAitor: H. Georgi
Abstract On a supersymmetric sigma model the covariantly constant fiwms are related to the conserved currents that are generators of a super W-algebra extending the supcrconformal algebra. The existence of covariantly constant forms restricts the holonomy group of the manifold. Via duality transformation we get new covariantly constant forms, thus restricting the holonomy group of thc new manifold.
I. Introduction One of the most intriguing properties of string theory is the existence o f a duality symmetry that relates strings propagating on different backgrounds to each other (see, e.g., I 1 l and references therein). Duality transformations play an important role in the study of moduli spaces of string vacua, and are presumably part of a large and fundamental symmetry of string theory. The purpose of this note is to investigate how duality acts on N = 1 sigma models with torsion possessing an extended algebra of conserved currents, i.e., a super W-algcbra. As was shown in [2], conserved chiral currents can in a natural way be associated to covariantly closed forms on the target manifold. Since covariantly closed forms imply certain constraints on the holonomy, the manilblds will in general have reduced holonomy. A simple example of such a manifold is a Calabi-Yau manifold, which has reduced holonomy S U ( 3 ) and whose covariantly constant complex E-mail address: [email protected].
structure enhances the superconformal algebra to an N = 2 algebra. In this letter we show explicitly how conserved currents associated to covariantly constant differential forms are under duality mapped to another set of conserved currents on the dual manifold. It will turn out that these latter currents are also associated to covariantly constant forms on the dual manifold, for which we present explicit expressions. From this we think that duality does, in general, leave the holonomy of the manifold invariant 2.
2. Super W-algebra and covariantly constant forms Given a supersymmetric nonlinear or-model lagrangian 2 Actually, as we shall see below, duality transformations make use of an isometry of the manifold, and the holonomy is preserved only if the isometry is compatible with the (covariantly constant) forms.
0370-2693/9,1/$07.00 ~ 1994 Elsevier Science B.V. All rights rescrvcd SSDI 037(1-2693 ( 9 4 1 0 0 8 9 2 -2
52
B. Kim / Physics Leners B 335 (1994) 51-55
Table 1 Holonomy groups and covariantly constant forms on nonsymmetric spaces. Dim.
Holonomy group
Covariantly constant forms (generators of the algebra)
n 2n 2n 4n 4n 7 8
SO(n) U(n) SU(n) Sp ( n ) Sp(n)Sp( I ) G2 Spin(7)
wn = volume form (w2) t' k = 1,.-.n; w2: KLihler form (w2) k k = 1..-.n; w,t = (vol) U2 wn: complex volume form, ~n w = awl + bw2 + cw3, a 2 + b 2 + 2
2
C2 :
Kfihler manifold Special K~.hler (Ricci I Hyperk~ihler Quaternionic K~.hler
1
2
fi4 = w I + w 2 + w 3 (O3, ¢04 ¢O4
The product of the algebra involves not only wedge products of forms but also contractions of two or more forms using the t" -7 I J
S=
D2(guuD+XUD_X~),
( 1)
+ p w , ~ , , : . . . , , p D _ D + X '~l D + X 'y2 • • • D t X '~' = cguw,~ , ...,~p D _ X u D F X Crl
forms
and
dualil
pogal,~2...,~, F p a D _ X
m • • • D . X ~'~' p
D ~ X " D _ X "2 • • • D + X '~t'
= (~7~,w,,,...,,~,)D_XUD+X a'..
. D . . X ~'p
( w h e r e we used the equation o f motion D _ D + X ~p
constant
N o w given a s u p e r s y m m e t r i c nonlinear o'grangian ( w e allow torsion for m o r e genera covariantly constant p - f o r m s w + (see be have the action
D - J,o = O u w , , ~..... ~ , D - X U D + X ('l " " D : - X ~p
I p,rD_XPD+X
Covariantly
transformation
for Jo, = w , , l . . . ~ p D - X '~' • • • D f.X ~'
_ _
3.
(2) ~ +
~r = 0 ) .
So D _ J , , = 0 if the form w is covariantly constant, i.e., V u W = 0. In this case J,o is a conserved current that is a generator o f a super W-algebra via Poisson Brackets, e x t e n d i n g the superconformal algebra [ 2 ]. The existence o f such covariantly constant forms is related to the h o l o n o m y group o f the target manifold. Table 1 shows the existence o f covariantly constant forms and the h o l o n o m y group o f the manifold ( f o r the n o n s y m m e t r i c spaces) [ 3 ] . For each h o l o n o m y group, there is an associated covariantly constant totally antisymmetric tensor, and this implies the existence o f an associated s y m m e t r y o f the c o r r e s p o n d i n g s u p e r s y m m e t r i c sigma model [ 2 ] . These currents generate a super W-algebra via Poisson Brackets and these algebras are extensions o f the superconformal algebra.
S = 1/D2((g~z~
+ bu~)D+XUD-X")
•
Suppose further there is an isometry g e n e r vector field Y with £ v ( d b ) = 0 and £ v w + one can c h o o s e ( l o c a l ) coordinates such a / o x ° and the metric g, the torsion potent the p - f o r m s w + are independent o f X°. U n d e r the transformations ( SX~' ) gu~ = a w +,l ..... , - ' D + X"' • • • D+ X "p
i.e., ~ X u .1 . . .[
~
UVog+ ~
I "''O'p--I
r~ x'r~
/'J+
' " "
D+X'rl, .I
( w h e n p = 2, SX u. = e g u v w ~+, ~ D + X ,r = a J,,+ # D _ X
cr
where w ,+, , = g , o J d+t, , the 2 form associatec c o m p l e x structure J + [ 8 ] ) , the variation grangian L is
B. Kim / Physics Letters B 335 (1994) 51-55
8 L = 2gm, S X " [ D _ D + X
+v
p
~" + F p o . D _ X
o-
D+X
(replace D ± X ° with D + X ° + V + and choosing X ° = 0 gauge), we get the first order lagrangian
]
= 2~:co;q.v..o-p_ 1 D + X °~ . . . D + X O p - I
X [ D _ D + X p + F .+P ~D_X
53
# D+X v ]
S = - 1/2 f D 2 ( e o o V + V - + e i o D + X i V _
1
= 2E--
+ eoiV+D_X i + eijD+XiD_X
P
+dp(O+V_
× [ D _ ( c o ; ~ l . . . . p-i D + X P D + X ' ~ "" " D + X % - ~ )
_ (. -W + c o p+c r l . . . o ' e _ -t~
1)
× D_XUD+X~D+X
'~ • • • D + X ~ - ~ ],
(7)
where = and
+ U5
j
(14)
+D_V+)).
H e r e { / z , u } = 0 , 1 .... 2 n - l , {i,j}= l .... 2n-l, eu, = gu, + bu,, and ¢ is the lagrange multiplier whose variation gives V± = D ± X ° and gives back the original action (3). The first order action (14) is invariant under
(8)
T u v p = ~1 ( b # v , p + bvp,/z + b p ~ , v )
Similarly,
[4,5].
8 , ¢ = e e ~ o g ~" [ W+o,~2...%_~ V+ D + X '~2 • • • D + X '~p-~ .31- . . . . . .
1
6 L = 2 7 1 - [ D + ( co;O.I." "lTp_l P × D_XPD_X
1.)-[~ vt:O'l PO'l...O-p_20~tJ+ A
°'1 . . . D _ X a p - ~
-- (V;co;O_I...O_p__
)
+
(.0+ I~ vkl vkI'"kp-1 L,.+..x
" • •
D + X °-p - z V+
• • • D + X kp-l ] ,
(15)
1)
x D+X~D_XPD_X
(9)
cr~ . . . D _ X °-e-I ]
,s, xJ = E g . [ C%o~,2...~,p_ + ~V + D + X '~2 . . . D + X ~ p - t
(where 27- = 0 + F - and FR-~" = F ~ - T ~ ) under the transformation
Ji- . . . . . .
co+ voh...trv_20 D + X °'1 ' " • D + X °'p-z V+ ( S X ~ ) g /~, = "rlco~,o_v..O_p_~D _ X °~ . . . D _ X % - ~ ,
+CO+k1 • " kp _ I D + X k l
(10)
" " D+ Xkp-1]
(16)
and i.e.,
8 X ~ = rlgU~ co~-~v..~,p_l D - X ~'l " • D - X ~p-I .
( 11)
6n¢ = -~Teo~g u~ [ co~-o,~z...,~,_~V - D - X ~ 2 +
So the action is invariant if we choose the form co + covariantly constant, i.e., V~-4-cov~,~-~ = O. In this case
. . . . . .
Wvoq ...o.p_2oD_XOq . . . D _ X ~ p
+ co~/q...k.-i D - X ~ I
" " "D - X k p - '
2 V_
]'
(17)
(12)
Jo)' = W;6rl...O-p_ I D + X P D + X °'1 • • • D + X '~p-'
8~ XJ = r/g/~ [co~-0,,2.--o-p_~V - D - X ~ ' 2 " " " D - X ~ ' P i
and Jo~-- ~" ('Op-Orl..,O'p__ I D
" "" D - X ' ~ " - ~
- X P D - X
~rl
" " " D - X
°"
1
(13)
are conserved (Noether) currents that are generators of a super W-algebra extending the superconformal algebra [ 2]. Now one can get a new target manifold with new metric g, and torsion potential b by dualizing as in [6-8]. Gauging the symmetry with gauge fields V±
+
. . . . . .
co~oq...o.p_zoD_XO'l . . . D _ X O ' e
2 V_
+ w~kv..kp_l D - X kl . . . D _ X k p - I ] .
(18) To find the dual model, we eliminate V± by the equations of motion and get
B. K i m / P h y s i c s Letters B 3 3 5 (1994) 5 1 - 5 5
54
V+ = eoo I (D+49 - eioD + X i) ,
gOj--kl...k~-l = O~j'-k,'"kp-, -- go01 W j k 1 ...kp_ 1 '
(19)
V_ = - e ~ o 1 ( D _ 4 9 + e o i D _ X i)
where
so that Wj-k, ...k,-i = eojWo-k,...kv , - eOkl ( ° O S k 2 k p --I
8,49 = eeuog ~ [ ( p - 1 )eoo WvOkv..kp_,
+ eOk2 ogOkljka...kl ' - l + " " "
× D + 4 9 D + X k2 . . . D+Xkp-, + -}- (O)b, kl...~p_ l --
+ ( - - 1 ) p - l e o k p iW0-k,...kp_2j. --1
+
( p -- 1 )eoo ek,O~OvOk2...ke_l
× D + X k~D + X k2 • . . D+Xk,, -l ], -1
Now the new currents in the dual space ar (20)
+
x D + 4 9 D + X k2 " " D + X kp-'
and
--1 + 1)eoo eklOC°vOk2...kp-i
× D + X k I D + X k2 . . . D+Xkp -t ],
x ° = 49)
Jco+ = &+ po.1...o_p_ I D + ivxp rL~* + Av o l "" .D+XO> ,
8~Xj = eg/v [ ( p - 1 )e00 w~0k2...k~_,
+ ( 0)+~k,...k~_, -- ( P -
1
+
](o-
(21)
and
= (O mrl...~rv_l D - X p D - X ° q
""
"D - X ~ e - 1
,
which generate a super W-algebra in the new Also substituting (19) into (14), we get 1 lagrangian [6,9]
--109-8~ 49 = - r l e o u g m" [ - ( p - 1 ) eoo ~0k2..%,_1
= _ 1/
X D _ 49D_ X k2 - - - D _ X ke- i -~- (O.)~l...kp_ 1 -- ( p -
× D_XkID_X t~,lsJ = ,rlgJV[-(p
-
- e ~ 1e i o D + X i D _ 4 9
1) eoo -1 eOkl°)vOk2...kp_l -
g2 . .. D _ X k , - 1 ],
D 2 (e~olD+49 D _ 49 + eooleojD+49D
(22)
+ (eij - e o J e i o e o j ) D + X i D - X
j)
and read off the dual metric ~ and torsion pot~ i.e.,
1)e~ol wvokz...kp ,
× D _ 49D _ X k2 • • • D _ X kp- l ( gooI guv = \ g~oI boi
-1
+ (w~-k,...g,_ , - ( p - l)eoo eoklW,,O~2...k,,_, ) × D _ X kJ D _ X k2 . .. D_X~P-1 ].
(23)
and we can read off the dual covariantly constant forms ~ + and & - : -+ + O)okl ""kp-1 = go01 O)Okl'"kp
1 '
(7)+ - 1 Wj~,..%, + jk,..-kp , = 09+ jkl..%,-i - go0 ,,
gffolboj gij - gool giogjo + g ~ l biobjo
(24) (25)
and b~v = (
0 --gool goi
gffol goj bij + go01(giobjo - gjobio)
with respect to the basis {d49, d X 1 .. d X 2n-I
where +1 ...kv
i --'-- e j o 09+ Okl...kp
+ -]- ek2OtOokljk3...kp -
4. Conclusions
i - - eklO 09+ Ojk2...kp_l
+...
-Jr- ( - - I ") p - I ekt,-lOt°Okr..kp_2j, +
(26)
and &0--kl.-.k,,_, = --g001 c00--k,---k~-i'
(27)
The existence of covariantly constant forms the holonomy group of the manifold. When 1 nection is the Levi-Civita connection, the cl tion was done by Berger [3]. In the case c model with a riemannian target manifold, v ity transformations we get another manifold v
B, Kim / Physics Letters B 335 (1994) 51-55
metric and torsion and the existence of the covariantly constant forms will restrict the holonomy of this new manifold. The classification of holonomy groups in the case of the connections with torsion, generalizing Berger's work, is an interesting area to be studied and the duality transformation will be very useful for that purpose.
Acknowledgments I thank Martin Ro~ek for suggesting this problem and for many discussions. I also thank Jan de Boer for helpful comments. Finally, I thank NSF Grant No. PHY 93-09888 for partial support.
55
References [ 1] A. Giveon, M. Porrati and E. Rabinovici, Target Space Duality in String Theory RI-I-94, NYU-TH.94/01/01, hepth/9401139. [2] P.S. Howe and G. Papadopoulos, Commun. Math. Phys.151 (1993) 467. [3] M. Berger, Bull. Soc. Math. France 83 (1953) 279. I4] S.J. Gates, C.M. Hull and M. Ro~ek, Nucl. Phys B 248 (1984) 157. [5] M. Rorek, Modified Calabi-Yau Manifolds with Torsion, in Essays on Mirror Manifolds, ed. S.T. Yau (International Press, Hong Kong, 1992) 480. [6] E.S. Fradkin and A.A. Tseytlin, Ann. Phys. 162 (1985) 31. [7] N. J. Hitchin, A. Karlhede, U. Lindstrrm and M. Ro~ek, Commun. Math. Phys. 108 (1987) 535. [8] B. Kim and M. Rorek, preprint ITP-SB-94-22 hepth/9406063. [9] T. Buscher, Phys. Lett. B 201 (1988) 466.
PHYSICS LETTERS B t-I XI.:VIt'R
Physics Letters B 335 (1994) 51-55
Super W-symmetries, covariantly constant forms and duality transformations Byungbae Kim 1 Institute fi~r Theoretical Physics. State Universi O' oJ New York at Stony Brook, Stony Brook. NY I 1794. USA Received 28 June 1994 FAitor: H. Georgi
Abstract On a supersymmetric sigma model the covariantly constant fiwms are related to the conserved currents that are generators of a super W-algebra extending the supcrconformal algebra. The existence of covariantly constant forms restricts the holonomy group of the manifold. Via duality transformation we get new covariantly constant forms, thus restricting the holonomy group of thc new manifold.
I. Introduction One of the most intriguing properties of string theory is the existence o f a duality symmetry that relates strings propagating on different backgrounds to each other (see, e.g., I 1 l and references therein). Duality transformations play an important role in the study of moduli spaces of string vacua, and are presumably part of a large and fundamental symmetry of string theory. The purpose of this note is to investigate how duality acts on N = 1 sigma models with torsion possessing an extended algebra of conserved currents, i.e., a super W-algcbra. As was shown in [2], conserved chiral currents can in a natural way be associated to covariantly closed forms on the target manifold. Since covariantly closed forms imply certain constraints on the holonomy, the manilblds will in general have reduced holonomy. A simple example of such a manifold is a Calabi-Yau manifold, which has reduced holonomy S U ( 3 ) and whose covariantly constant complex E-mail address: [email protected].
structure enhances the superconformal algebra to an N = 2 algebra. In this letter we show explicitly how conserved currents associated to covariantly constant differential forms are under duality mapped to another set of conserved currents on the dual manifold. It will turn out that these latter currents are also associated to covariantly constant forms on the dual manifold, for which we present explicit expressions. From this we think that duality does, in general, leave the holonomy of the manifold invariant 2.
2. Super W-algebra and covariantly constant forms Given a supersymmetric nonlinear or-model lagrangian 2 Actually, as we shall see below, duality transformations make use of an isometry of the manifold, and the holonomy is preserved only if the isometry is compatible with the (covariantly constant) forms.
0370-2693/9,1/$07.00 ~ 1994 Elsevier Science B.V. All rights rescrvcd SSDI 037(1-2693 ( 9 4 1 0 0 8 9 2 -2
52
B. Kim / Physics Leners B 335 (1994) 51-55
Table 1 Holonomy groups and covariantly constant forms on nonsymmetric spaces. Dim.
Holonomy group
Covariantly constant forms (generators of the algebra)
n 2n 2n 4n 4n 7 8
SO(n) U(n) SU(n) Sp ( n ) Sp(n)Sp( I ) G2 Spin(7)
wn = volume form (w2) t' k = 1,.-.n; w2: KLihler form (w2) k k = 1..-.n; w,t = (vol) U2 wn: complex volume form, ~n w = awl + bw2 + cw3, a 2 + b 2 + 2
2
C2 :
Kfihler manifold Special K~.hler (Ricci I Hyperk~ihler Quaternionic K~.hler
1
2
fi4 = w I + w 2 + w 3 (O3, ¢04 ¢O4
The product of the algebra involves not only wedge products of forms but also contractions of two or more forms using the t" -7 I J
S=
D2(guuD+XUD_X~),
( 1)
+ p w , ~ , , : . . . , , p D _ D + X '~l D + X 'y2 • • • D t X '~' = cguw,~ , ...,~p D _ X u D F X Crl
forms
and
dualil
pogal,~2...,~, F p a D _ X
m • • • D . X ~'~' p
D ~ X " D _ X "2 • • • D + X '~t'
= (~7~,w,,,...,,~,)D_XUD+X a'..
. D . . X ~'p
( w h e r e we used the equation o f motion D _ D + X ~p
constant
N o w given a s u p e r s y m m e t r i c nonlinear o'grangian ( w e allow torsion for m o r e genera covariantly constant p - f o r m s w + (see be have the action
D - J,o = O u w , , ~..... ~ , D - X U D + X ('l " " D : - X ~p
I p,rD_XPD+X
Covariantly
transformation
for Jo, = w , , l . . . ~ p D - X '~' • • • D f.X ~'
_ _
3.
(2) ~ +
~r = 0 ) .
So D _ J , , = 0 if the form w is covariantly constant, i.e., V u W = 0. In this case J,o is a conserved current that is a generator o f a super W-algebra via Poisson Brackets, e x t e n d i n g the superconformal algebra [ 2 ]. The existence o f such covariantly constant forms is related to the h o l o n o m y group o f the target manifold. Table 1 shows the existence o f covariantly constant forms and the h o l o n o m y group o f the manifold ( f o r the n o n s y m m e t r i c spaces) [ 3 ] . For each h o l o n o m y group, there is an associated covariantly constant totally antisymmetric tensor, and this implies the existence o f an associated s y m m e t r y o f the c o r r e s p o n d i n g s u p e r s y m m e t r i c sigma model [ 2 ] . These currents generate a super W-algebra via Poisson Brackets and these algebras are extensions o f the superconformal algebra.
S = 1/D2((g~z~
+ bu~)D+XUD-X")
•
Suppose further there is an isometry g e n e r vector field Y with £ v ( d b ) = 0 and £ v w + one can c h o o s e ( l o c a l ) coordinates such a / o x ° and the metric g, the torsion potent the p - f o r m s w + are independent o f X°. U n d e r the transformations ( SX~' ) gu~ = a w +,l ..... , - ' D + X"' • • • D+ X "p
i.e., ~ X u .1 . . .[
~
UVog+ ~
I "''O'p--I
r~ x'r~
/'J+
' " "
D+X'rl, .I
( w h e n p = 2, SX u. = e g u v w ~+, ~ D + X ,r = a J,,+ # D _ X
cr
where w ,+, , = g , o J d+t, , the 2 form associatec c o m p l e x structure J + [ 8 ] ) , the variation grangian L is
B. Kim / Physics Letters B 335 (1994) 51-55
8 L = 2gm, S X " [ D _ D + X
+v
p
~" + F p o . D _ X
o-
D+X
(replace D ± X ° with D + X ° + V + and choosing X ° = 0 gauge), we get the first order lagrangian
]
= 2~:co;q.v..o-p_ 1 D + X °~ . . . D + X O p - I
X [ D _ D + X p + F .+P ~D_X
53
# D+X v ]
S = - 1/2 f D 2 ( e o o V + V - + e i o D + X i V _
1
= 2E--
+ eoiV+D_X i + eijD+XiD_X
P
+dp(O+V_
× [ D _ ( c o ; ~ l . . . . p-i D + X P D + X ' ~ "" " D + X % - ~ )
_ (. -W + c o p+c r l . . . o ' e _ -t~
1)
× D_XUD+X~D+X
'~ • • • D + X ~ - ~ ],
(7)
where = and
+ U5
j
(14)
+D_V+)).
H e r e { / z , u } = 0 , 1 .... 2 n - l , {i,j}= l .... 2n-l, eu, = gu, + bu,, and ¢ is the lagrange multiplier whose variation gives V± = D ± X ° and gives back the original action (3). The first order action (14) is invariant under
(8)
T u v p = ~1 ( b # v , p + bvp,/z + b p ~ , v )
Similarly,
[4,5].
8 , ¢ = e e ~ o g ~" [ W+o,~2...%_~ V+ D + X '~2 • • • D + X '~p-~ .31- . . . . . .
1
6 L = 2 7 1 - [ D + ( co;O.I." "lTp_l P × D_XPD_X
1.)-[~ vt:O'l PO'l...O-p_20~tJ+ A
°'1 . . . D _ X a p - ~
-- (V;co;O_I...O_p__
)
+
(.0+ I~ vkl vkI'"kp-1 L,.+..x
" • •
D + X °-p - z V+
• • • D + X kp-l ] ,
(15)
1)
x D+X~D_XPD_X
(9)
cr~ . . . D _ X °-e-I ]
,s, xJ = E g . [ C%o~,2...~,p_ + ~V + D + X '~2 . . . D + X ~ p - t
(where 27- = 0 + F - and FR-~" = F ~ - T ~ ) under the transformation
Ji- . . . . . .
co+ voh...trv_20 D + X °'1 ' " • D + X °'p-z V+ ( S X ~ ) g /~, = "rlco~,o_v..O_p_~D _ X °~ . . . D _ X % - ~ ,
+CO+k1 • " kp _ I D + X k l
(10)
" " D+ Xkp-1]
(16)
and i.e.,
8 X ~ = rlgU~ co~-~v..~,p_l D - X ~'l " • D - X ~p-I .
( 11)
6n¢ = -~Teo~g u~ [ co~-o,~z...,~,_~V - D - X ~ 2 +
So the action is invariant if we choose the form co + covariantly constant, i.e., V~-4-cov~,~-~ = O. In this case
. . . . . .
Wvoq ...o.p_2oD_XOq . . . D _ X ~ p
+ co~/q...k.-i D - X ~ I
" " "D - X k p - '
2 V_
]'
(17)
(12)
Jo)' = W;6rl...O-p_ I D + X P D + X °'1 • • • D + X '~p-'
8~ XJ = r/g/~ [co~-0,,2.--o-p_~V - D - X ~ ' 2 " " " D - X ~ ' P i
and Jo~-- ~" ('Op-Orl..,O'p__ I D
" "" D - X ' ~ " - ~
- X P D - X
~rl
" " " D - X
°"
1
(13)
are conserved (Noether) currents that are generators of a super W-algebra extending the superconformal algebra [ 2]. Now one can get a new target manifold with new metric g, and torsion potential b by dualizing as in [6-8]. Gauging the symmetry with gauge fields V±
+
. . . . . .
co~oq...o.p_zoD_XO'l . . . D _ X O ' e
2 V_
+ w~kv..kp_l D - X kl . . . D _ X k p - I ] .
(18) To find the dual model, we eliminate V± by the equations of motion and get
B. K i m / P h y s i c s Letters B 3 3 5 (1994) 5 1 - 5 5
54
V+ = eoo I (D+49 - eioD + X i) ,
gOj--kl...k~-l = O~j'-k,'"kp-, -- go01 W j k 1 ...kp_ 1 '
(19)
V_ = - e ~ o 1 ( D _ 4 9 + e o i D _ X i)
where
so that Wj-k, ...k,-i = eojWo-k,...kv , - eOkl ( ° O S k 2 k p --I
8,49 = eeuog ~ [ ( p - 1 )eoo WvOkv..kp_,
+ eOk2 ogOkljka...kl ' - l + " " "
× D + 4 9 D + X k2 . . . D+Xkp-, + -}- (O)b, kl...~p_ l --
+ ( - - 1 ) p - l e o k p iW0-k,...kp_2j. --1
+
( p -- 1 )eoo ek,O~OvOk2...ke_l
× D + X k~D + X k2 • . . D+Xk,, -l ], -1
Now the new currents in the dual space ar (20)
+
x D + 4 9 D + X k2 " " D + X kp-'
and
--1 + 1)eoo eklOC°vOk2...kp-i
× D + X k I D + X k2 . . . D+Xkp -t ],
x ° = 49)
Jco+ = &+ po.1...o_p_ I D + ivxp rL~* + Av o l "" .D+XO> ,
8~Xj = eg/v [ ( p - 1 )e00 w~0k2...k~_,
+ ( 0)+~k,...k~_, -- ( P -
1
+
](o-
(21)
and
= (O mrl...~rv_l D - X p D - X ° q
""
"D - X ~ e - 1
,
which generate a super W-algebra in the new Also substituting (19) into (14), we get 1 lagrangian [6,9]
--109-8~ 49 = - r l e o u g m" [ - ( p - 1 ) eoo ~0k2..%,_1
= _ 1/
X D _ 49D_ X k2 - - - D _ X ke- i -~- (O.)~l...kp_ 1 -- ( p -
× D_XkID_X t~,lsJ = ,rlgJV[-(p
-
- e ~ 1e i o D + X i D _ 4 9
1) eoo -1 eOkl°)vOk2...kp_l -
g2 . .. D _ X k , - 1 ],
D 2 (e~olD+49 D _ 49 + eooleojD+49D
(22)
+ (eij - e o J e i o e o j ) D + X i D - X
j)
and read off the dual metric ~ and torsion pot~ i.e.,
1)e~ol wvokz...kp ,
× D _ 49D _ X k2 • • • D _ X kp- l ( gooI guv = \ g~oI boi
-1
+ (w~-k,...g,_ , - ( p - l)eoo eoklW,,O~2...k,,_, ) × D _ X kJ D _ X k2 . .. D_X~P-1 ].
(23)
and we can read off the dual covariantly constant forms ~ + and & - : -+ + O)okl ""kp-1 = go01 O)Okl'"kp
1 '
(7)+ - 1 Wj~,..%, + jk,..-kp , = 09+ jkl..%,-i - go0 ,,
gffolboj gij - gool giogjo + g ~ l biobjo
(24) (25)
and b~v = (
0 --gool goi
gffol goj bij + go01(giobjo - gjobio)
with respect to the basis {d49, d X 1 .. d X 2n-I
where +1 ...kv
i --'-- e j o 09+ Okl...kp
+ -]- ek2OtOokljk3...kp -
4. Conclusions
i - - eklO 09+ Ojk2...kp_l
+...
-Jr- ( - - I ") p - I ekt,-lOt°Okr..kp_2j, +
(26)
and &0--kl.-.k,,_, = --g001 c00--k,---k~-i'
(27)
The existence of covariantly constant forms the holonomy group of the manifold. When 1 nection is the Levi-Civita connection, the cl tion was done by Berger [3]. In the case c model with a riemannian target manifold, v ity transformations we get another manifold v
B, Kim / Physics Letters B 335 (1994) 51-55
metric and torsion and the existence of the covariantly constant forms will restrict the holonomy of this new manifold. The classification of holonomy groups in the case of the connections with torsion, generalizing Berger's work, is an interesting area to be studied and the duality transformation will be very useful for that purpose.
Acknowledgments I thank Martin Ro~ek for suggesting this problem and for many discussions. I also thank Jan de Boer for helpful comments. Finally, I thank NSF Grant No. PHY 93-09888 for partial support.
55
References [ 1] A. Giveon, M. Porrati and E. Rabinovici, Target Space Duality in String Theory RI-I-94, NYU-TH.94/01/01, hepth/9401139. [2] P.S. Howe and G. Papadopoulos, Commun. Math. Phys.151 (1993) 467. [3] M. Berger, Bull. Soc. Math. France 83 (1953) 279. I4] S.J. Gates, C.M. Hull and M. Ro~ek, Nucl. Phys B 248 (1984) 157. [5] M. Rorek, Modified Calabi-Yau Manifolds with Torsion, in Essays on Mirror Manifolds, ed. S.T. Yau (International Press, Hong Kong, 1992) 480. [6] E.S. Fradkin and A.A. Tseytlin, Ann. Phys. 162 (1985) 31. [7] N. J. Hitchin, A. Karlhede, U. Lindstrrm and M. Ro~ek, Commun. Math. Phys. 108 (1987) 535. [8] B. Kim and M. Rorek, preprint ITP-SB-94-22 hepth/9406063. [9] T. Buscher, Phys. Lett. B 201 (1988) 466.