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Compactification of type IIB string theory on Calabi–Yau threefolds
Nuclear Physics B 569 Ž2000. 229–246 www.elsevier.nlrlocaternpe
Compactification of type IIB string theory on Calabi– Yau threefolds a Robert Bohm , Carl Herrmann b, Jan Louis a ¨ a, Holger Gunther ¨ b
a Martin – Luther – UniÕersitat ¨ Halle – Wittenberg, FB Physik, D-06099 Halle, Germany Centre de Physique Theorique, CNRS – Luminy Case 907, F-13288 Marseille Cedex 9, France ´
Received 16 August 1999; accepted 15 December 1999
Abstract We study compactifications of type IIB supergravity on Calabi–Yau threefolds. The resulting low energy effective Lagrangian is displayed in the large volume limit and its symmetry properties – with specific emphasis on the SLŽ2,Z. – are discussed. The explicit map to type IIA string theory compactified on a mirror Calabi–Yau is derived. We argue that strong coupling effects on the world-sheet break the SLŽ2,Z.. q 2000 Elsevier Science B.V. All rights reserved.
1. Introduction Compactifications of string theories on six-dimensional Calabi–Yau manifolds result in string vacua with four flat Minkowskian dimensions Ž d s 4. and, depending on the type of Calabi–Yau manifold, a fixed number of preserved supercharges. Specifically, string vacua with N s 2 supersymmetry in d s 4 are obtained by compactifying type II strings on Calabi–Yau threefolds Y3 , the heterotic string on K 3 = T 2 or the type I string on K 3 = T 2 . It is believed that all of the resulting string vacua are perturbative descriptions of different regions in one and the same moduli space w1,2x. In this paper we confine our attention to the region of the moduli space where type II string compactifications are the appropriate perturbative theories.
E-mail addresses: [email protected] ŽR. Bohm ¨ ., [email protected] ŽH. Gunther ., [email protected] ŽC. Herrmann., [email protected] ŽJ. Louis.. ¨ 0550-3213r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 5 5 0 - 3 2 1 3 Ž 9 9 . 0 0 7 9 6 - 8
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R. Bohm ¨ et al.r Nuclear Physics B 569 (2000) 229–246
In d s 10 type II string theories come in two different versions, the non-chiral type IIA and the chiral type IIB string theory but upon compactification on Calabi–Yau threefolds they are related by mirror symmetry. More precisely, type IIB compactified on Y3 is equivalent to IIA compactified on the mirror manifold Y˜3 w3,4x. Due to this equivalence one has the choice to study either compactification but since there is no simple covariant action of type IIB supergravity in d s 10 w5–7x type IIA compactifications appear to be easier w8x. In terms of the underlying conformal field theory the two theories in d s 4 only differ by the GSO projection and thus can be discussed on the same footing w3,4x. However, it is believed that the type IIB theory in d s 10 has an exact non-perturbative SLŽ2,Z. symmetry w9–11x which is not shared by type IIA Žin d s 10.. Thus, it is of interest to study the ‘fate’ of this SLŽ2,Z. symmetry upon compactification to d s 4 w12x and this is one of the goals of this paper. For that reason we focus on geometrical compactifications Žand not the conformal field theory versions. of the effective type IIB theory for most parts of this paper. In addition, the general type IIB effective Lagrangian and the explicit map to the type IIA vacua has only been given in a special case w13,14x and as a byproduct we close this gap. This paper is organized as follows. In Section 2 we recall some of the properties of type IIB supergravity in d s 10. In Section 3 we discuss the compactification on Calabi–Yau threefolds and display the four-dimensional low energy effective action in the large volume limit. Apart from the gravitational multiplet one finds hŽ1,1. tensor multiplets, hŽ1,2. vector multiplets and one ‘‘double-tensor’’ multiplet. We concentrate on the couplings of the tensor multiplets since the couplings of the vector multiplets have been given before w4,15–20x. In addition the SLŽ2,Z. – inherited from the d s 10 theory – acts on the tensor multiplets while the vector multiplets are left invariant. In Section 4 the explicit map between type IIB compactified on Y3 and type IIA compactified on the mirror manifold Y˜3 is displayed. This map enables us to also discuss the geometry of the tensor multiplets in terms of a holomorphic prepotential and obtain the world-sheet instanton corrections. We find that the SLŽ2,Z. is manifest at the string tree level in the large volume limit of the compactification but broken once strong coupling effects on the world-sheet Žworld-sheet instantons. are taken into account. Furthermore, the SLŽ2,Z. is also present in the large complex structure limit of type IIA vacua but rather obscure in the standard field variables. Section 5 contains our conclusions and some of the technical aspects of this paper are collected in an appendix.
2. Type IIB Supergravity in d s 10 The bosonic massless modes of the type IIB string are the graviton g M N Ž M, N s 0, . . . ,9., a doublet of antisymmetric tensors BMI N Ž I s 1,2., two real scalar fields f ,l and a 4-form DM N P Q with a self-dual field strength. g M N , BM1 N and f arise in the NS–NS sector while BM2 N ,l and DM N P Q reside in the R–R sector; f is the dilaton of type IIB string theory. The difficulty of constructing a covariant low energy effective action for these massless modes is due to the presence of the self-dual 5-form field strength. The field equations of type IIB supergravity on the other hand are well known w5–7x. However, if
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one does not impose the self-duality condition a covariant action Žin the Einstein frame. can be given w21,22x1 S s y 12 d 10 x y g
'
H
1 q
'
96 y g
ž
R y 14 Tr Ž E ME My1 . q 34 H I MI J H J q 56 F 2
/
eI J D n H I n H J ,
Ž 1.
where R is the scalar curvature and we abbreviated MI J '
1
< l< 2 Im l yRe l
ž
yRe l , 1
/
l ' l q ieyf ,
HMI N P ' E w M BNI P x s 13 Ž E M BNI P q E N BPI M q E P BMI N . , FM N P Q R ' E w M DN P Q R x q 34 e I J BwIM N E P BQJ R x .
Ž 2.
In Eq. Ž1. we suppress space-time indices; in particular, the topological term D n H I n H J is contracted with the ten-dimensional e-tensor while all other terms are contracted with the ten-dimensional metric. Finally, we choose e 12 s q1. The type IIB supergravity is obtained as the Euler–Lagrange equations of the action Ž1. together with the self-duality condition FM N P Q R s
1
e M N P Q R ST U V W F ST U V W .
'
5! y g
Ž 3.
This theory has a number of symmetries. First of all there are the gauge transformations of the 4-form Žwith parameters SN P Q .
d DM N P Q s E w M SN P Q x ,
Ž 4.
under which all other fields are invariant. Secondly, there are the gauge transformations of the two antisymmetric tensors Žwith parameters V NI . which also transform the 4-form
d BMI N s E w M V NI x , d DM N P Q s y 34 e I J V wIM HNJ P Q x .
Ž 5.
Finally, there is an SLŽ2,Z. acting as follows: 2
¨ l s acll qq db , ¨H sL H H X
l
I M NP
XI MNP
I J
J MNP ,
Ž 6.
1 Recently a covariant action including the self-dual 4-form has been constructed w23–25x but here we choose to follow the procedure outlined in Refs. w21,22x for compactifying type IIB supergravity and use Ž1. instead. 2 In fact at the classical level one has the larger group SLŽ2,R. which is broken by quantum corrections to Ž SL 2,Z..
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where
Ls
ž
d b
c , a
/
ad y bc s 1,
a,b,c,d g Z.
Ž 7.
Using Ž2. one shows that the matrix M transforms according to M
¨M sL X
y1T
M Ly1 .
Ž 8.
3. Calabi–Yau compactifications of type IIB supergravity 3.1. The spectrum Let us now study the compactification of the type IIB low energy effective theory on Calabi–Yau threefolds Y3 . The resulting theory in d s 4 has N s 2 supersymmetry and the low energy spectrum comes in appropriate N s 2 supermultiplets. More specifically one finds that the ten-dimensional metric g M N decomposes into the four-dimensional metric gmn Ž m , n s 0, . . . ,3., 2 = hŽ1,2. deformations of the complex structure d g ab , d g a b Ž a , b s 1,2,3. and hŽ1,1. deformations of the Kahler class d g a b . The Hodge ¨ numbers hŽ1,1.Ž hŽ1,2. . count the harmonic Ž1,1.-forms ŽŽ1,2.-forms. on Y3 . The antisymI metric tensors BMI N decompose into a doublet of antisymmetric tensors Bmn and I 2 = hŽ1,1. scalar modes Ba b . Finally the 4-form decomposes into hŽ1,1. Žreal. antisymmetric tensors Dmn a b and hŽ1,2. q 1 real vectors Dm a bg , Dm a bg .3 Together with the two scalars f and l these fields assemble into the following N s 2 supermultiplets: gravitational multiplet
Ž gmn , Dm a bg . ,
double-tensor multiplet
Ž BmnI , f ,l . , Ž Dm a bg , d ga b , d ga b . , Ž Dmn a b , d ga b , BaI b . ,
hŽ1,2. vector multiplets hŽ1,1. tensor multiplets
where we only display the bosonic components. The gravitational multiplet contains two gravitini while all the other multiplets contain two Weyl fermions.4 In d s 4 an antisymmetric tensor describes one physical degree of freedom and thus can always be dualized to a scalar. At the level of supermultiplets this duality relates both the tensor and the double-tensor multiplet to a hypermultiplet which contains four real scalar degrees of freedom. In this dual basis the low energy spectrum features hŽ1,1. q 1 hypermultiplets and hŽ1,2. vector multiplets apart from the gravitational multiplet. The couplings of the vector multiplets have been obtained before w4,15–20x and thus we exclusively focus on the couplings of the hŽ1,1. tensor and the universal double-tensor multiplet. ŽOr, in other words, we consider Calabi–Yau threefolds with hŽ1,2. s 0.. On a 3
Note that the dimensionally reduced self-duality condition Ž3. relates Daba b to Dmngg , Dm a bg to Dm a bg and Dma bg to Dm a bg . 4 The double-tensor multiplet has not been constructed as an off-shell multiplet but we expect that it exists since it appears in the low energy limit of type IIB string theory.
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Calabi–Yau threefold the Kahler deformations of the metric can be expanded in terms of ¨ a harmonic Ž1,1.-forms vab according to BaI b s bˆ Ia Ž x . vaab ,
i d g ab s Õˆ a Ž x . vaab , a Dmn a b s Dˆmn Ž x . vaab ,
a s 1, . . . ,hŽ1 ,1. .
Ž 9.
In terms of these variables the tensor multiplets consist of ˆ ˆ ˆ Before we display the four-dimensional Lagrangian let us collect a few more properties of Calabi–Yau threefolds w26,27x. The Kahler form J is defined as ¨ a Ž Dmn ,Õ a,b Ia ..
J s i d g ab d j a n d j b ,
Ž 10 .
a
where the j are the complex coordinates on Y3 . The volume V can be expressed in terms of J according to 1 6
6
H'g d j s HJ n J n J.
Vsi
Ž 11 .
On the space of Ž1,1.-forms one defines a metric i b Ga b s y vaab vgd g a d g gb g d 6j . V The intersection numbers of Y3 are given by
'
H
Ž 12 .
kabc s v a n v b n v c.
Ž 13 .
H
Both, the metric and the volume, can be expressed in terms of the moduli Õˆ a and the intersection numbers V s 16 k a b c Õˆ a Õˆ b Õˆ c , Ga b s yVy1k a b c Õˆ c q 14 Vy2k ac d Õˆ c Õˆ dk b e f Õˆ e Õˆ f .
Ž 14 .
3.2. The Lagrangian The next step is the construction of the dimensionally reduced type IIB low energy effective theory. We follow the procedure outlined in Refs. w21,22x and reduce the ten-dimensional action Ž1. imposing the self-duality condition on the field equations by hand. The resulting field equations in d s 4 can be shown to be the Euler–Lagrange equations of a four-dimensional action. The technical details of the reduction procedure are deferred to the appendix and we only give the final result
'y g y1 L s y
1 2
2
R y 14 e 2 f 4 V Ž Õ . Ž Em l . y Ž Em f4 .
2
I y 61 ey3 f 4 V 1r2 Ž Õ . Hmnr MI J H Jmnr y Ga b Ž Õ . Em Õ aE m Õ b
y e f 4 V 1r2 Ž Õ . Ga b Ž Õ . Em b Ia MI J E m b J b y 32 ey2 f 4 V Ž Õ . Ga b Ž Õ . a J = Ž Fmnr q e I J Hmnr b Ia . Ž F bmnr q e I J H Jmnr b I b .
i y
'
12 y g
a e mnrsk a b c e I J Ž Fmnr b I b Es b Jc
I y 23 e K L Hmnr b J a b K b Es b L c . ,
Ž 15 .
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where we redefined the field variables compared to the expansion parameters of Eqs. Ž9. according to a a I Dmn ' Dˆmn q 12 e I J Bmn b Ja,
b Ia ' 23 bˆ Ia .
a Fmnr ' E w m Dnra x ,
Ž 16 .
a
Furthermore, the kinetic terms for the Ž1,1.-moduli Õˆ and the dilaton decoupled due to the definition of the four-dimensional dilaton w20x ey2 f 4 s V Ž Õ . ey2 f ,
Ž 17 .
and ‘rotated’ moduli fields Õ a s Õˆ a e f r2 .
Ž 18 .
Note that Ž14. and Ž15. are only valid in the limit where the ten-dimensional theory is a good approximation, i.e. when V is large. This limit is termed the ‘large volume limit’ but generically subleading Ž a X . corrections are also important. These are briefly discussed in Subsection 4.3. As we already discussed, an antisymmetric tensor in four space-time dimensions is a I dual to a scalar and so we can dualize the hŽ1,1. q 2 antisymmetric tensor fields Dmn , Bmn in the action Ž15.. This is done most conveniently by adding hŽ1,1. q 2 Lagrange multipliers g a ,h I to the Lagrangian I a L X s L q i e mnrs Hmnr Es h I q i e mnrs Fmnr Es g a .
Ž 19 .
a I The Žalgebraic. equations of motion of L X for Fmnr and Hmnr can then be used to eliminate these fields in favour of the scalars g a and h I . The resulting Lagrangian reads
'y g y1 L X s y
1 2
2
2
R y 14 e 2 f 4 V Ž Em l . y Ž Em f4 . y Ga b Em b 1 a E m b 1 b
y Ga b Em Õ a E m Õ b y e 2 f 4 VGa b Ž Em b 2 a y l Em b 1 a . Ž E m b 2 b y l E m b 1 b . e2f4 y 16V
Ž Gy1 .
ab
Ž Em g a y
1 2
k ac d e I J b Ic Em b J d .
= Ž E m g b y 12 k b e f e K L b K e E m b L f . y 14 e 2 f 4 Vy1 Em h 2 y b 1 a Ž Em g a y 16 k a b c e I J b I b Em b Jc .
ž
2
/
y 14 e 4f 4 Ž Em h1 q l Em h 2 y Ž lb 1 a y b 2 a . 2
= Ž Em g a y 16 k a b c e I J b I b Em b Jc . ,
/
Ž 20 .
where in addition the f4-dependence is displayed more explicitly. 3.3. Symmetry properties The symmetries of the ten-dimensional theory discussed in Section 2 are also manifest in d s 4. The action Ž15. is invariant under hŽ1,1. q 2 gauge transformations Žwith parameters Sna, VnI . of the antisymmetric tensors a d Dmn s E w m Snax ,
I d Bmn s E w m VnIx .
Ž 21 .
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In addition we have 2 = hŽ1,1. continuous Peccei–Quinn ŽPQ. symmetries Žwith paramea ters c Ia . acting on the scalar fields b Ia and the antisymmetric tensors Dmn
d b Ia s c Ia ,
a J d Dmn s ye I J c Ia Bmn .
Ž 22 .
These symmetries are ‘inherited’ from the ten-dimensional symmetries Ž4., Ž5.. Note that in the dual basis the gauge transformations Ž21. manifest themselves as hŽ1,1. q 2 additional continuous PQ symmetries Žwith parameters c˜a ,cˆI .
d g a s c˜a ,
d h I s cˆI ,
Ž 23 .
while Ž22. transmogrifies into
d b Ia s c Ia ,
d g a s 12 k a b c e I J c I b b Jc ,
d h I s ye I J Ž c J a g a q 16 k a b c e K L b J a c K b b L c . .
Ž 24 .
One of the distinct features of the type IIB theory in d s 10 is its SLŽ2,Z. invariance Ž6. – Ž8.. This symmetry is of importance since it includes a strong-weak coupling duality as one of its generators. Thus it is of interest to study the fate of this symmetry in the d s 4 action. Since Eq. Ž15. was obtained as a straight dimensional reduction of the action Ž1. the SLŽ2,Z. will also be manifest in Ž15. but due to the field redefinitions Ž17., Ž18. it also acts on the Calabi–Yau moduli Õ a and altogether becomes more involved. More precisely, using Ž6. – Ž8., Ž17. and Ž18., one finds Õ
a
¨Õ
a<
cl q d <,
y2 f 4
e
¨ < cl q d < , ey2 f 4
l
¨
ac < l < 2 q Ž bd q ad . l q bd < cl q d < 2
,
Ž 25 . I where l s l q i Vy1 r2 eyf 4 . The antisymmetric tensors Bmn and the scalars b Ia inherit the transformation law Ž6. and obey I Bmn
¨L B I J
J mn
,
b Ia
¨L b I J
Ja
,
Ž 26 .
a while the Dmn remain invariant. From Ž19. one infers that the dual scalars g a are invariant whereas the h I transform with the inverse matrix
hI
¨L
y1 J hJ . I
Ž 27 .
It can be explicitly checked that the Lagrangian Ž15. is invariant under the transformations specified in Ž25. and Ž26. and thus at the string tree level and in the large volume limit the SLŽ2,Z. is present in d s 4 w12x.
4. The Map between IIA and IIB 4.1. Preliminaries Type IIA string theory compactified on Calabi–Yau threefolds features hŽ1,1. vector multiplets, hŽ1,2. hypermultiplets and one tensor multiplet in its low energy spectrum. The tensor multiplet can be dualized to an additional hypermultiplet. A type IIB compactification on Y3 is equivalent to a type IIA compactification on the mirror
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manifold Y˜3 once the appropriate Žworld-sheet. quantum corrections are taken into account. This equivalence is a property of the underlying conformal field theory w28,29x but has also been demonstrated for the geometrical compactifications on Calabi–Yau manifolds w30–32x. On the mirror manifold the Hodge numbers are reversed and one has hŽ1,1. Ž Y3 . s hŽ1 ,2. Ž Y˜3 . ,
hŽ1 ,2. Ž Y3 . s hŽ1 ,1. Ž Y˜3 . .
Ž 28 .
The low energy spectrum of the two theories is most easily compared in the dual basis where the hŽ1,1. q 1 hypermultiplets of IIB are identified with the hŽ1,2. q 1 hypermultiplets of type IIA. The purpose of this section is to explicitly display the map between the field variables of type IIB and type IIA. For hŽ1,1.Ž Y3 . s 1 this map was given in Ref. w13x. Let us start by recalling the low energy effective action of type IIA. The universal hypermultiplet of IIA which is the dual of the tensor multiplet contains the dilaton fA , the dual f˜ of an antisymmetric tensor and two scalars z 0 , z˜ 0 from the R–R sector. The scalars of the remaining hŽ1,2. hypermultiplets are denoted by z a, z a, z a, z˜ a where a s 1, . . . ,hŽ1,2. .5 The z a, z a arise from the NS–NS sector while the z a, z˜ a come from the R–R sector. The tree level Lagrangian for these hypermultiplets is known to be w20,33,34x
'y g y1 L s y
1 2
2
R y Ž Em fA . y Ga b Ž z , z . E m z a Em z b 2
y 14 e 4fA Em f˜ q z Ei m z˜i y z˜ Ei m z i q 12 e 2 fA R i j Ž z , z . E mz Ei m z j
ž
/
q 12 e 2 fA Ry1 i j Ž z , z . Ž Ii k Ž z , z . E mz k q E mz˜i . Ijl Ž z , z . Em z l q Em z˜j ,
ž
/
Ž 29 . where i, j,k,l s 0, . . . ,hŽ1,2. . The scalar manifold spanned by the 4 = Ž hŽ1,2. q 1. scalars is constrained by N s 2 supergravity to be a quaternionic manifold w35x and this has been explicitly verified for Ž29. in Refs. w33,34x. However, this scalar manifold is not the most general quaternionic manifold but at the string tree level further constrained by the c-map w4x. That is, the submanifold spanned by the deformations of the complex structure z a has to be a special Kahler manifold w15–19x.6 More precisely, the metric ¨ Ga b Ž z, z . is Kahler and furthermore determined by a holomorphic prepotential F Ž X . ¨ which is a homogenous function of degree two Ž F Ž l X . s l2 F Ž X ... Specifically, one has
E Ga b s
E a
Ez Ezb
K,
Ž 30 .
5 Here we use a slightly inconsistent notation since we reserved the index a to label Ž1,1.-forms. However, we are anticipating the mirror map between type IIA and type IIB which identifies the Ž1,1.-forms on Y3 with the Ž1,2.-forms on the mirror manifold Y˜3 and for that reason we use the same indices already at this stage. 6 The reason is that in type IIB vacua the Ž1,2.-forms reside in vector multiplets and the geometry of their scalars is constrained to be special Kahler. ¨
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where K s ylog X i Fi Ž X . q X i Fi Ž X . , Fi Ž X . '
EF
zis
,
EXi
Xi
Ž z 0 s 1. .
X0
Ž 31 .
Furthermore, the matrices R and I in Ž29. are also determined by the prepotential according to R i j s Re Ni j , Ni j s 14 Fi j y
Ii j s Im Ni j ,
Ž Nz . i Ž zN . j , Ž zNz .
Ž 32 .
where Fi j s
E 2F E X Ei X j
Ni j s 14 Fi j q Fi j ,
,
ž
Ž Nz . i s Ni j z j ,
/
Ž zNz . s z i Ni j z j .
Ž 33 .
4.2. The map The geometrical compactification of the effective low energy IIB supergravity from d s 10 to d s 4 is only valid in the large volume limit of Y3 . In this limit the Calabi–Yau volume V is a cubic function of the Ž1,1. moduli Õ a and the intersection numbers k a b c are constant. In order to display the map between IIA and IIB one has to take a similar limit on the IIA side. This limit – termed the ‘large complex structure limit’ – has been studied in the literature w30–32,36x. For us the important point is that such a limit exists and in this limit the prepotential of the complex structure moduli is given by Fs
i 3!
kabc
X aX b X c X
0
2
' Ž X 0. f Ž z. ,
f Ž z. s
i 3!
kabc z az b z c.
Ž 34 .
Using Ž30., Ž31., Ž34. and z a s x a q iy a one obtains the Kahler potential and the metric ¨ K s yln Ž k yyy . , Ga b Ž y . s y
Ž k y . a b 3 Ž k yy . a Ž k yy . b y , 2 2 Ž k yyy . 2 Ž k yyy . 3
ž
/
Ž 35 .
where we abbreviated
Ž k yyy . s k a b c y a y b y c , Ž k yy . a s k a b c y b y c , Ž k y . ab s k a b c y c .
Ž 36 .
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The matrices defined in Ž32. simplify considerably and are given by Is
1 y 13 Ž k xxx . 4
ž
1 2
1 2
Ž k xx . b , yŽ k x . ab
Ž k xx . a
R s y 241 Ž k yyy .
ž
/
1 y 4Ga b x a x b 4Ga b x a
Ry1 s y24 Ž k yyy .
y1
ž
4Ga b x b , y4Ga b
/
xb
1 xa
1 4
y1 a b
y G
qxaxb
/
.
Ž 37 .
Inserting Ž37. into Ž29. yields
'y g y1 L s y
1 2
2
R y Ž Em fA . y Ga b Em y a E m y b y Gab Em x a E m x b 2
y 18 e 2 fA V Ž Em z 0 . y 12 e 2 fA VGa b Ž x a Em z 0 y Em z a . Ž x bE mz 0 y E mz
b
.
y e 2 fA Vy1 Gy1 a b Em z˜a q Ž k xx . a Em z 0 1 2
ž
1 8
y 41 Ž k x . ac Em z c . Ž E mz˜b q 81 Ž k xx . b E mz 0 y 41 Ž k x . b d E mz d . y 2 e 2 fA Vy1 Em z˜ 0 q x aEm z˜a q 241 Ž k xxx . Em z 0 y 18 Ž k xx . a Em z a
ž
2
/
2
y 14 e 4fA Em f˜ q z Ei m z˜i y z˜ Ei m z i ,
ž
Ž 38 .
/
where in anticipation of the map to type IIB we use V s 16 Ž k yyy . Žalthough this does not correspond to the volume of type IIA.. The map of the field variables is obtained by comparing the two Lagrangians Ž20. and Ž38.. First of all one learns that the dilatons of the two theories have to be identified Ž fA s f4 . which is in accord with the fact that mirror symmetry is already valid in string perturbation theory. Furthermore, one is led to the straightforward identification y a sÕ a,
x a s b1 a .
Ž 39 .
The remaining terms can be systematically compared due to their specific dependence on y a and fA . One obtains
Em z 0 s "'2 Em l, b 1 a Em z 0 y Em z a s "'2 Ž l Em b 1 a y Em b 2 a . ,
Em z˜ a q 18 Ž k b 1 b 1 . a Em z 0 y 14 Ž k b 1 . a b Em z b s"
'2 4
1
žE
m ga y 2 kabc
Ž b1 b Em b 2 c y b 2 b Em b1 c . / ,
Em z˜ 0 q b 1 a Em z˜a q 241 Ž k b 1 b 1 b1 . Em z 0 y 18 Ž k b1 b1 . a Em z a s"
'2 4
žE h yb m
2
1a
Em g a q 16 k a b c b 1 a Ž b 1 b Em b 2 c y b 2 b Em b 1 c . ,
/
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Em f˜ y z˜ Ei m z i q z Ei m z˜i s " Em h1 q l Em h 2 q b 2 aEm g a y lb1 aEm g a y 16 k a b c b 2 a Ž b 1 bEm b 2 c y b 2 bEm b 1 c .
ž
q 16 l k a b c b 1 a Ž b 1 bEm b 2 c y b 2 bEm b 1 c . .
Ž 40 .
/
This system of equations is solved by
z 0 s '2 l, z a s '2 Ž lb1 a y b 2 a . ,
'2
z˜a s y z˜0 s
4
'2
ga q
h2 y
4
'2 8
'2 24
l Ž k b1 b1 . a y
l Ž k b1 b1 b1 . q
'2 8
'2 24
Ž k b1 b 2 . a ,
Ž k b1 b1 b 2 . ,
f˜ s h1 q 12 lh 2 q 12 b 2 a g a y 12 lb1 a g a y 121 Ž k b 1 b 2 b 2 . q 121 l Ž k b 1 b 1 b 2 . ,
Ž 41 .
which gives the desired relation between the type IIA and type IIB field variables. It is somewhat surprising that this relation is so involved. Remarkably Ž41. can be inverted ls
1
z0 ,
'2
b2 a s
1
a a '2 Ž x z 0 y z . ,
ga s y h2 s
4
'2
4
'2
z˜a q
z˜0 q
1 2'2
Ž k xz . a ,
1
Ž k xx z . , 6'2
h1 s f˜ y z zi ˜i q 16 Ž k x zz . y 121 z 0 Ž k xx z . .
Ž 42 .
4.3. World-sheet instanton corrections After having established the map between the type IIA and type IIB variables in the large volumerlarge complex structure limit we can discuss the corrections away from this limit. Mirror symmetry assures the equivalence of IIA and IIB compactified on mirror manifolds and it becomes a matter of convenience in what variables one describes the action. The a X corrections to the large volume limit are most easily discussed in terms of type IIA variables and the holomorphic prepotential f Ž z .. In the large volume limit f Ž z . is cubic Žcf. Ž34.. while sub-leading Ž a X . corrections are known to be w30–32,37–39x f Ž z. s
i 3!
a
k a b c z a z b z c q c xz Ž 3 . q Ý n d aLi 3 Ž ey2 p d a z . . da
Ž 43 .
R. Bohm ¨ et al.r Nuclear Physics B 569 (2000) 229–246
240
Here c is a normalization factor, x s 2Ž hŽ1,1. y hŽ1,2. . is the Euler number, d a is an hŽ1,1.-dimensional summation index, n d a are hŽ1,1. model dependent integer coefficients and xj
`
Li 3 Ž x . s
Ý js1
.
j3
Ž 44 .
In order to obtain the type IIB effective action Žin terms of type IIA variables. one inserts Ž43. into Ž29. using Ž30. – Ž33.. The same action can be expressed in terms of type IIB variables by using the transformations Ž39. and Ž41.. This last step is necessary in order to discuss the fate of the SLŽ2,Z. symmetry away from the large volume limit. It is this aspect we now turn to. 4.4. Symmetry properties Let us first identify the PQ symmetries of type IIB in the type IIA Lagrangian Ž29.. It is invariant under 2 = hŽ1,2. q 3 continuous PQ symmetries Žwith parameters g i ,g˜ i , a . which act on the fields as follows:
dz i s g i ,
dz˜i s g˜ i ,
df˜ s a q g˜ i z i y g i z˜ i .
Ž 45 . The 2 = Ž hŽ1,2. q 1. symmetries of the first two terms are due to the fact that z i and z˜i arise in the R–R sector while the last symmetry is the PQ symmetry which any scalar dual to an antisymmetric tensor has. In the large complex structure limit the Lagrangian Ž38. is invariant under hŽ1,2. additional continuous PQ symmetries which act on x a but also on z i and z˜ i w40–43x 7
d x a s gˆ a ,
d y a s 0,
dz˜ 0 s ygˆa z˜ a ,
dz 0 s 0,
dz a s gˆ az 0 ,
dz˜ a s 14 k a b c gˆ bz c .
Ž 46 .
The total number of Ž3 = hŽ1,2. q 3. PQ symmetries in the type IIA theory correspond to the Ž3 = hŽ1,2. q 2. PQ symmetries of type IIB displayed in Ž23., Ž24. together with one of the generators of the SLŽ2,Z.. More precisely, the SLŽ2,Z. of Eqs. Ž6. can be generated by
l
¨ l q 1,
l
¨ y l1 ,
Ž 47 .
where the first transformation is nothing but the ‘missing’ PQ symmetry.8 World-sheet instantons break Ž46. to a discrete subgroup. It is a generic feature of perturbative string theory that continuous PQ symmetries of scalars arising in the NS–NS sector are broken to a discrete subgroup by strong coupling effects on the world-sheet.9 On the other hand PQ symmetries of fields in the R–R sector are protected and survive the perturbative expansion of string theory. This discussion also applies for 7
Note that this symmetry is only given in its infinitesimal form. Since at the tree level the symmetry group is really SLŽ2,R. this PQ symmetry is also continuous. 9 An exception to this rule are the scalars which are dual to antisymmetric tensors of the NS–NS sector. For example the PQ symmetry of f˜ in Eq. Ž45. is preserved in string perturbation theory. 8
R. Bohm ¨ et al.r Nuclear Physics B 569 (2000) 229–246
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the SLŽ2,Z.; the first transformation of Ž47. shifts l of the R–R sector and is preserved in perturbation theory. The second transformation of Ž47. transforms the dilaton and in Subsection 3.3 we observed that the entire Ž47. is a symmetry in the large volume limit. In order to discuss the fate of the symmetry away from this limit we need the transformation properties of the IIA variables in slightly more detail. Using Ž39., Ž41., Ž42. and Ž25. – Ž27. one derives for the second generator of Ž47. the following transformation law of the type IIA variables: ey2 f 4
¨ (z
'2 ey2 f
4
2 y1 y2 f 4 e 0 q 2V
ya
¨ '12 (z
xa
¨ y '12 Ž x z y z . ,
z0
¨y Ž z
,
2 y1 y2 f 4 e 0 q 2V
a
y a,
a
0
2 z0 2 y1 y2 f 4 e 0 q 2V
.
,
¨ y'2 x q '2 Ž zz qŽ x2Vz yez . a
za
0 2 0
a
a 0 y1 y2 f 4
.
,
z˜ a
¨ z˜ y
z˜ 0
¨ y '42 ž f˜ y z z˜ / q '482 Ž Ž k xzz . y z Ž k xxz . q z
a
1 4
z 0 k a b c Ž x bz 0 y z b . Ž x cz 0 y z c . 8 Ž z 02 q 2Vy1 ey2 f 4 .
i
i
0
'2
z 0 k a b c Ž x az 0 y z a . Ž x bz 0 y z b . Ž x cz 0 y z c .
48
Ž z 02 q 2Vy1 ey2 f .
y
f˜
Ž k x z . a q 18 z 0 Ž k xx . a y
4
¨'2 Ž 2 z˜ y x z˜ y Ž k xxz .q 0
a
a
q
1 6
1 24
2 0
,
Ž k xxx . .
,
z 0 Ž k xxx . .
z 0 Ž f˜ y z iz˜i q2 Ž x az 0 y z a . z˜a q 13 Ž k x zz . y 16 z 0 Ž k xx z . y 121 z 02 Ž k xxx . .
'2 Ž z 02 q2Vy1 ey2 f . 4
.
Ž 48 .
Note that the Calabi–Yau moduli and the four-dimensional dilaton mix in a complicated way and thus the complex z a s x a q iy a s b 1 a q iÕ a transform not into themselves but into a linear combination involving the z i. Eqs. Ž48. show that the relatively simple transformations Ž25. – Ž27. become rather involved in terms of type IIA variables and as a consequence the SLŽ2,Z. is hidden in the large complex structure limit of type IIA. Or in other words, the type IIA variables are not an appropriate basis to display the SLŽ2,Z. symmetry.
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From Ž48. one also learns that world-sheet instantons break the second generator of SLŽ2,Z.. Inserting Ž48. into Ž43. shows that the prepotential transforms into a function involving arbitrarily high powers of the dilaton ey2 f 4 which cannot be cancelled by any other term at the string tree level. Thus strong coupling Žnon-perturbative. effects on the string world-sheet do not respect the full SLŽ2,Z.. However, it is possible that once non-perturbative effects in the four-dimensional space-time are also taken into account w44–51x the SLŽ2,Z. is restored. Work along these lines is in progress.
5. Conclusion In this paper we compactified ten-dimensional IIB supergravity on a Calabi–Yau threefold Y3 with an arbitrary number of harmonic Ž1,1.-forms. In the four-dimensional Lagrangian this leads to hŽ1,1. tensor multiplets coupled to supergravity and a doubletensor multiplet. The SLŽ2,Z. symmetry of ten-dimensional type IIB theory is also a symmetry of the d s 4 action in the large volume limit where the ten-dimensional description is a good approximation. The SLŽ2,Z. acts naturally on the d s 10 field variables and therefore mixes the four-dimensional dilaton and the Calabi–Yau moduli in a non-trivial way. By mirror symmetry the type IIB theory is equivalent to IIA supergravity compactified on the mirror manifold Y˜3 . We verified this equivalence in the large complex structure limit of type IIA and displayed explicitly the relation between the two sets of field variables. Via this map we were able to obtain the action of the SLŽ2,Z. on the type IIA field variables. Finally, we noticed that for small Calabi–Yau manifolds the SLŽ2,Z. is broken by strong coupling effects on the world-sheet Žworld-sheet instantons..
Acknowledgements This work is supported in part by the French–German binational program PROCOPE. C.H. thanks J.L. and his group for the hospitality in Halle and H.G. thanks R. Grimm and his group for the hospitality in Marseille. This work is additionally supported by GIF – the German–Israeli Foundation for Scientific Research. J.L. thanks S. Yankielowicz and J. Sonnenschein for the hospitality in Tel Aviv and Y. Nir for the hospitality at the Weizmann Institute. We thank I. Antoniadis, C. Bachas, B. de Wit, M. Haack, T. Mohaupt and V. Kaplunovsky for useful conversations.
Appendix A Throughout this paper we use the following notations and conventions: the signature of the ten-dimensional metric is chosen as Žyqq . . . q., capital Latin indices are ten-dimensional indices, Greek indices from the middle of the alphabet are four-dimensional space-time indices and the real coordinates of the Calabi–Yau threefold are denoted by y aˆ ; aˆ s 1, . . . ,6. We also use a set of complex coordinates j a , j a Ž a , a s 1,2,3., being defined as j 1 s Ž y 1 q iy 2 .r '2 , j 2 s Ž y 3 q iy 4 .r '2 , j 3 s Ž y 5 q
R. Bohm ¨ et al.r Nuclear Physics B 569 (2000) 229–246
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iy 6 .r '2 together with their complex conjugates. The conventions used for Christoffel symbols and the various curvature tensors are GNMP s 12 g M Q Ž E P g N Q q E N g P Q y EQ g N P . , M RNMP Q s EQ GNMP y E P GNMQ q GNSP GSQ y GNSQ GSMP ,
RM N s RMP P N , R s g M N RM N . Ž A.1 . The purpose of this appendix is to supply more details on the derivation of the Lagrangian given in Ž15.. It is constructed by compactifying the d s 10 effective theory of Eq. Ž1. on a Calabi–Yau threefold. First of all the terms which do not involve the 4-form DM N P Q can be treated with standard methods. That is, the ten-dimensional metric g M N is decomposed into a four-dimensional space-time part gmn and a six-dimensional internal Calabi–Yau metric g ab ˆ ˆ . This in turn forces a decomposition of the ten-dimensional Ricci scalar R10 aˆ bˆ gˆ R10 s R4 q g mn Rmaˆan Raˆmm bˆ q Ragb ˆ qg ˆˆ ˆ .
ž
Ž A.2 .
/
The Calabi–Yau metric g ab ˆ ˆ is expanded around a background metric deformations d g ab ˆˆ
g a0ˆ bˆ
with small
0 g ab ˆ ˆ s g aˆ bˆ q d g aˆ bˆ .
Ž A.3 .
In d s 4 massless scalar fields arise as the inequivalent harmonic deformations of the Calabi–Yau metric w27,52x. More precisely one expands i d g ab s Õˆ a Ž x . vaab , a s 1, . . . ,hŽ1 ,1. ,
d g ab s z A Ž x . xaAb , A s 1, . . . ,hŽ1 ,2. , Ž A.4 . a A where vab are harmonic Ž1,1.-forms while xa b are related to harmonic Ž1,2.-forms and Õˆ a and z A are the scalar moduli. The z A turn out to be members of vector multiplets and therefore will be omitted in the following discussion. Inserting ŽA.3. and ŽA.4. into ŽA.2. and keeping up to quadratic terms in v a results in w20x10 b b R10 s R4 y 12 Em Õˆ aE m Õˆ b vaaa vbb g 0 a b g 0 ba q Em Õˆ aE m Õˆ b vaaa vbb g 0 a a g 0 bb .
Ž A.5 . BMI N
The antisymmetric tensors BMI N s
I Bmn
0
0
I Bab ˆˆ
0
decompose as
.
Ž A.6 .
Since there are no harmonic Ž2,0. forms on Y3 the BaˆI bˆ are expanded only in terms of harmonic Ž1,1.-forms according to Eqs. Ž9.. Finally, the ten-dimensional integration measure splits according to 10
Hd x(y g
10
s d 4 x y g4
H (
6
Hd j i(g
6
,
Ž A.7 .
and the integration over the Calabi–Yau threefold can be performed using Ž12. and Ž13.. 10
There is a slight inconsistency in the derivation of Eq. ŽA.5. in Ref. w20x in that not all terms contributing at quadratic order are properly taken into account. However, this merely affects the coefficients of Eq. ŽA.5.. We thank M. Haack for communicating the correct formula prior to publication. A more detailed derivation of Eq. ŽA.5. will be given in Ref. w53x.
R. Bohm ¨ et al.r Nuclear Physics B 569 (2000) 229–246
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The two remaining terms in Eq. Ž1. contain the 4-form DM N P Q . As we discussed in Subsection 3.1 the massless modes in d s 4 arising from dimensional reduction of DM N P Q are vectors Dm aˆ bg ˆ ˆ and tensors Dmn aˆ bˆ . Not all of them are independent but related by the self-duality condition Ž3.. The vectors are of no concern here and so in the reduction procedure we only focus on the tensor fields. In order to proceed we make an Ansatz for the couplings of the tensor fields where the individual terms are dictated by the reduction of the last terms in Ž1. 2
ˆ J L s y g 10 k 1 Ž Fmnraˆ bˆ . q k 2 e I J BaˆI bˆ Hmnr F mnraˆ b
(
ž
ˆ
ˆ ˆ ˆ
J ˆ ˆ e q k 3 e I J e K L BaˆI bˆ Hmnr B K aˆ b H L mnr q e mnrse aˆ bgdef IJ
/
I J I J K L = k 4 Fmnraˆ bˆ Bgd ˆ ˆ Es Bef ˆ ˆ q k 5 e K L Hmnr Baˆ bˆ Bgd ˆ ˆ Es Bef ˆˆ ,
ž
/
Ž A.8 .
where Fmnraˆ bˆ ' 301 Em D˜nraˆ bˆ q Er D˜mn aˆ bˆ q En D˜rm aˆ bˆ ,
ž
/
I D˜mn aˆ bˆ ' Dmn aˆ bˆ q 34 e I J Bmn BaˆJbˆ ,
Ž A.9 .
and k 1 , . . . ,k 5 are constants to be determined. This is done by comparing the equations of motion of ŽA.8. with the dimensionally reduced ten-dimensional field equations. Since we only need to fix a few constants we simplify the task by doing this comparison in the limit l s 0, f s const. and gmn s hmn . For our purpose the important field equations turn out to be
E P GM N P s y 103 iFM N P QS G P QS .
Ž A.10 .
The general definition of GM N P can be found in w21,22x while here we only record its components in this specific limit 1 2 Gmnr s e f r2 Ž eyf Hmnr q iHmnr .,
Gmaˆ bˆ s 13 e f r2 ey fEm Ba1ˆ bˆ q i Em Baˆ2bˆ ,
ž
/
Gamn ˆ ˆ s 0. ˆ s Gm an ˆ s Gmn aˆ s Gaˆ bg
Ž A.11 .
As a consequence ŽA.10. simplifies to
E r Grmn s y10iFmnraˆ bˆ G raˆ bˆ , 10 mnr mgd ˆˆ . E m Gabm q 3Fm aˆ bgd ˆˆ ˆ G ˆ ˆ s y 3 i Fmnraˆ bˆ G
ž
Ž A.12 .
/
I The first equation is merely an equation for Bmn and hence not of immediate interest for mgd nrsef ˆ ˆs 1 ˆ ˆ G mgd ˆ ˆ the us. Using the self-duality condition Fmaˆ bgd emnrs eaˆ bgdef ˆˆ ˆ G ˆ ˆ ˆˆ ˆ F 12'y g 1 second equation in ŽA.12. yields for Bab ˆˆ J E mEm Ba1ˆ bˆ s 10 e f Fmnraˆ bˆ q 203 e I J BaˆI bˆ Hmnr H 2 mnr
ž
5 q
'
6 yg
/
ˆ
ˆ
ˆ
nrsef ˆ q 3 e B Ief ˆ H Jnrs E m B 2 gd ˆ . e femnrs eaˆ bgdef ˆ ˆ ˆˆ ˆ Ž F . 20 I J
Ž A.13 .
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1 This equation has to be compared with the equation of motion for Bab ˆ ˆ obtained from ŽA.8. J E mEm Ba1ˆ bˆ s y4 e f k 2 Fmnraˆ bˆ q 2 k 3 e I J BaˆI bˆ Hmnr H 2 mnr
ž
4 y
'y g
/
ˆ
ˆ
ˆ
mnrgd ˆ y k e B Igd ˆ H J mnr E s B 2 ef ˆ . e femnrs eaˆ bgdef ˆ ˆ ˆˆ ˆ Ž2 k 4 F . 5 IJ
Ž A.14 . Comparing ŽA.13. with ŽA.14. yields k 1 s y 253 ,
k 2 s y 52 ,
k 3 s y 163 ,
k 4 s 485 ,
k 5 s y 321 ,
Ž A.15 .
where the overall normalization is determined by using the gauge invariance of Eqs. Ž21. 2 and Ž22.. ŽThe analogous equations for Bab ˆ ˆ are consistent with the same set of coefficients.. Finally one adds ŽA.8. to the dimensionally reduced action obtained from the first three terms in Ž1. using ŽA.5., ŽA.6. and ŽA.7., performs a Weyl rescaling gmn Vy1 gmn and integrates over the Calabi–Yau threefold using the formulae given in Subsection 3.1. This results in the Lagrangian Ž15..
™
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Compactification of type IIB string theory on Calabi– Yau threefolds a Robert Bohm , Carl Herrmann b, Jan Louis a ¨ a, Holger Gunther ¨ b
a Martin – Luther – UniÕersitat ¨ Halle – Wittenberg, FB Physik, D-06099 Halle, Germany Centre de Physique Theorique, CNRS – Luminy Case 907, F-13288 Marseille Cedex 9, France ´
Received 16 August 1999; accepted 15 December 1999
Abstract We study compactifications of type IIB supergravity on Calabi–Yau threefolds. The resulting low energy effective Lagrangian is displayed in the large volume limit and its symmetry properties – with specific emphasis on the SLŽ2,Z. – are discussed. The explicit map to type IIA string theory compactified on a mirror Calabi–Yau is derived. We argue that strong coupling effects on the world-sheet break the SLŽ2,Z.. q 2000 Elsevier Science B.V. All rights reserved.
1. Introduction Compactifications of string theories on six-dimensional Calabi–Yau manifolds result in string vacua with four flat Minkowskian dimensions Ž d s 4. and, depending on the type of Calabi–Yau manifold, a fixed number of preserved supercharges. Specifically, string vacua with N s 2 supersymmetry in d s 4 are obtained by compactifying type II strings on Calabi–Yau threefolds Y3 , the heterotic string on K 3 = T 2 or the type I string on K 3 = T 2 . It is believed that all of the resulting string vacua are perturbative descriptions of different regions in one and the same moduli space w1,2x. In this paper we confine our attention to the region of the moduli space where type II string compactifications are the appropriate perturbative theories.
E-mail addresses: [email protected] ŽR. Bohm ¨ ., [email protected] ŽH. Gunther ., [email protected] ŽC. Herrmann., [email protected] ŽJ. Louis.. ¨ 0550-3213r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 5 5 0 - 3 2 1 3 Ž 9 9 . 0 0 7 9 6 - 8
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In d s 10 type II string theories come in two different versions, the non-chiral type IIA and the chiral type IIB string theory but upon compactification on Calabi–Yau threefolds they are related by mirror symmetry. More precisely, type IIB compactified on Y3 is equivalent to IIA compactified on the mirror manifold Y˜3 w3,4x. Due to this equivalence one has the choice to study either compactification but since there is no simple covariant action of type IIB supergravity in d s 10 w5–7x type IIA compactifications appear to be easier w8x. In terms of the underlying conformal field theory the two theories in d s 4 only differ by the GSO projection and thus can be discussed on the same footing w3,4x. However, it is believed that the type IIB theory in d s 10 has an exact non-perturbative SLŽ2,Z. symmetry w9–11x which is not shared by type IIA Žin d s 10.. Thus, it is of interest to study the ‘fate’ of this SLŽ2,Z. symmetry upon compactification to d s 4 w12x and this is one of the goals of this paper. For that reason we focus on geometrical compactifications Žand not the conformal field theory versions. of the effective type IIB theory for most parts of this paper. In addition, the general type IIB effective Lagrangian and the explicit map to the type IIA vacua has only been given in a special case w13,14x and as a byproduct we close this gap. This paper is organized as follows. In Section 2 we recall some of the properties of type IIB supergravity in d s 10. In Section 3 we discuss the compactification on Calabi–Yau threefolds and display the four-dimensional low energy effective action in the large volume limit. Apart from the gravitational multiplet one finds hŽ1,1. tensor multiplets, hŽ1,2. vector multiplets and one ‘‘double-tensor’’ multiplet. We concentrate on the couplings of the tensor multiplets since the couplings of the vector multiplets have been given before w4,15–20x. In addition the SLŽ2,Z. – inherited from the d s 10 theory – acts on the tensor multiplets while the vector multiplets are left invariant. In Section 4 the explicit map between type IIB compactified on Y3 and type IIA compactified on the mirror manifold Y˜3 is displayed. This map enables us to also discuss the geometry of the tensor multiplets in terms of a holomorphic prepotential and obtain the world-sheet instanton corrections. We find that the SLŽ2,Z. is manifest at the string tree level in the large volume limit of the compactification but broken once strong coupling effects on the world-sheet Žworld-sheet instantons. are taken into account. Furthermore, the SLŽ2,Z. is also present in the large complex structure limit of type IIA vacua but rather obscure in the standard field variables. Section 5 contains our conclusions and some of the technical aspects of this paper are collected in an appendix.
2. Type IIB Supergravity in d s 10 The bosonic massless modes of the type IIB string are the graviton g M N Ž M, N s 0, . . . ,9., a doublet of antisymmetric tensors BMI N Ž I s 1,2., two real scalar fields f ,l and a 4-form DM N P Q with a self-dual field strength. g M N , BM1 N and f arise in the NS–NS sector while BM2 N ,l and DM N P Q reside in the R–R sector; f is the dilaton of type IIB string theory. The difficulty of constructing a covariant low energy effective action for these massless modes is due to the presence of the self-dual 5-form field strength. The field equations of type IIB supergravity on the other hand are well known w5–7x. However, if
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one does not impose the self-duality condition a covariant action Žin the Einstein frame. can be given w21,22x1 S s y 12 d 10 x y g
'
H
1 q
'
96 y g
ž
R y 14 Tr Ž E ME My1 . q 34 H I MI J H J q 56 F 2
/
eI J D n H I n H J ,
Ž 1.
where R is the scalar curvature and we abbreviated MI J '
1
< l< 2 Im l yRe l
ž
yRe l , 1
/
l ' l q ieyf ,
HMI N P ' E w M BNI P x s 13 Ž E M BNI P q E N BPI M q E P BMI N . , FM N P Q R ' E w M DN P Q R x q 34 e I J BwIM N E P BQJ R x .
Ž 2.
In Eq. Ž1. we suppress space-time indices; in particular, the topological term D n H I n H J is contracted with the ten-dimensional e-tensor while all other terms are contracted with the ten-dimensional metric. Finally, we choose e 12 s q1. The type IIB supergravity is obtained as the Euler–Lagrange equations of the action Ž1. together with the self-duality condition FM N P Q R s
1
e M N P Q R ST U V W F ST U V W .
'
5! y g
Ž 3.
This theory has a number of symmetries. First of all there are the gauge transformations of the 4-form Žwith parameters SN P Q .
d DM N P Q s E w M SN P Q x ,
Ž 4.
under which all other fields are invariant. Secondly, there are the gauge transformations of the two antisymmetric tensors Žwith parameters V NI . which also transform the 4-form
d BMI N s E w M V NI x , d DM N P Q s y 34 e I J V wIM HNJ P Q x .
Ž 5.
Finally, there is an SLŽ2,Z. acting as follows: 2
¨ l s acll qq db , ¨H sL H H X
l
I M NP
XI MNP
I J
J MNP ,
Ž 6.
1 Recently a covariant action including the self-dual 4-form has been constructed w23–25x but here we choose to follow the procedure outlined in Refs. w21,22x for compactifying type IIB supergravity and use Ž1. instead. 2 In fact at the classical level one has the larger group SLŽ2,R. which is broken by quantum corrections to Ž SL 2,Z..
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where
Ls
ž
d b
c , a
/
ad y bc s 1,
a,b,c,d g Z.
Ž 7.
Using Ž2. one shows that the matrix M transforms according to M
¨M sL X
y1T
M Ly1 .
Ž 8.
3. Calabi–Yau compactifications of type IIB supergravity 3.1. The spectrum Let us now study the compactification of the type IIB low energy effective theory on Calabi–Yau threefolds Y3 . The resulting theory in d s 4 has N s 2 supersymmetry and the low energy spectrum comes in appropriate N s 2 supermultiplets. More specifically one finds that the ten-dimensional metric g M N decomposes into the four-dimensional metric gmn Ž m , n s 0, . . . ,3., 2 = hŽ1,2. deformations of the complex structure d g ab , d g a b Ž a , b s 1,2,3. and hŽ1,1. deformations of the Kahler class d g a b . The Hodge ¨ numbers hŽ1,1.Ž hŽ1,2. . count the harmonic Ž1,1.-forms ŽŽ1,2.-forms. on Y3 . The antisymI metric tensors BMI N decompose into a doublet of antisymmetric tensors Bmn and I 2 = hŽ1,1. scalar modes Ba b . Finally the 4-form decomposes into hŽ1,1. Žreal. antisymmetric tensors Dmn a b and hŽ1,2. q 1 real vectors Dm a bg , Dm a bg .3 Together with the two scalars f and l these fields assemble into the following N s 2 supermultiplets: gravitational multiplet
Ž gmn , Dm a bg . ,
double-tensor multiplet
Ž BmnI , f ,l . , Ž Dm a bg , d ga b , d ga b . , Ž Dmn a b , d ga b , BaI b . ,
hŽ1,2. vector multiplets hŽ1,1. tensor multiplets
where we only display the bosonic components. The gravitational multiplet contains two gravitini while all the other multiplets contain two Weyl fermions.4 In d s 4 an antisymmetric tensor describes one physical degree of freedom and thus can always be dualized to a scalar. At the level of supermultiplets this duality relates both the tensor and the double-tensor multiplet to a hypermultiplet which contains four real scalar degrees of freedom. In this dual basis the low energy spectrum features hŽ1,1. q 1 hypermultiplets and hŽ1,2. vector multiplets apart from the gravitational multiplet. The couplings of the vector multiplets have been obtained before w4,15–20x and thus we exclusively focus on the couplings of the hŽ1,1. tensor and the universal double-tensor multiplet. ŽOr, in other words, we consider Calabi–Yau threefolds with hŽ1,2. s 0.. On a 3
Note that the dimensionally reduced self-duality condition Ž3. relates Daba b to Dmngg , Dm a bg to Dm a bg and Dma bg to Dm a bg . 4 The double-tensor multiplet has not been constructed as an off-shell multiplet but we expect that it exists since it appears in the low energy limit of type IIB string theory.
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Calabi–Yau threefold the Kahler deformations of the metric can be expanded in terms of ¨ a harmonic Ž1,1.-forms vab according to BaI b s bˆ Ia Ž x . vaab ,
i d g ab s Õˆ a Ž x . vaab , a Dmn a b s Dˆmn Ž x . vaab ,
a s 1, . . . ,hŽ1 ,1. .
Ž 9.
In terms of these variables the tensor multiplets consist of ˆ ˆ ˆ Before we display the four-dimensional Lagrangian let us collect a few more properties of Calabi–Yau threefolds w26,27x. The Kahler form J is defined as ¨ a Ž Dmn ,Õ a,b Ia ..
J s i d g ab d j a n d j b ,
Ž 10 .
a
where the j are the complex coordinates on Y3 . The volume V can be expressed in terms of J according to 1 6
6
H'g d j s HJ n J n J.
Vsi
Ž 11 .
On the space of Ž1,1.-forms one defines a metric i b Ga b s y vaab vgd g a d g gb g d 6j . V The intersection numbers of Y3 are given by
'
H
Ž 12 .
kabc s v a n v b n v c.
Ž 13 .
H
Both, the metric and the volume, can be expressed in terms of the moduli Õˆ a and the intersection numbers V s 16 k a b c Õˆ a Õˆ b Õˆ c , Ga b s yVy1k a b c Õˆ c q 14 Vy2k ac d Õˆ c Õˆ dk b e f Õˆ e Õˆ f .
Ž 14 .
3.2. The Lagrangian The next step is the construction of the dimensionally reduced type IIB low energy effective theory. We follow the procedure outlined in Refs. w21,22x and reduce the ten-dimensional action Ž1. imposing the self-duality condition on the field equations by hand. The resulting field equations in d s 4 can be shown to be the Euler–Lagrange equations of a four-dimensional action. The technical details of the reduction procedure are deferred to the appendix and we only give the final result
'y g y1 L s y
1 2
2
R y 14 e 2 f 4 V Ž Õ . Ž Em l . y Ž Em f4 .
2
I y 61 ey3 f 4 V 1r2 Ž Õ . Hmnr MI J H Jmnr y Ga b Ž Õ . Em Õ aE m Õ b
y e f 4 V 1r2 Ž Õ . Ga b Ž Õ . Em b Ia MI J E m b J b y 32 ey2 f 4 V Ž Õ . Ga b Ž Õ . a J = Ž Fmnr q e I J Hmnr b Ia . Ž F bmnr q e I J H Jmnr b I b .
i y
'
12 y g
a e mnrsk a b c e I J Ž Fmnr b I b Es b Jc
I y 23 e K L Hmnr b J a b K b Es b L c . ,
Ž 15 .
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where we redefined the field variables compared to the expansion parameters of Eqs. Ž9. according to a a I Dmn ' Dˆmn q 12 e I J Bmn b Ja,
b Ia ' 23 bˆ Ia .
a Fmnr ' E w m Dnra x ,
Ž 16 .
a
Furthermore, the kinetic terms for the Ž1,1.-moduli Õˆ and the dilaton decoupled due to the definition of the four-dimensional dilaton w20x ey2 f 4 s V Ž Õ . ey2 f ,
Ž 17 .
and ‘rotated’ moduli fields Õ a s Õˆ a e f r2 .
Ž 18 .
Note that Ž14. and Ž15. are only valid in the limit where the ten-dimensional theory is a good approximation, i.e. when V is large. This limit is termed the ‘large volume limit’ but generically subleading Ž a X . corrections are also important. These are briefly discussed in Subsection 4.3. As we already discussed, an antisymmetric tensor in four space-time dimensions is a I dual to a scalar and so we can dualize the hŽ1,1. q 2 antisymmetric tensor fields Dmn , Bmn in the action Ž15.. This is done most conveniently by adding hŽ1,1. q 2 Lagrange multipliers g a ,h I to the Lagrangian I a L X s L q i e mnrs Hmnr Es h I q i e mnrs Fmnr Es g a .
Ž 19 .
a I The Žalgebraic. equations of motion of L X for Fmnr and Hmnr can then be used to eliminate these fields in favour of the scalars g a and h I . The resulting Lagrangian reads
'y g y1 L X s y
1 2
2
2
R y 14 e 2 f 4 V Ž Em l . y Ž Em f4 . y Ga b Em b 1 a E m b 1 b
y Ga b Em Õ a E m Õ b y e 2 f 4 VGa b Ž Em b 2 a y l Em b 1 a . Ž E m b 2 b y l E m b 1 b . e2f4 y 16V
Ž Gy1 .
ab
Ž Em g a y
1 2
k ac d e I J b Ic Em b J d .
= Ž E m g b y 12 k b e f e K L b K e E m b L f . y 14 e 2 f 4 Vy1 Em h 2 y b 1 a Ž Em g a y 16 k a b c e I J b I b Em b Jc .
ž
2
/
y 14 e 4f 4 Ž Em h1 q l Em h 2 y Ž lb 1 a y b 2 a . 2
= Ž Em g a y 16 k a b c e I J b I b Em b Jc . ,
/
Ž 20 .
where in addition the f4-dependence is displayed more explicitly. 3.3. Symmetry properties The symmetries of the ten-dimensional theory discussed in Section 2 are also manifest in d s 4. The action Ž15. is invariant under hŽ1,1. q 2 gauge transformations Žwith parameters Sna, VnI . of the antisymmetric tensors a d Dmn s E w m Snax ,
I d Bmn s E w m VnIx .
Ž 21 .
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In addition we have 2 = hŽ1,1. continuous Peccei–Quinn ŽPQ. symmetries Žwith paramea ters c Ia . acting on the scalar fields b Ia and the antisymmetric tensors Dmn
d b Ia s c Ia ,
a J d Dmn s ye I J c Ia Bmn .
Ž 22 .
These symmetries are ‘inherited’ from the ten-dimensional symmetries Ž4., Ž5.. Note that in the dual basis the gauge transformations Ž21. manifest themselves as hŽ1,1. q 2 additional continuous PQ symmetries Žwith parameters c˜a ,cˆI .
d g a s c˜a ,
d h I s cˆI ,
Ž 23 .
while Ž22. transmogrifies into
d b Ia s c Ia ,
d g a s 12 k a b c e I J c I b b Jc ,
d h I s ye I J Ž c J a g a q 16 k a b c e K L b J a c K b b L c . .
Ž 24 .
One of the distinct features of the type IIB theory in d s 10 is its SLŽ2,Z. invariance Ž6. – Ž8.. This symmetry is of importance since it includes a strong-weak coupling duality as one of its generators. Thus it is of interest to study the fate of this symmetry in the d s 4 action. Since Eq. Ž15. was obtained as a straight dimensional reduction of the action Ž1. the SLŽ2,Z. will also be manifest in Ž15. but due to the field redefinitions Ž17., Ž18. it also acts on the Calabi–Yau moduli Õ a and altogether becomes more involved. More precisely, using Ž6. – Ž8., Ž17. and Ž18., one finds Õ
a
¨Õ
a<
cl q d <,
y2 f 4
e
¨ < cl q d < , ey2 f 4
l
¨
ac < l < 2 q Ž bd q ad . l q bd < cl q d < 2
,
Ž 25 . I where l s l q i Vy1 r2 eyf 4 . The antisymmetric tensors Bmn and the scalars b Ia inherit the transformation law Ž6. and obey I Bmn
¨L B I J
J mn
,
b Ia
¨L b I J
Ja
,
Ž 26 .
a while the Dmn remain invariant. From Ž19. one infers that the dual scalars g a are invariant whereas the h I transform with the inverse matrix
hI
¨L
y1 J hJ . I
Ž 27 .
It can be explicitly checked that the Lagrangian Ž15. is invariant under the transformations specified in Ž25. and Ž26. and thus at the string tree level and in the large volume limit the SLŽ2,Z. is present in d s 4 w12x.
4. The Map between IIA and IIB 4.1. Preliminaries Type IIA string theory compactified on Calabi–Yau threefolds features hŽ1,1. vector multiplets, hŽ1,2. hypermultiplets and one tensor multiplet in its low energy spectrum. The tensor multiplet can be dualized to an additional hypermultiplet. A type IIB compactification on Y3 is equivalent to a type IIA compactification on the mirror
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manifold Y˜3 once the appropriate Žworld-sheet. quantum corrections are taken into account. This equivalence is a property of the underlying conformal field theory w28,29x but has also been demonstrated for the geometrical compactifications on Calabi–Yau manifolds w30–32x. On the mirror manifold the Hodge numbers are reversed and one has hŽ1,1. Ž Y3 . s hŽ1 ,2. Ž Y˜3 . ,
hŽ1 ,2. Ž Y3 . s hŽ1 ,1. Ž Y˜3 . .
Ž 28 .
The low energy spectrum of the two theories is most easily compared in the dual basis where the hŽ1,1. q 1 hypermultiplets of IIB are identified with the hŽ1,2. q 1 hypermultiplets of type IIA. The purpose of this section is to explicitly display the map between the field variables of type IIB and type IIA. For hŽ1,1.Ž Y3 . s 1 this map was given in Ref. w13x. Let us start by recalling the low energy effective action of type IIA. The universal hypermultiplet of IIA which is the dual of the tensor multiplet contains the dilaton fA , the dual f˜ of an antisymmetric tensor and two scalars z 0 , z˜ 0 from the R–R sector. The scalars of the remaining hŽ1,2. hypermultiplets are denoted by z a, z a, z a, z˜ a where a s 1, . . . ,hŽ1,2. .5 The z a, z a arise from the NS–NS sector while the z a, z˜ a come from the R–R sector. The tree level Lagrangian for these hypermultiplets is known to be w20,33,34x
'y g y1 L s y
1 2
2
R y Ž Em fA . y Ga b Ž z , z . E m z a Em z b 2
y 14 e 4fA Em f˜ q z Ei m z˜i y z˜ Ei m z i q 12 e 2 fA R i j Ž z , z . E mz Ei m z j
ž
/
q 12 e 2 fA Ry1 i j Ž z , z . Ž Ii k Ž z , z . E mz k q E mz˜i . Ijl Ž z , z . Em z l q Em z˜j ,
ž
/
Ž 29 . where i, j,k,l s 0, . . . ,hŽ1,2. . The scalar manifold spanned by the 4 = Ž hŽ1,2. q 1. scalars is constrained by N s 2 supergravity to be a quaternionic manifold w35x and this has been explicitly verified for Ž29. in Refs. w33,34x. However, this scalar manifold is not the most general quaternionic manifold but at the string tree level further constrained by the c-map w4x. That is, the submanifold spanned by the deformations of the complex structure z a has to be a special Kahler manifold w15–19x.6 More precisely, the metric ¨ Ga b Ž z, z . is Kahler and furthermore determined by a holomorphic prepotential F Ž X . ¨ which is a homogenous function of degree two Ž F Ž l X . s l2 F Ž X ... Specifically, one has
E Ga b s
E a
Ez Ezb
K,
Ž 30 .
5 Here we use a slightly inconsistent notation since we reserved the index a to label Ž1,1.-forms. However, we are anticipating the mirror map between type IIA and type IIB which identifies the Ž1,1.-forms on Y3 with the Ž1,2.-forms on the mirror manifold Y˜3 and for that reason we use the same indices already at this stage. 6 The reason is that in type IIB vacua the Ž1,2.-forms reside in vector multiplets and the geometry of their scalars is constrained to be special Kahler. ¨
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where K s ylog X i Fi Ž X . q X i Fi Ž X . , Fi Ž X . '
EF
zis
,
EXi
Xi
Ž z 0 s 1. .
X0
Ž 31 .
Furthermore, the matrices R and I in Ž29. are also determined by the prepotential according to R i j s Re Ni j , Ni j s 14 Fi j y
Ii j s Im Ni j ,
Ž Nz . i Ž zN . j , Ž zNz .
Ž 32 .
where Fi j s
E 2F E X Ei X j
Ni j s 14 Fi j q Fi j ,
,
ž
Ž Nz . i s Ni j z j ,
/
Ž zNz . s z i Ni j z j .
Ž 33 .
4.2. The map The geometrical compactification of the effective low energy IIB supergravity from d s 10 to d s 4 is only valid in the large volume limit of Y3 . In this limit the Calabi–Yau volume V is a cubic function of the Ž1,1. moduli Õ a and the intersection numbers k a b c are constant. In order to display the map between IIA and IIB one has to take a similar limit on the IIA side. This limit – termed the ‘large complex structure limit’ – has been studied in the literature w30–32,36x. For us the important point is that such a limit exists and in this limit the prepotential of the complex structure moduli is given by Fs
i 3!
kabc
X aX b X c X
0
2
' Ž X 0. f Ž z. ,
f Ž z. s
i 3!
kabc z az b z c.
Ž 34 .
Using Ž30., Ž31., Ž34. and z a s x a q iy a one obtains the Kahler potential and the metric ¨ K s yln Ž k yyy . , Ga b Ž y . s y
Ž k y . a b 3 Ž k yy . a Ž k yy . b y , 2 2 Ž k yyy . 2 Ž k yyy . 3
ž
/
Ž 35 .
where we abbreviated
Ž k yyy . s k a b c y a y b y c , Ž k yy . a s k a b c y b y c , Ž k y . ab s k a b c y c .
Ž 36 .
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The matrices defined in Ž32. simplify considerably and are given by Is
1 y 13 Ž k xxx . 4
ž
1 2
1 2
Ž k xx . b , yŽ k x . ab
Ž k xx . a
R s y 241 Ž k yyy .
ž
/
1 y 4Ga b x a x b 4Ga b x a
Ry1 s y24 Ž k yyy .
y1
ž
4Ga b x b , y4Ga b
/
xb
1 xa
1 4
y1 a b
y G
qxaxb
/
.
Ž 37 .
Inserting Ž37. into Ž29. yields
'y g y1 L s y
1 2
2
R y Ž Em fA . y Ga b Em y a E m y b y Gab Em x a E m x b 2
y 18 e 2 fA V Ž Em z 0 . y 12 e 2 fA VGa b Ž x a Em z 0 y Em z a . Ž x bE mz 0 y E mz
b
.
y e 2 fA Vy1 Gy1 a b Em z˜a q Ž k xx . a Em z 0 1 2
ž
1 8
y 41 Ž k x . ac Em z c . Ž E mz˜b q 81 Ž k xx . b E mz 0 y 41 Ž k x . b d E mz d . y 2 e 2 fA Vy1 Em z˜ 0 q x aEm z˜a q 241 Ž k xxx . Em z 0 y 18 Ž k xx . a Em z a
ž
2
/
2
y 14 e 4fA Em f˜ q z Ei m z˜i y z˜ Ei m z i ,
ž
Ž 38 .
/
where in anticipation of the map to type IIB we use V s 16 Ž k yyy . Žalthough this does not correspond to the volume of type IIA.. The map of the field variables is obtained by comparing the two Lagrangians Ž20. and Ž38.. First of all one learns that the dilatons of the two theories have to be identified Ž fA s f4 . which is in accord with the fact that mirror symmetry is already valid in string perturbation theory. Furthermore, one is led to the straightforward identification y a sÕ a,
x a s b1 a .
Ž 39 .
The remaining terms can be systematically compared due to their specific dependence on y a and fA . One obtains
Em z 0 s "'2 Em l, b 1 a Em z 0 y Em z a s "'2 Ž l Em b 1 a y Em b 2 a . ,
Em z˜ a q 18 Ž k b 1 b 1 . a Em z 0 y 14 Ž k b 1 . a b Em z b s"
'2 4
1
žE
m ga y 2 kabc
Ž b1 b Em b 2 c y b 2 b Em b1 c . / ,
Em z˜ 0 q b 1 a Em z˜a q 241 Ž k b 1 b 1 b1 . Em z 0 y 18 Ž k b1 b1 . a Em z a s"
'2 4
žE h yb m
2
1a
Em g a q 16 k a b c b 1 a Ž b 1 b Em b 2 c y b 2 b Em b 1 c . ,
/
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Em f˜ y z˜ Ei m z i q z Ei m z˜i s " Em h1 q l Em h 2 q b 2 aEm g a y lb1 aEm g a y 16 k a b c b 2 a Ž b 1 bEm b 2 c y b 2 bEm b 1 c .
ž
q 16 l k a b c b 1 a Ž b 1 bEm b 2 c y b 2 bEm b 1 c . .
Ž 40 .
/
This system of equations is solved by
z 0 s '2 l, z a s '2 Ž lb1 a y b 2 a . ,
'2
z˜a s y z˜0 s
4
'2
ga q
h2 y
4
'2 8
'2 24
l Ž k b1 b1 . a y
l Ž k b1 b1 b1 . q
'2 8
'2 24
Ž k b1 b 2 . a ,
Ž k b1 b1 b 2 . ,
f˜ s h1 q 12 lh 2 q 12 b 2 a g a y 12 lb1 a g a y 121 Ž k b 1 b 2 b 2 . q 121 l Ž k b 1 b 1 b 2 . ,
Ž 41 .
which gives the desired relation between the type IIA and type IIB field variables. It is somewhat surprising that this relation is so involved. Remarkably Ž41. can be inverted ls
1
z0 ,
'2
b2 a s
1
a a '2 Ž x z 0 y z . ,
ga s y h2 s
4
'2
4
'2
z˜a q
z˜0 q
1 2'2
Ž k xz . a ,
1
Ž k xx z . , 6'2
h1 s f˜ y z zi ˜i q 16 Ž k x zz . y 121 z 0 Ž k xx z . .
Ž 42 .
4.3. World-sheet instanton corrections After having established the map between the type IIA and type IIB variables in the large volumerlarge complex structure limit we can discuss the corrections away from this limit. Mirror symmetry assures the equivalence of IIA and IIB compactified on mirror manifolds and it becomes a matter of convenience in what variables one describes the action. The a X corrections to the large volume limit are most easily discussed in terms of type IIA variables and the holomorphic prepotential f Ž z .. In the large volume limit f Ž z . is cubic Žcf. Ž34.. while sub-leading Ž a X . corrections are known to be w30–32,37–39x f Ž z. s
i 3!
a
k a b c z a z b z c q c xz Ž 3 . q Ý n d aLi 3 Ž ey2 p d a z . . da
Ž 43 .
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Here c is a normalization factor, x s 2Ž hŽ1,1. y hŽ1,2. . is the Euler number, d a is an hŽ1,1.-dimensional summation index, n d a are hŽ1,1. model dependent integer coefficients and xj
`
Li 3 Ž x . s
Ý js1
.
j3
Ž 44 .
In order to obtain the type IIB effective action Žin terms of type IIA variables. one inserts Ž43. into Ž29. using Ž30. – Ž33.. The same action can be expressed in terms of type IIB variables by using the transformations Ž39. and Ž41.. This last step is necessary in order to discuss the fate of the SLŽ2,Z. symmetry away from the large volume limit. It is this aspect we now turn to. 4.4. Symmetry properties Let us first identify the PQ symmetries of type IIB in the type IIA Lagrangian Ž29.. It is invariant under 2 = hŽ1,2. q 3 continuous PQ symmetries Žwith parameters g i ,g˜ i , a . which act on the fields as follows:
dz i s g i ,
dz˜i s g˜ i ,
df˜ s a q g˜ i z i y g i z˜ i .
Ž 45 . The 2 = Ž hŽ1,2. q 1. symmetries of the first two terms are due to the fact that z i and z˜i arise in the R–R sector while the last symmetry is the PQ symmetry which any scalar dual to an antisymmetric tensor has. In the large complex structure limit the Lagrangian Ž38. is invariant under hŽ1,2. additional continuous PQ symmetries which act on x a but also on z i and z˜ i w40–43x 7
d x a s gˆ a ,
d y a s 0,
dz˜ 0 s ygˆa z˜ a ,
dz 0 s 0,
dz a s gˆ az 0 ,
dz˜ a s 14 k a b c gˆ bz c .
Ž 46 .
The total number of Ž3 = hŽ1,2. q 3. PQ symmetries in the type IIA theory correspond to the Ž3 = hŽ1,2. q 2. PQ symmetries of type IIB displayed in Ž23., Ž24. together with one of the generators of the SLŽ2,Z.. More precisely, the SLŽ2,Z. of Eqs. Ž6. can be generated by
l
¨ l q 1,
l
¨ y l1 ,
Ž 47 .
where the first transformation is nothing but the ‘missing’ PQ symmetry.8 World-sheet instantons break Ž46. to a discrete subgroup. It is a generic feature of perturbative string theory that continuous PQ symmetries of scalars arising in the NS–NS sector are broken to a discrete subgroup by strong coupling effects on the world-sheet.9 On the other hand PQ symmetries of fields in the R–R sector are protected and survive the perturbative expansion of string theory. This discussion also applies for 7
Note that this symmetry is only given in its infinitesimal form. Since at the tree level the symmetry group is really SLŽ2,R. this PQ symmetry is also continuous. 9 An exception to this rule are the scalars which are dual to antisymmetric tensors of the NS–NS sector. For example the PQ symmetry of f˜ in Eq. Ž45. is preserved in string perturbation theory. 8
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the SLŽ2,Z.; the first transformation of Ž47. shifts l of the R–R sector and is preserved in perturbation theory. The second transformation of Ž47. transforms the dilaton and in Subsection 3.3 we observed that the entire Ž47. is a symmetry in the large volume limit. In order to discuss the fate of the symmetry away from this limit we need the transformation properties of the IIA variables in slightly more detail. Using Ž39., Ž41., Ž42. and Ž25. – Ž27. one derives for the second generator of Ž47. the following transformation law of the type IIA variables: ey2 f 4
¨ (z
'2 ey2 f
4
2 y1 y2 f 4 e 0 q 2V
ya
¨ '12 (z
xa
¨ y '12 Ž x z y z . ,
z0
¨y Ž z
,
2 y1 y2 f 4 e 0 q 2V
a
y a,
a
0
2 z0 2 y1 y2 f 4 e 0 q 2V
.
,
¨ y'2 x q '2 Ž zz qŽ x2Vz yez . a
za
0 2 0
a
a 0 y1 y2 f 4
.
,
z˜ a
¨ z˜ y
z˜ 0
¨ y '42 ž f˜ y z z˜ / q '482 Ž Ž k xzz . y z Ž k xxz . q z
a
1 4
z 0 k a b c Ž x bz 0 y z b . Ž x cz 0 y z c . 8 Ž z 02 q 2Vy1 ey2 f 4 .
i
i
0
'2
z 0 k a b c Ž x az 0 y z a . Ž x bz 0 y z b . Ž x cz 0 y z c .
48
Ž z 02 q 2Vy1 ey2 f .
y
f˜
Ž k x z . a q 18 z 0 Ž k xx . a y
4
¨'2 Ž 2 z˜ y x z˜ y Ž k xxz .q 0
a
a
q
1 6
1 24
2 0
,
Ž k xxx . .
,
z 0 Ž k xxx . .
z 0 Ž f˜ y z iz˜i q2 Ž x az 0 y z a . z˜a q 13 Ž k x zz . y 16 z 0 Ž k xx z . y 121 z 02 Ž k xxx . .
'2 Ž z 02 q2Vy1 ey2 f . 4
.
Ž 48 .
Note that the Calabi–Yau moduli and the four-dimensional dilaton mix in a complicated way and thus the complex z a s x a q iy a s b 1 a q iÕ a transform not into themselves but into a linear combination involving the z i. Eqs. Ž48. show that the relatively simple transformations Ž25. – Ž27. become rather involved in terms of type IIA variables and as a consequence the SLŽ2,Z. is hidden in the large complex structure limit of type IIA. Or in other words, the type IIA variables are not an appropriate basis to display the SLŽ2,Z. symmetry.
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From Ž48. one also learns that world-sheet instantons break the second generator of SLŽ2,Z.. Inserting Ž48. into Ž43. shows that the prepotential transforms into a function involving arbitrarily high powers of the dilaton ey2 f 4 which cannot be cancelled by any other term at the string tree level. Thus strong coupling Žnon-perturbative. effects on the string world-sheet do not respect the full SLŽ2,Z.. However, it is possible that once non-perturbative effects in the four-dimensional space-time are also taken into account w44–51x the SLŽ2,Z. is restored. Work along these lines is in progress.
5. Conclusion In this paper we compactified ten-dimensional IIB supergravity on a Calabi–Yau threefold Y3 with an arbitrary number of harmonic Ž1,1.-forms. In the four-dimensional Lagrangian this leads to hŽ1,1. tensor multiplets coupled to supergravity and a doubletensor multiplet. The SLŽ2,Z. symmetry of ten-dimensional type IIB theory is also a symmetry of the d s 4 action in the large volume limit where the ten-dimensional description is a good approximation. The SLŽ2,Z. acts naturally on the d s 10 field variables and therefore mixes the four-dimensional dilaton and the Calabi–Yau moduli in a non-trivial way. By mirror symmetry the type IIB theory is equivalent to IIA supergravity compactified on the mirror manifold Y˜3 . We verified this equivalence in the large complex structure limit of type IIA and displayed explicitly the relation between the two sets of field variables. Via this map we were able to obtain the action of the SLŽ2,Z. on the type IIA field variables. Finally, we noticed that for small Calabi–Yau manifolds the SLŽ2,Z. is broken by strong coupling effects on the world-sheet Žworld-sheet instantons..
Acknowledgements This work is supported in part by the French–German binational program PROCOPE. C.H. thanks J.L. and his group for the hospitality in Halle and H.G. thanks R. Grimm and his group for the hospitality in Marseille. This work is additionally supported by GIF – the German–Israeli Foundation for Scientific Research. J.L. thanks S. Yankielowicz and J. Sonnenschein for the hospitality in Tel Aviv and Y. Nir for the hospitality at the Weizmann Institute. We thank I. Antoniadis, C. Bachas, B. de Wit, M. Haack, T. Mohaupt and V. Kaplunovsky for useful conversations.
Appendix A Throughout this paper we use the following notations and conventions: the signature of the ten-dimensional metric is chosen as Žyqq . . . q., capital Latin indices are ten-dimensional indices, Greek indices from the middle of the alphabet are four-dimensional space-time indices and the real coordinates of the Calabi–Yau threefold are denoted by y aˆ ; aˆ s 1, . . . ,6. We also use a set of complex coordinates j a , j a Ž a , a s 1,2,3., being defined as j 1 s Ž y 1 q iy 2 .r '2 , j 2 s Ž y 3 q iy 4 .r '2 , j 3 s Ž y 5 q
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iy 6 .r '2 together with their complex conjugates. The conventions used for Christoffel symbols and the various curvature tensors are GNMP s 12 g M Q Ž E P g N Q q E N g P Q y EQ g N P . , M RNMP Q s EQ GNMP y E P GNMQ q GNSP GSQ y GNSQ GSMP ,
RM N s RMP P N , R s g M N RM N . Ž A.1 . The purpose of this appendix is to supply more details on the derivation of the Lagrangian given in Ž15.. It is constructed by compactifying the d s 10 effective theory of Eq. Ž1. on a Calabi–Yau threefold. First of all the terms which do not involve the 4-form DM N P Q can be treated with standard methods. That is, the ten-dimensional metric g M N is decomposed into a four-dimensional space-time part gmn and a six-dimensional internal Calabi–Yau metric g ab ˆ ˆ . This in turn forces a decomposition of the ten-dimensional Ricci scalar R10 aˆ bˆ gˆ R10 s R4 q g mn Rmaˆan Raˆmm bˆ q Ragb ˆ qg ˆˆ ˆ .
ž
Ž A.2 .
/
The Calabi–Yau metric g ab ˆ ˆ is expanded around a background metric deformations d g ab ˆˆ
g a0ˆ bˆ
with small
0 g ab ˆ ˆ s g aˆ bˆ q d g aˆ bˆ .
Ž A.3 .
In d s 4 massless scalar fields arise as the inequivalent harmonic deformations of the Calabi–Yau metric w27,52x. More precisely one expands i d g ab s Õˆ a Ž x . vaab , a s 1, . . . ,hŽ1 ,1. ,
d g ab s z A Ž x . xaAb , A s 1, . . . ,hŽ1 ,2. , Ž A.4 . a A where vab are harmonic Ž1,1.-forms while xa b are related to harmonic Ž1,2.-forms and Õˆ a and z A are the scalar moduli. The z A turn out to be members of vector multiplets and therefore will be omitted in the following discussion. Inserting ŽA.3. and ŽA.4. into ŽA.2. and keeping up to quadratic terms in v a results in w20x10 b b R10 s R4 y 12 Em Õˆ aE m Õˆ b vaaa vbb g 0 a b g 0 ba q Em Õˆ aE m Õˆ b vaaa vbb g 0 a a g 0 bb .
Ž A.5 . BMI N
The antisymmetric tensors BMI N s
I Bmn
0
0
I Bab ˆˆ
0
decompose as
.
Ž A.6 .
Since there are no harmonic Ž2,0. forms on Y3 the BaˆI bˆ are expanded only in terms of harmonic Ž1,1.-forms according to Eqs. Ž9.. Finally, the ten-dimensional integration measure splits according to 10
Hd x(y g
10
s d 4 x y g4
H (
6
Hd j i(g
6
,
Ž A.7 .
and the integration over the Calabi–Yau threefold can be performed using Ž12. and Ž13.. 10
There is a slight inconsistency in the derivation of Eq. ŽA.5. in Ref. w20x in that not all terms contributing at quadratic order are properly taken into account. However, this merely affects the coefficients of Eq. ŽA.5.. We thank M. Haack for communicating the correct formula prior to publication. A more detailed derivation of Eq. ŽA.5. will be given in Ref. w53x.
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The two remaining terms in Eq. Ž1. contain the 4-form DM N P Q . As we discussed in Subsection 3.1 the massless modes in d s 4 arising from dimensional reduction of DM N P Q are vectors Dm aˆ bg ˆ ˆ and tensors Dmn aˆ bˆ . Not all of them are independent but related by the self-duality condition Ž3.. The vectors are of no concern here and so in the reduction procedure we only focus on the tensor fields. In order to proceed we make an Ansatz for the couplings of the tensor fields where the individual terms are dictated by the reduction of the last terms in Ž1. 2
ˆ J L s y g 10 k 1 Ž Fmnraˆ bˆ . q k 2 e I J BaˆI bˆ Hmnr F mnraˆ b
(
ž
ˆ
ˆ ˆ ˆ
J ˆ ˆ e q k 3 e I J e K L BaˆI bˆ Hmnr B K aˆ b H L mnr q e mnrse aˆ bgdef IJ
/
I J I J K L = k 4 Fmnraˆ bˆ Bgd ˆ ˆ Es Bef ˆ ˆ q k 5 e K L Hmnr Baˆ bˆ Bgd ˆ ˆ Es Bef ˆˆ ,
ž
/
Ž A.8 .
where Fmnraˆ bˆ ' 301 Em D˜nraˆ bˆ q Er D˜mn aˆ bˆ q En D˜rm aˆ bˆ ,
ž
/
I D˜mn aˆ bˆ ' Dmn aˆ bˆ q 34 e I J Bmn BaˆJbˆ ,
Ž A.9 .
and k 1 , . . . ,k 5 are constants to be determined. This is done by comparing the equations of motion of ŽA.8. with the dimensionally reduced ten-dimensional field equations. Since we only need to fix a few constants we simplify the task by doing this comparison in the limit l s 0, f s const. and gmn s hmn . For our purpose the important field equations turn out to be
E P GM N P s y 103 iFM N P QS G P QS .
Ž A.10 .
The general definition of GM N P can be found in w21,22x while here we only record its components in this specific limit 1 2 Gmnr s e f r2 Ž eyf Hmnr q iHmnr .,
Gmaˆ bˆ s 13 e f r2 ey fEm Ba1ˆ bˆ q i Em Baˆ2bˆ ,
ž
/
Gamn ˆ ˆ s 0. ˆ s Gm an ˆ s Gmn aˆ s Gaˆ bg
Ž A.11 .
As a consequence ŽA.10. simplifies to
E r Grmn s y10iFmnraˆ bˆ G raˆ bˆ , 10 mnr mgd ˆˆ . E m Gabm q 3Fm aˆ bgd ˆˆ ˆ G ˆ ˆ s y 3 i Fmnraˆ bˆ G
ž
Ž A.12 .
/
I The first equation is merely an equation for Bmn and hence not of immediate interest for mgd nrsef ˆ ˆs 1 ˆ ˆ G mgd ˆ ˆ the us. Using the self-duality condition Fmaˆ bgd emnrs eaˆ bgdef ˆˆ ˆ G ˆ ˆ ˆˆ ˆ F 12'y g 1 second equation in ŽA.12. yields for Bab ˆˆ J E mEm Ba1ˆ bˆ s 10 e f Fmnraˆ bˆ q 203 e I J BaˆI bˆ Hmnr H 2 mnr
ž
5 q
'
6 yg
/
ˆ
ˆ
ˆ
nrsef ˆ q 3 e B Ief ˆ H Jnrs E m B 2 gd ˆ . e femnrs eaˆ bgdef ˆ ˆ ˆˆ ˆ Ž F . 20 I J
Ž A.13 .
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1 This equation has to be compared with the equation of motion for Bab ˆ ˆ obtained from ŽA.8. J E mEm Ba1ˆ bˆ s y4 e f k 2 Fmnraˆ bˆ q 2 k 3 e I J BaˆI bˆ Hmnr H 2 mnr
ž
4 y
'y g
/
ˆ
ˆ
ˆ
mnrgd ˆ y k e B Igd ˆ H J mnr E s B 2 ef ˆ . e femnrs eaˆ bgdef ˆ ˆ ˆˆ ˆ Ž2 k 4 F . 5 IJ
Ž A.14 . Comparing ŽA.13. with ŽA.14. yields k 1 s y 253 ,
k 2 s y 52 ,
k 3 s y 163 ,
k 4 s 485 ,
k 5 s y 321 ,
Ž A.15 .
where the overall normalization is determined by using the gauge invariance of Eqs. Ž21. 2 and Ž22.. ŽThe analogous equations for Bab ˆ ˆ are consistent with the same set of coefficients.. Finally one adds ŽA.8. to the dimensionally reduced action obtained from the first three terms in Ž1. using ŽA.5., ŽA.6. and ŽA.7., performs a Weyl rescaling gmn Vy1 gmn and integrates over the Calabi–Yau threefold using the formulae given in Subsection 3.1. This results in the Lagrangian Ž15..
™
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