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arXiv:1007.5156v2 [hep-th] 8 Oct 2010 ∗ eateto nomto ytm cec,Usnmy Uni Utsunomiya Science, Systems of Department -al [email protected] e-mail: he iemrhs ssont nuetedltncoupling. the induce to with shown associated is diffeomorphism number sheet The membrane. wrapped erivsiaetedltnculn ntesrn , the in coupling dilaton the reinvestigate We iao opigrevisited coupling Dilaton tuoia3188,Japan 321-8585, Utsunomiya hz Uehara Shozo Abstract 1 ∗ hc oe rma from comes which h tigworld- string the versity, 1 Introduction

Supermembrane in eleven [1] is an important object in M-theory and the relation with superstring is well known. In fact, through the double [2] the wrapped supermembrane on R10 S1 is reduced to type IIA superstring on R10. Furthermore, it was explicitly shown that× the (p, q)-string [3, 4] in type IIB theory is obtained from the wrapped supermembrane on the T 2-compactified target space through the double dimensional reduction [5, 6, 7]. This also indicates that the duality in type IIB theory is naturally understood in M-theory [3, 8]. Recently, the coupling of the string Euler character χ to the dilaton φ has been studied in M-theory [9]. It was presented that the χφ-term in type IIA theory arises from the measure of the membrane partition function. This may also indicate that the membrane explains the properties of . In this paper we reinvestigate the membrane origin of the dilaton term by the Fujikawa method [10, 11] and we will see that the path integral measure to the coupling of the dilaton φ to the Ricci scalar R(2) of the string worldsheet metric γ

d2σ√ γ R(2)φ . (1.1) − ZΣ This paper is organized as follows. In the next section, we deduce the string action from the wrapped supermembrane by the double dimensional reduction. In section 3, we seek for the dilation coupling with the partition function of the reduced string action. The final section is devoted to discussion.

2 Double dimensional reduction

The action of a supermembrane coupled to an eleven-dimensional back- ground is given by [1]

3 1 ˆiˆj A B 1 1 ˆiˆjkˆ Mˆ Nˆ Pˆ S = T d σ γˆ γˆ Πˆ Πˆ η + γˆ ǫ ∂ˆZ ∂ˆZ ∂ˆZ Cˆ ˆ ˆ ˆ , (2.1) −2 − ˆi ˆj AB 2 − − 3! i j k P NM   Z p p where ˆ Aˆ Mˆ ˆ Aˆ Πˆi = ∂ˆiZ EMˆ , (2.2) 1 ˆ ˆ ˆ T is the of the supermembrane, CMˆ NˆPˆ(Z) is the super three-form,γ ˆˆiˆj (i, j = 0, 1, 2) is the worldvolume metric,γ ˆ = detγ ˆˆiˆj, and the target space is a supermanifold with the coordinates ZMˆ = (XM , θα) (M = 0, , 10, α = 1, , 32). Fur- ··· ··· thermore, with the tangent superspace index Aˆ = (A, a), Eˆ Aˆ is the supervielbein and Mˆ ηAB is the tangent space metric in eleven dimensions. The of the world- ˆi volume parameter σ and eleven-dimensional background fields (GMN , CˆMNP ) is 0, while that of the worldvolume metricγ ˆˆˆ is 2. ij − We shall investigate the origin of the R(2)φ-term in the dimensionally reduced string theory. In string theory, the anomaly of the reparametrization ghost number current gives √ γ R(2), which is (a local version of) the Riemann-Roch theorem, and hence, similarly to− Ref.[9], we may focus on the bosonic degrees of freedom to investigate the R(2)φ-term from the supermembrane theory.

1 −2 −3 The eleven-dimensional l11 is defined by T = (2π) l11 .

2 The bosonic background fields are included in the superfields as

ˆ A A ˆ ˆ EM (Z) =e ˆM (X) , CMNP (Z) = AMNP (X) . (2.3) =0 fermions=0

Then, the action (2.1) is reduced to

3 1 ˆiˆj M N 1 S = T d σ γˆ γˆ ∂ˆX ∂ˆX G (X)+ γˆ −2 − i j MN 2 −  Z p p 1 ˆiˆjkˆ M N P ǫ ∂ˆX ∂ˆX ∂ˆX Aˆ (X) . (2.4) − 3! i j k PNM 

Note that the variation w.r.t.γ ˆˆiˆj yields the induced metric,

M N γˆˆˆ = ∂ˆX ∂ˆX G (X) Gˆˆ , (2.5) ij i j MN ≡ ij and plugging it back into the original action leads to the one in the Nambu-Goto form

(NG) 3 1 ˆiˆjkˆ M N P S = T d σ det Gˆˆ ǫ ∂ˆX ∂ˆX ∂ˆX Aˆ (X) . (2.6) − − ij − 3! i j k PNM Z  q  In Ref.[9], they examined the membrane partition function

X γˆ S Z = D D e− , (2.7) Vol(Diff0) topologiesX Z under the assumption that the membrane was wrapped once around the S1-compactified 11th direction of the target space and they truncated to the zero-mode sector of the circle. That is, the worldvolume was assumed to be S1 Σ with Σ being some Riemann surface and the target space topology was M 10 S1 with× S1 being the 11th direction of M-theory or the M-theory circle, and the double× dimensional reduction [2] was applied. They fixed some variables

2 11 σ = X , γˆ22 =1 , γˆ02 =γ ˆ12 =0 , G1010 =1 , (2.8) and analyzed the path integral measure of the partition function (2.7). They found that the norm of the variation of the worldvolume metric leads to the relation

δγˆ = R δγ , (2.9) || ij|| 11 || ij||

where R11 is the radius of the M-theory circle.p Then, (2.9) is followed by the relation between the space (or the conformal Killing vectors) measures of the string and the dimensionally reduced wrapped membrane, which leads to the χφ-term to the string action. In this paper, we reinvestigate the partition function (2.7). Similarly to Ref.[9], since we are interested in the coupling between the dilaton and the worldsheet in the reduced string theory, we consider the S1-compactified wrapped supermembrane, for 10 1 simplicity. We take the X -directions to be compactified on S of radius L1 and hence 1 the worldvolume of the membrane is at least locally Σws S , where Σws is a Riemann surface and S1 is to be parametrized by σ2. Then, we shall× represent the wrapping of the supermembrane as

˚ 10 ˆi ˚ 10 ˆi G1010 X (σ +2πδˆi2)=2πw1L1 + G1010 X (σ ) , (2.10) q q 3 where G˚1010 stands for the asymptotic values of the corresponding component of the target space metric and w1 N is a wrapping number. For simplicity, we put w1 = 1 hereafter. From eq.(A.1) we∈ shall see

L1 2φ0/3 R1 = L1 e− , (2.11) ≡ G˚1010

where φ0 is the asymptotic constantp value of the type IIA dilaton background and hence the M/IIA-relation or 11d/IIA-SUGRA-relation leads to

R1 = l11 . (2.12)

We parametrize the worldvolume metricγ ˆˆiˆj as

1/2 2 h− γij + hViVj hVi γˆˆiˆj = l11 , (2.13) hVj h     2 where γij, h and Vi are dimensionless, and we have

6 det(ˆγˆiˆj) = l11 det(γij) , (2.16) 1 1/2 ij 1/2 ik − ˆiˆj 2 h γ h γ Vk γˆˆiˆj = γˆ = l11− 1/2 jk 1− 1/2 kl . (2.17) h γ Vk h− + h Vkγ Vl      −  Then, the action (2.4) is rewritten as

T 3 1/2 ij 2 3 1 S = d σ l √ γ h γ (∂ X ∂ X l hˆVˆ Vˆ ) l √ γ (h− hˆ 1) 2 − 11 − i · j − 11 i j − 11 − − Z  l3 √ γ h1/2hγˆ ij(V Vˆ )(V Vˆ )+ ǫij ∂ XM ∂ XN ∂ XP Aˆ , (2.18) − 11 − i − i j − j i j 2 MNP  where

M N 2 ∂iX ∂2X ∂ˆX ∂ˆX = ∂ˆX ∂ˆX G , hˆ = l− ∂ X ∂ X, Vˆ = · . (2.19) i · j i j MN 11 2 · 2 i ∂ X ∂ X 2 · 2 We shall make a gauge choice for the worldvolume diffeomorphism.3 We adopt the 2 ˆ ˆ Note that for an arbitrary worldvolume vector Aˆi and a Aˆiˆj , we have

ˆi ˆikˆ i −2 1/2 ik Aˆ =γ ˆ Aˆˆ Aˆ = l11 h γ (Aˆ V Aˆ2) , k → k − k ˆiˆj ˆikˆ ˆjlˆ ij −4 ik jl Aˆ =γ ˆ γˆ Aˆˆˆ Aˆ = l11 hγ γ (Aˆ V Aˆ2 Aˆ 2V + V V Aˆ22) . (2.14) kl → kl − k l − k l k l ˆ Meanwhile, the first two components ǫi of a parameter ǫi for the worldvolume diffeomorphism can be a parameter of the worldsheet diffeomorphism because, for example, (µ =0, 1, , 9) ··· M ˆi M µ i µ δX = ǫ ∂ˆX δX = ǫ ∂iX . (2.15) i DDR→

3Under the diffeomorphism, the components in (2.13) are transformed as

ˆi 2 k k k δγ = ǫ ∂ˆγ + γ ∂2ǫ + (γ V γ V γ V )∂2ǫ + γ ∂ ǫ + γ ∂ ǫ , (2.20) ij i ij ij ij k − jk i − ki j ik j jk i ˆi j −3/2 j 2 2 δV = ǫ ∂ˆV + V ∂ ǫ + (h γ V V ) ∂2ǫ + ∂ ǫ V ∂2ǫ , (2.21) i i i j i ij − i j i − i ˆi 2 i δh = ǫ ∂ˆih +2h ∂2ǫ +2hVi ∂2ǫ . (2.22)

4 following gauge conditions4

10 2 X = R1 σ , γ01 = γ00 + γ11 =0 . (2.23) Under the gauge condition (2.23) the action (2.4) or (2.18) becomes

1/2 ij T 3 h η µ ν G1010 µ ν ρ SGfd = d σ l11∂iX ∂jX gµν 2 ∂iX ∂jX ∂2X gµρAν 2 − G1/2 − hˆ Z  1010 n G g g + 1010 ∂ Xµ∂ Xν∂ Xρ∂ Xσ A A g 2A A g µρ νσ ˆ i j 2 2 µ ν ρσ − ν σ µρ − 3/2 l11h G1010 ij µ ν ij µ  ν ρ o + l11ǫ ∂iX ∂jX Bµν + ǫ ∂iX ∂jX ∂2X Aµνρ

3 1 3 1/2 ij l ρh− (hˆ h) l h hˆ η (V Vˆ )(V Vˆ ) , (2.24) − 11 − − 11 i − i j − j  where γij has been fixed to be the fiducial metricγ ¯ij by the gauge condition (2.23) γ¯ ρη , (2.25) ij ≡ ij and use has been made of the (10+1)-decomposition of the background fields in (A.1) and (A.2). The FP-determinant ∆FP, or the Fadeev-Popov ghost action SFP, associated with the gauge condition (2.23) is given by (see appendix B and also [15])

+ ∆FP(ˆγ,X) = ˆb++ ˆb cˆ cˆ− exp iSFP , (2.26) D D −− D D Z   i 3 1/2 + SFP = d σ h ˆb (∂+ V+∂2)ˆc− + ˆb++(∂ V ∂2)ˆc , (2.27) −2π2 −− − − − − Z n o where ˆb andc ˆ± are Grassmann odd ghosts with ghost numbers 1 and +1, respectively, ±± − and 1 1 ∂ = (∂0 ∂1), V = (V0 V1) . (2.28) ± 2 ± ± 2 ± The total action is given by ST = SGfd + SFP. (2.29) Now that we make the double dimensional reduction by imposing the conditions to deduce the type IIA string (µ =0, 1, , 9) ··· µ µ ∂ ∂ ∂ X (= ∂ 2 X )=0 , G = Aˆ =0 . (2.30) 2 σ ∂X10 MN ∂X10 MNP By the double dimensional reduction the total action is reduced to

Ts 2 h ij µ ν ij µ ν ST Sst = d σ η ∂iX ∂jX gµν + ǫ ∂iX ∂jX Bµν DDR→ 2 − h¯ Z  r ¯ 1 h 1 1/2 ij ρ 1 h h¯ η (Vi V¯i)(Vj V¯j) − 2πTs h − − 2πTs − −   2i 1/2 + h (b ∂+c− + b++∂ c ) , (2.31) − πT −− − s  4 10 10 2 We may adopt ∂2X = R1 instead of X = R1σ to see the conformal transformation explicitly in the dimensionally reduced string theory;[14] however, it does not make an essential difference in our analysis.

5 where

2 1 Ts =2πR1T = (2πl11)− , (2.32) µ Gµ10∂iX hˆ G1010 h¯ , Vˆi V¯i , (2.33) DDR→ ≡ DDR→ R1G1010 ≡

2 and the variables in Sst are understood to be independent of σ . Note that Vi can be integrated out straightforwardly, while the field equation for ρ is

h = h¯ . (2.34)

As was pointed out in [14], substituting this algebraic condition back into Sst yields a ρ-independent action. However, ρ is not free but is related, for example, to Xµ through the equation of motion for h. The Euclidean action corresponding to Sst is

h 1 h¯ SE = T d2z ∂ Xµ∂ Xνg + ∂ Xµ∂ XνB + ρ 1 st s h¯ z z¯ µν z z¯ µν 8πT h − Z r s 1 1   + h1/2h¯ (V V¯ )(V V¯ )+ h1/2(b ∂ cz + b ∂ cz¯) . (2.35) 2πT z − z z¯ − z¯ 2πT zz z¯ z¯z¯ z s s  3 Dilaton coupling

In this section we examine the partition function with the double dimensional reduced membrane action. First, we shall make the BRS invariant path integral measure according to the Fujikawa method [10]. In this case, the integration variables for the path integral are obtained as follows: For a worldvolume scalar Xµ we have

M 2 3 M N X˚ exp Tl− d σ γXˆ X G D 11 − MN Z  Z  p µ 2 M 3 µ ν gµν 10 G10µX = X˚ exp Tl11 d σ √ γ X X + G1010 X + . D − √G G Z  Z  1010 1010    (3.1)

This implies (we shall set det g = 1) µν − X˚µ = ( γ/h¯)1/4Xµ . (3.2) D D −   Meanwhile, we have the following decomposition for a worldvolume vector,

2 ˆi ˆj 1/2 i j 2 i 2 l11− γˆˆiˆjA A = h− γijA A + h(A + A Vi) . (3.3)

a a This implies (γij =e ˜i e˜ja, (˜ei : zweibein))

aˆ 1/4 a i 2 1/4 2 1/2 1/4 i 1/4 1/2 2 A˚ = ( γ/h) e˜ A ( γh ) A = ( γ) h− A ( γ) h A . D D − i D − D − D −        (3.4) Similarly, for a symmetric tensor we have

1/4 1/2 1/4 1/4 1 ˚Φ ˆ = ( γ)− h Φ h− Φ ( γ) h− Φ . (3.5) D aˆb D − ij D 2i D − 22       6 Let us rewrite Sst with the path integral variables X,˚ ˚b and ˚c. We have (cf. (3.2), (3.4), (3.5))

µ 1/2 1/4 µ X˚ = ρ h¯− X , z 1/4 z ˚c = ρ h− c , ˚ 1/2 1/2 bzz = ρ− h bzz . (3.6)

E Thus, Sst becomes h X˚µ X˚ν SE = d2z T ∂ ∂ g st s h¯ z ρ1/2h¯ 1/4 z¯ ρ1/2h¯ 1/4 µν Z  r  −   −  h X˚µ X˚ν 1 h¯ + T ∂ ∂ B + ρ 1 s ¯ z 1/2¯ 1/4 z¯ 1/2¯ 1/4 µν h ρ h− ρ h− 8π h − r       1 1/2 1 1/4 z 1 1/2 1 1/4 z¯ + ρ ˚b ∂ ρ− h ˚c + ρ ˚b ∂ ρ− h ˚c , (3.7) 2π zz z¯ 2π z¯z¯ z     and the partition function is

µ z z¯ SE Z X˚ ˚c ˚c ˚b ˚b e− st . (3.8) ∼ D D D D zzD z¯z¯ Z E ¯ We find that ρ and also h can be removed from the action Sst by putting h = h and making the following rescalings,

αc/2 αc/2 αc αc 1/2 (X,˚ ˚b,˚c) (e X,˚ e− ˚b, e ˚c), ( e = ρh¯− ) (3.9) → αg αg αg 2 1/4 (X,˚ ˚b,˚c) (X,˚ e− ˚b, e ˚c). ( e =(h¯ /h) ) (3.10) → However, the rescalings in (3.9) and (3.10) shall generate the jacobians from the path integral measure [10]. We can see that (3.9) is a conformal transformation and (3.10) is a ghost number transformation for the worldsheet reparametrization ghosts. Since the superstring induced from the wrapped supermembrane is in the , the jacobian coming from (3.9) should be trivial, or the should be can- celed.5 On the other hand, the jacobian from (3.10) corresponds to the reparametrization ghost number anomaly and it is well known that the anomaly equation gives the local version of the Riemann-Roch theorem, 3 3 ∂ jgh = ∂ ∂ ln ρ = γER(2), (3.11) z¯ z −4π z z¯ 8π gh p (2) where jz is the worldsheet reparametrization ghost number current and R is the world- sheet curvature. Thus, we have

αg αg ln J (e− ˚b) (e ˚c) = ˚b ˚cJ(α )= ˚b ˚c e D D D D g D D 3 = ˚b ˚c exp d2z Φ(α ) γE R(2) , (3.12) D D −8π g h Z p i where 1 h¯2 1 1 Φ(α ) = ln α = ln ln G = φ , (3.13) g g 4 h ≃ 4 1010 3   5Of course, we should recover the fermionic coordinates and the bosonic ghosts for the supertransfor- mation in to cancel out the conformal anomaly (see section 4).

7 and φ is the background dilaton (cf. (A.1)). Then, we have (˚ is omitted)

1 αg αg αg z αg z¯ z z¯ 2 E (2) (e− b ) (e− b ) (e c ) (e c )= b b c c exp d z γ φR . D zz D z¯z¯ D D D zzD z¯z¯D D −4π h Z p (3.14)i We have seen that this dilaton-curvature term has appeared due to h, which indicates that the term originated with the membrane.

4 Discussion

In this paper, we have studied the dilaton coupling in the string theory, which is in- duced from a wrapped membrane. We have seen that the dilaton coupling to the string worldsheet curvature comes out from the path integral measure due to the ghost number anomaly, which originates from the fact that the string is reduced from the membrane. There remains some points to be discussed here. The induced string action (2.31) or (2.35) has the conformal mode ρ dependence. This action is obtained from the Polyakov- type action (2.4) through the double dimensional reduction. On the other hand, once we start with the Nambu-Goto action (2.6) and adopt essentially the same gauge conditions as (2.23), 10 2 X X X X = R1σ , γ01 = γ00 + γ11 =0, (4.1) X where γij is the induced metric (cf. (2.5), (2.13))

X 1 1/2 2 1 γ = l− G G l− G− G G , (4.2) ij 11 22 ij − 11 22 i2 j2 the total action in this case becomes

3 (NG) Tl11 3 X X 2 ij µ ν 3 ij µ ν ρ S = d σ (γ γ ) l− ǫ ∂ X ∂ X B l− ǫ ∂ X ∂ X ∂ X A T 2 − 00 − 11 − 11 i j µν − 11 i j 2 µνρ Z  ˆ1/2 ˆ + ˆ 4ih b++(∂ Vˆ ∂2)ˆc + b (∂+ Vˆ+∂2)ˆc− . (4.3) − − − − −− − n o (NG) ˆ ˆ We shall see that ST corresponds to ST (2.29) with h = h and Vi = Vi. Thus, the dilaton-curvature coupling term (3.14) will be followed by a similar analysis. In this paper we have omitted the fermionic coordinates and hence the worldvolume local . However, we have used the fact that the jacobian coming from (3.9) is trivial, which is to be explicitly shown with the fermionic coordinates. Since the wrapped supermembrane is reduced to the Green-Schwarz type of superstring, but not the Neveu-Schwarz-Ramond type, such a gauge fixing as in [16] will be desired. This should be studied further. We have considered the S1-compactification of the supermembrane. In the meantime, it was shown explicitly that the wrapped supermembrane on a 2-torus induces the (p, q)- string [7, 5]. Then, the dilaton-curvature coupling term in the type IIB superstring would be directly presented with the wrapped supermembrane. We adopted the double dimensional reduction for the wrapped supermembrane and did not investigate the Kaluza-Klein modes. Meanwhile, quantum mechanical justifica- tions of the double dimensional reduction were studied in [17, 18]. In order to study the coupling between ρ and the Kaluza-Klein modes, such quantum mechanical treatment should be investigated further.

8 Acknowledgments: The author would like to thank Hiroyuki Okagawa and Satoshi Yamada for their collaboration at the early stage of this work. This work is supported in part by MEXT Grant-in-Aid for Scientific Research #20540249.

A 11d vs. 10d background fields

The 11-dimensional metric can be written as 2φ 2φ 2 gµν + e AµAν e Aµ 3 φ GMN e− ≡ 2φ 2φ e Aν e ! 1 g + 1 G G G √G1010 µν G1010 µ10 ν10 µ10 = , (A.1) Gν10 G1010 ! and the third-rank AˆMNP is decomposed as Aˆ = Aˆ , Aˆ = A , B . (A.2) { MNP } { µνρ µν10} { µνρ µν} B FP determinant (2.27)

Let us calculate the FP determinant. The gauge condition (2.23) 10 2 X = R1 σ , γ01 = γ00 + γ11 =0, (B.1) leads to (cf. (2.13))

1 10 ζˆ 2 ζˆ ∆ (ˆγ,X)− = ζˆ δ((X ) R σ ) δ((ˆγˆˆ γ¯ˆˆˆ) ) FP D − 1 ij − ij Z 2 1/2 = ζˆ δ(R ζˆ ) δ(h− δˆ(∆γ )) D 1 ζ ij Z 2 1/2 ij = ζˆ δ(R ζˆ ) λˆ exp 2πi γˆ h− λˆ δˆ(∆γ ) D 1 D − ζ ij Z Z h Z i 2 p 1/2 ij = ζˆ δ(R ζˆ ) λˆ exp 2πi √ γ¯ h λˇ δˆ(∆γ ) , (B.2) D 1 D − ζ ij Z Z h Z i where γ¯ˆˆiˆj is given by substituting the fiducial metricγ ¯ij (2.25) for γij inγ ˆˆiˆj, ∆γij is the difference between γij andγ ¯ij, ∆γ = γ γ¯ , (B.3) ij ij − ij ij ij λˇ is traceless, λˇ γ¯ = 0, and δˆ(∆γ )= δˆ(γ γ¯ ) . Thus, eq.(2.27) is followed ij ζ ij ζ ij − ij |γij =¯γij by ij 01 1 0 0 1 λˇ δˆ(∆γ ) = 2λˇ γ¯ (V ∂ ζˆ V ∂ ζˆ + ∂ ζˆ ∂ ζˆ ) ζ ij 00 0 2 − 1 2 1 − 0 +2λˇ00γ¯ (V ∂ ζˆ1 V ∂ ζˆ0 + ∂ ζˆ0 ∂ ζˆ1) 00 1 2 − 0 2 0 − 1 = 2¯γ11λˇ (V ∂ ζˆ1 V ∂ ζˆ0 + ∂ ζˆ0 ∂ ζˆ1) 01 0 2 − 1 2 1 − 0 +2¯γ00λˇ (V ∂ ζˆ1 V ∂ ζˆ0 + ∂ ζˆ0 ∂ ζˆ1) 00 1 2 − 0 2 0 − 1 4 + = λˇ++(V ∂2 ∂ )ζˆ + λˇ (V+∂2 ∂+)ζˆ− , (B.4) √ detγ ¯ − − − −− − − n o where

0 1 1 1 1 ζˆ± = ζˆ ζˆ , ∂ = (∂0 ∂1), V = (V0 V1), λˇ = (λˇ00 λˇ01). (B.5) ± ± 2 ± ± 2 ± ±± 2 ±

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