Dilaton coupling revisited Shozo Uehara∗ Department of Information Systems Science, Utsunomiya University, Utsunomiya 321-8585, Japan Abstract We reinvestigate the dilaton coupling in the string theory, which comes from a wrapped membrane. The ghost number anomaly associated with the string world- sheet diffeomorphism is shown to induce the dilaton coupling. arXiv:1007.5156v2 [hep-th] 8 Oct 2010 ∗e-mail: [email protected] 1 1 Introduction Supermembrane in eleven dimensions [1] is an important object in M-theory and the relation with superstring is well known. In fact, through the double dimensional reduction [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 worldsheet 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 string theory. 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 leads 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 supergravity 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 tension 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 superspace 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 mass dimension 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 Planck length 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) fermions=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 topology 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 moduli 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 curvature 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 tensor 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).