Conformal Moduli and b c Pictures for NSR Strings − 5 0 0 2 n a J Dimitri Polyakov† 8 1 Center for Advanced Mathematical Studies† 3 and Department of Physics v American University of Beirut 9 0 Beirut, Lebanon 0 1 0 4 0 Abstract / h t We explore the geometry of superconformal moduli of the NSR superstring theory in - p order to construct the consistent sigma-model for NSR strings, free of picture-changing e h complications.The sigma-model generating functional is constructed by the integration : v i over the bosonic and fermionic moduli, corresponding to insertions of the vertex operators X in scattering amplitudes.While the integration over the supermoduli leads to the standard r a picture-changing insertions, the integration over the bosonic moduli results in the appearance of picture-changing operators for the b c fermionic ghosts with the ghost number − 1. Important example of the b c ghost pictures involves the vertex operators in inte- − − grated and unintegrated forms. We obtain the BRST-invariant expressions for the b c − picture-changing operators for open and closed strings and study some of their properties. We also show that the superconformal moduli spaces of the NSR theory contain the global singularities, leading to the phenomenon of ghost-matter mixing and the appearance of non-perturbative D-brane creation operators. Keywords: PACS:04.50.+h;11.25.Mj. January 2004 † dp02@aub.edu.lb † associate member 1. Introduction Despite the fact that the NSR (Neveu-Schwarz-Ramond) superstring theory appears to be the best explored, elegant and relatively simple model describing the superstring dynamics, it still suffers a number of serious complications. One, and perhaps the most important of them is related to the problem of picture-changing and the difficulties in constructing the generating functional for Ramond and Ramond-Ramond states, because of the picture-changing ambiguity [1]. Since the Ramond vertex operators have fractional ghost-numbers, one always has to consider their combinations at different ghost pictures in order to calculate their scattering amplitudes (except for the 4-point function where all the operators can be taken at the picture 1/2). As a result, constructing the gen- − erating functional for the Ramond scattering amplitudes becomes a confusing question. Actually, the problem of the picture-changing is not limited to the Ramond sector.Due to the ghost number anomaly cancellation condition, vertex operators of any correlation function in NSR string theory must have total superconformal ghost number 2 and total − b c ghost number 3. This means that we cannot limit ourselves to just integrated and − picture 0 vertices in the generating functional of the NSR model, since both integrated and unintegrated vertex operators, at various superconformal pictures must be involved in the sigma-model action. Therefore, due to thispicture ambiguity, weface the problem building the generating functional for the consistent perturbation theory for NSR strings. In this paper we show how this problem can be resolved if one accurately takes into account the dependence of the superstring correlators on the superconformal moduli and explores the geometry of supermoduli spaces, related to the vertex operator insertions on the sphere or the disc. The paper is organized as follows. In the first section, we construct the basis for the supermoduli related to the vertex operator insertions and study the geometry of the related supermoduli spaces. In the second section we perform the integration over the supermoduli leading to the appearance of the bosonic and b c ghost picture changing. − Integrated and unintegrated vertex operators are shown to be the particular examples of different b c pictures. The general BRST-invariant expression for the b c picture- − − changing is derived. We show that if one takes all the vertex operators of the sigma-model unintegrated and at the picture 1, the supermoduli integration insures the correct b c − − and β γ ghost number balance. In the third section we study the global singularities − of the supermoduli spaces, leading to the phenomenon of the ghost-matter mixing [2], [3] and the appearance of the non-perturbative vertices carrying the RR-charges, which can be interpreted as the D-brane creation operators. We conclude by introducing the 1 general expression for the NSR sigma-model, free of picture-changing ambiguity,by using the relevant superconformal moduli structures. In the concluding section we discuss some possible implications of our results. NSR sigma-model and moduli integration Consider the string scattering amplitudes in the NSR formalism.To be certain, let us first consider the closed string case, the open strings can be treated similarly. The scattering amplitude on a sphere for N vertex operators in the NSR superstring theory is given by: M(N) P(N) ¯ ¯ < V (z ,z¯)...V (z ,z¯ ) >= dm dm¯ dθ dθ DXDψDψD[ghosts] 1 1 1 N N N Z i iZ a aZ Y Y i=1 a=1 e−SNSR+mi<ξi|Tm+Tgh>+m¯i<ξ¯i|T¯m+T¯gh>+θa<χa|Gm+Ggh>+θ¯a<χ¯a|G¯m+G¯gh> M(N) P(N) δ(< χa β >)δ(< χ¯a β¯ >) < ξi b >< ξ¯i ¯b > V (z ,z¯ )...V (z ,z¯ ) 1 1 1 N N N | | | | Y Y a=1 i=1 (1) Here z ,...,z are the points of the vertex operator insertions on the sphere and 1 N S d2z ∂X ∂¯Xm +ψ ∂¯ψm +ψ¯ ∂ψ¯m +b∂¯c+¯b∂c¯+β∂¯γ +β¯∂γ¯ NSR ∼ Z { m m m } (2) m = 0,...,9 is the NSR superstring action in the superconformal gauge. Next, (m ,θ ) are the holo- i a morphic even and odd coordinates in the moduli superspace and (ξi,χa) are their dual super Beltrami differentials (similarly for (m¯ ,θ¯ ) and (ξ¯i,χ¯a) The < ... ... > symbol i a | stands for the scalar product in the Hilbert space and the delta-functions δ(< χa β >) | and δ(< ξi b >) =< ξi b > are needed to insure that the basis in the moduli space is nor- | | mal to variations along the superconformal gauge slices (similarly for the antiholomorphic counterparts). The dimensions of the moduli and supermoduli spaces are related to the number N of the vertex operators present and are different for the NS and Ramond sectors (see the discussion below). The standard bosonization relations for the b,c,β and γ ghosts are given by [1] c(z) = eσ(z);b(z) = e−σ(z) γ(z) = eφ−χ(z);β(z) = eχ−φ∂χ(z) (3) < σ(z)σ(w) >=< χ(z)χ(w) >= < φ(z)φ(w) >= log(z w) − − 2 The explanation is needed for the formula (1). In NSR superstring theory, while the partition function on the sphere does not have of course any modular dependence, the integration over the moduli and the supermoduli does appear for the sphere scattering amplitudes, for all the N-point correlators with N 3. This is related to the important fact ≥ that in string theory the BRST and the local gauge invariances are not isomorphic to each other [4],[5]. Namely, the physical vertex operators while being BRST-invariant, are not necessarily invariant under the local supersymmetry. Moreover, while the integrated vertices are invariant under the local reparametrizations, the unintegrated ones are not. E.g. the infinitezimal conformal transformation of an unintegrated photon at zero momentum gives: dw δ V (z) = ǫ(w)T(w)(c∂Xm +γψm)(z) ǫ ph I 2iπ (4) dw 1 = ǫ(w) ∂(c∂Xm +γψm)(z)+O((z w)0) = ǫ(z)∂(c∂Xm +γψm)(z) I 2iπ {z w − } − More generally, under the infinitezimal conformal transformations z z +ǫ(z) any di- → mension 0 primary field transforms as Φ(z) Φ(z +ǫ(z)) = Φ(z)+ǫ∂Φ(z) → Similarly, applying the local supersymmetry generator dwǫ(w)G(w) to a photon at pic- 2iπ H ture 1: V = ce−φψm with G being a full matter+ghost supercurrent and ǫ now a ph − fermionic parameter, it is easy to calculate 1 δ V = ǫ(z)( ce−φ∂Xm +c∂ceχ∂χe−2φψm) = 0. ǫ ph −2 6 Note, however, that the photon vertex operator at picture 0 is supersymmetric. The problem is that, due to the b c and β γ ghost number anomalies on the sphere we cannot − − limit ourself to just integrated vertices or those at the picture 0 but must always consider a combination of pictures in the amplitudes. This combination, however, is not arbitrary but must follow from the consistent superstring perturbation theory, or the expansion in the appropriate sigma-model terms. These terms are typically of the form λV where V is a vertex operator of some physical state and λ is the space-time field corresponding to this vertex operator.This term must be added to the original action as a string emits the λ- field. Then the string partitionfunction must be expanded inλ leading to the perturbation theory series and the effective action for λ, upon computing the correlators [6]. But the 3 important question is - which vertex operator should we take? Should it be integrated or unintegrated, which superconformal ghost picture should be chosen? If we choose the integrated and picture 0 vertices as the sigma-model terms, the local superconformal symmetry is preserved, but we face the problem of the ghost number anomaly cancellation. For this reason these vertex operators are not suitable as the sigma-model terms. If, on the other hand, we choose unintegrated vertices at the picture -1, then one faces the question of the gauge non-invariance - though these operators are BRST-invariant, they are not invariant under local superconformal gauge transformations. Therefore, in order to insure the gauge invariance of the scattering amplitudes, it is necessary to impose some restrictions on the parameter ǫ(z) of the superconformal transformations; namely, it must vanish at all the insertion points of the vertex operators. With such a restriction on the gauge parameter (effectively reducing the superconformal gauge group) the scattering amplitudes will be gauge invariant. Below we will prove that the consistent superstring perturbation theory must be defined as follows: 1) All the vertex operators must be taken unintegrated and at the picture 1. − 2)The restrictions on the gauge parameter: ǫ(z ) = 0;k = 1,...N (5) k must be imposed at the insertion points of the vertex operators (where ǫ is either a bosonic infinitezimal reparametrization or a fermionic local supersymmetry transformation on the worldsheet). The restrictions (5) on the gauge parameter effectively reduce the gauge group of su- perdiffeomorphisms on the worldsheet. As a result, it is in general no longer possible to choose the superconformal gauge, fully eliminating the functional integrals over the worldsheet metric γab and the worldsheet gravitino field χa in the partition function.Instead α these functional integrals are reduced to integration over the finite number of the conformal moduli m ;i = 1,...,M(N) and the anticommuting moduli θa;a = 1,...,P(N) of the i gravitino while the gauge-fixed superstring action is modified by the supermoduli terms according to (1). The final remark about the general expression forthe correlationfunction (1) concerns the dimensionalities of the odd and even moduli spaces as functions of the number N of the vertices. These dimensionalities are equal to the numbers of independent holomorphic super Beltrami differentials χa and ξi, dual to the basic vectors θ (z) and a 4 m (z) in the supermoduli space. These numbers are different for the NS and Ramond sec- i tors. Namely, the OPE’s of the stress-tensor and the supercurrent with the unintegrated perturbtive vertex operators (e.g. a photon) are given by: 1 T(z)V(w) ∂V(w)+O(z w)0 ∼ z w − − (6) 1 G(z)V(w) W(w)+O(z w)0 ∼ z w − − where W is some normally ordered operator of conformal dimension 1/2. E.g. for a picture zero photon at zero momentum: V = c∂Xm+γψm it is easy to check that W(w) = 1∂cψm 2 Now since the induced worldsheet metric and the induced worldsheet gravitino field can be expressed as γ (z) =: ∂ Xm∂ X : (z)+... T (z) ab a b m ab ∼ χ (z) =: ψm∂ Xm : (z)+... G (z) (7) aα α a ∼ aα a,α = 1,2 it is clear that the χ (z), γ (z) and hence the corresponding superconformal moduli aα aα θ (z) and m (z) behave as (z z )−1;i = 1,...N when approaching a NS vertex point. For a i i − this reason the natural choice of the basis for m and θ, consistent with their pole structures and the holomorphic properties, is given by N θ (z) = za (z z )−1 a j − Y j=1 N (8) m (z) = zi (z z )−1 i j − Y j=1 a = 1,...,M(N);i= 1,...,P(N) The holomorphy condition requires that at the infinity the θa vectors must go to zero not slower than the supercurrent’s two-point function lim < G(0)G(z) > 1 while the z→∞ ∼ z3 mi(z) must decay as lim < T(0)T(z) > 1 or faster. To see this, note that the z→∞ ∼ z4 expansions of T and G in terms of their normal modes are L n T(z) = zn+2 X n (9) G n G(z) = zn+3/2 X n 5 therefore the normal orgering of these operators at zero point implies n 2 for T and ≤ − n 3 for G. Under conformal transformation z u = 1 mapping the zero point to ≤ −2 → z infinity, G and T transform as G(z) iu3G(u) → − (10) T(z) u4T(u)+... → therefore the normal ordering or the regularity at the infinity requires 1 1 T(u) +O( ) ∼ u4 u5 (11) 1 1 G(u) +O( ) ∼ u3 u4 implying the same asymptotic behaviour for θ and m , in the light of (7). This condition a i immediately implies that M(N) = N 3 − (12) P(N) = N 2 − for the scattering amplitudes in the NS-sector. For any N unintegrated Ramond operators V the OPE of T and V remains the same as in (6), therefore the basis and and the R number of the bosonic moduli do not change in the Ramond sector. However, the OPE of GSO-projected Ramond vertex operators with the supercurrent is given by G(z)VR(w) (z w)−12U(w)+... (13) ∼ − where U(w) is some dimension 1 operator. For this reason the supermoduli of the gravitini, approaching theinsertionpointsofV (z ), behaveasχa(z) (z z )−1. Forthescattering R i i 2 ∼ − amplitudes on the sphere involving the N NS vertices V(z ),i = 1,...N and N Ramond 1 i 1 2 operators V(w ),j = 1,...,N the basis in the moduli space (involving the combinations j 2 of quadratic and 3/2-differentials) should be chosen as N1,N2 1 θa(z) = za (14) (z z )√z w Y i j i=1,j=1 − − The holomorphy condition (11) for these differentials gives the total dimension of the supermoduli space for the NSR N-point scattering amplitudes: M(N) = N 3 − (15) N 2 P(N) = N + 2;N +N = N 1 1 2 2 − 6 This concludes the explanation of the formula (1) for the NSR scattering amplitudes. Performing the integration over m and θ in (1) using (15) we obtain i a < VNS−NS(z ,z¯ )...VNS−NS(z ,z¯ )VRR(w ,w¯ )...VRR(w ,w¯ ) > 1 1 1 N1 N1 N1 1 1 1 N2 N2 N2 = DXDψDψ¯D[ghosts]e−SNSR Z N1+12N2−2 N1+N2−3 (16) δ(< χa β >) < χa G +G > 2 < ξi b > δ(< ξi T >) 2 m gh | | | | | | | | Y Y a=1 i=1 VNS−NS(z ,z¯ )...VNS−NS(z ,z¯ )VRR(w ,w¯ )...VRR(w ,w¯ ) 1 1 1 N1 N1 N1 1 1 1 N2 N2 N2 andsimilarlyfortheopenstringcase. Eachoftheoperatorsδ(< χa β >) < χa G +G > m gh | | has the superconformal ghost number +1. These operators are the standard operators of picture-changing. Indeed, for the particular choice of χa = δ(z z ) we have a − : Γ : (z ) = δ(< χa β >) < χa G +G > a m gh | | (17) =: δ(β)(G +G )(z ) =: eφ(G +G ) : (z ) m ghost a m ghost a i.e. the standard expression for the picture-changing operator and similarly for Γ¯(z¯) a The total ghost number of the left or right picture changing insertions following from the supermoduli integrationis therefore equal toN +1N 2. Thus if one takes all the NS-NS 1 2 2− vertex operators, entering the correlator (16) or the sigma-model terms, at the canonical ( 1, 1)-picture and all the RR operators at picture ( 1, 1) , the integration over the − − −2 −2 supermoduli insures the correct ghost number of the correlation function to cancel the ghost number anomaly on the sphere. Similarly, the operator Z =:< ξi b > δ(< ξi T >) < ξ¯i ¯b > δ(< ξ¯i T¯ >) : (18) | | | | resulting from the integration over the m -moduli has left and right fermionic ghost num- i bers 1. Since we found that the total number of these operators is equal to N 3, one has − − to take all the vertex operators unintegrated (i.e. with the left and right +1 b c fermionic − ghost number) so that the fermionic ghost number anomaly, equal to 3 on the sphere, − is precisely cancelled by the moduli integration. The Z-operator (18) is a straightforward generalization of the picture-changing transformation (17) for the case of the fermionic b c pictures. In particular, the integrated and the unintegrated vertex operators in NSR − string theory are simply two different b c ghost picture representations of the vertex − 7 operator. The Z-operator must therefore reduce the b c ghost number of a vertex opera- − tor by one unit, at the same time transforming the local operators into the non-local (i.e. the dimension (1,1) operators integrated over the worldsheet). Namely, if cc¯V(z,z¯) is an unintegrated vertex of a conformal dimension zero, the Z-transformation should give : Z(cc¯V) : d2zV(z,z¯)+... (19) ∼ Z i.e. thecc¯V-operatoristransformedintothetheworldsheetintegralofV pluspossiblysome other termsinsuring the BRST-invariance. The non-locality ofthe Z-operatorfollows from the non-locality of the delta-function of the full stress-energy tensor in the definition (18). As previously one can choose the conformal coordinate patches on the Riemann surface excluding N 3 points corresponding to the basis ξi(z) = δ(z z ), so the Z-operator (18) i − − becomes Z =: b¯bδ(T)δ(T¯) : (z,z¯) (20) The expression (20) for the Z-operator is still not quite convenient for practical calcula- tions. Using the BRST invariance of Z its OPE properties we shall try to derive a suitable representation for δ(T(z)) (with the help of the arguments similar to those one uses to derive the exponential representations for δ(γ) = e−φ and δ(β) = eφ) Since the Z operator must have the conformal dimension 0, the operator δ(T) has conformal dimension 2. As − the full matter+ghost stress tensor satisfies T(z)T(w) 2(z w)−2T(w)+(z w)−1∂T(w)+ : TT : +..., (21) ∼ − − the corresponding OPE for δ(T) must be given by : (z w)2δ(T(z))δ(T(w)) : δ(T(w))+... (22) − ∼ In addition, the δ(T)-operator must satisfy [Q ,Z] =: Tδ(T) := 0 (23) brst since Z is BRST-closed. Moreover, as one can formally write : δ(T(z)) : (T(z) iǫ)−1 : ∼ − : (T(z) + iǫ)−1 : where the “inverse” of :T(z): can be represented as a dimension 2- − − operator 2 dz T−1(w) = (w z)3T(z) α I 2iπ − 8 (where α is a central charge) so that [T−1(z),T(z)] = 1 we shall be looking for the representation of Z in the form bδ(T(z)) : T−1A : (z) (24) ∼ where A issome dimension 2 operator chosen so that Z satisfies (21),(22)Withsome effort, one finds A(z) = b 4ce2χ−2φ(T +b∂c) (25) − and hence dz : bδ(T(w)) := (z w)3 bT 4ce2χ−2φT(T b∂c) I 2iπ − { − − } (26) Z(w,w¯) = d2z z w 6 (bT 4ce2χ−2φT(T b∂c))(¯bT¯ 4c¯e2χ¯−2φ¯T¯(T¯ ¯b∂¯c¯)) Z | − | { − − − − By simple calculation, using Q ,b = T and Q ,ce2χ−2φ = 1 ∂cce2χ−2χ it { brst } { brst } 4 − is easy to check that the expression (26) for the fermionic ghost picture-changing operator is BRST-invariant. The operator Z(w,w¯) particularly maps the unintegrated vertices into integrated ones. The rules of how Z(w,w¯) acts on unintegrated vertices are as follows. Let cc¯V(w,w¯) be an unintegrated vertex operator at w, where V is the dimension (1,1) operator. Writing Z(w,w¯) d2z z w 6R(z)R¯(z¯) (27) ≡ Z | − | where R(z) = bT(z) 4ce2χ−2φTT(z) 4bc∂ce2χ−2φT(z) (28) − − is defined according to (25), one has to calculate the OPE between R(z) and cV(w) around the z-point. For elementary perturbative vertices, such as a graviton, the relevant contri- butions (up to total derivatives and integrations by parts) will be of the order of (z w)−3, − cancelling the factor of (z w)3 in (25) and removing any dependence on w. The operator − W(z) from the (z w)−3 of the OPE of the conformal dimension 1 will then be the inte- − grand of the vertex in the integrated form. As for the possible more singular terms of the OPE , one can show that for the perturbative superstring vertices, such as a graviton, are generally the total derivatives; higher order terms will be either total derivatives or BRST trivial (being of the general form TL where L is BRST-closed). The Z-operator (26) is defined for the closed string vertices. Analogously, the operator of the Z-transformations in the open string case is given by dz Z (w) = (z w)3R(z) (29) open I 2iπ − C 9

Conformal Moduli and b-c Pictures for NSR Strings

Similar Conformal Moduli and b-c Pictures for NSR Strings