Notes on reductions of superstring theory to bosonic string theory
aa r X i v : . [ h e p - t h ] J u l IPMU 13-0067UT-13-12
Notes on reductions of superstring theoryto bosonic string theory
Kantaro Ohmori ♭ and Yuji Tachikawa ♭,♯♭ Department of Physics, Faculty of Science,University of Tokyo, Bunkyo-ku, Tokyo 133-0022, Japan, ♯ Kavli Institute for the Physics and Mathematics of the Universe,University of Tokyo, Kashiwa, Chiba 277-8583, Japan
Abstract
It is in general very subtle to integrate over the odd moduli of super Riemann surfaces inperturbative superstring computations. We study how these subtleties go away in favorablecases, including the embedding of N = 0 string to N = 1 string by Berkovits and Vafa, andthe relation of the graviphoton amplitude and the topological string amplitude by Antoniadis,Gava, Narain and Taylor and Bershadsky, Cecotti, Ooguri and Vafa. The Poincar´e dual ofthe moduli space of Riemann surfaces in the moduli space of super Riemann surfaces playsan important role. Introduction and Summary
Perturbative superstring amplitudes are given by an integral over the moduli space M ofsuper Riemann surfaces [1–4]. As clarified in a series of papers last year [5–7], generalproperties of superstring amplitudes can be formulated and be understood directly in termsof M . That said, practical computations were often done by first reducing the integral on M to an integral on the moduli space M red of Riemann surfaces , using a projection p : M → M red . (1.1)A main source of the difficulties is that p cannot in general be globally holomorphic whenthe genus is sufficiently high [8]. This fact introduces various subtle global issues when onetries to first integrate over the odd moduli. This problem was often called the ambiguityof the integrand in the literature, see e.g. [3]. The viewpoint emphasizing the supermodulispace clarifies this ambiguity, and also provides a way to deal with it in a consistent manner,as reviewed below.Yet there are favorable cases where superstring amplitudes can be reduced to bosonicstring amplitudes by showing a relation schematically of the form Z M F = Z M red F ′ . (1.2)Two examples are • the embedding of an arbitrary bosonic string to N = 1 superstring by Berkovits andVafa [9], and • the reduction of the graviphoton amplitudes in Calabi-Yau compactifications to thetopological string amplitudes by Antoniadis, Gava, Narain and Taylor [10] and byBershadsky, Cecotti, Ooguri and Vafa [11]. Note that topological strings are bosonicstrings as far as the integration over the moduli space is concerned.The aim of this short note is to show that the so-called ambiguity of the integrand doesnot affect these equivalences, because only the holomorphic inclusion ι : M red → M (1.3)is used in the analysis. Namely, we will find below that the integrand F on the superstringside automatically has the form F = F ′ ∧ PD[ M red ] (1.4)where PD[ M red ] denotes the Poincar´e dual of the bosonic moduli space in the super modulispace. This structure is a consequence of a simple argument involving charge conservationon the world sheet. In this article we use X red to denote the bosonic part of the supermanifold X , following [5].
1n the formalism using the picture changing operators (PCOs), [1–3], our observation canbe summarized as follows. In a patch of the supermoduli space, we can integrate out the oddmoduli, producing the insertion of PCOs. Changing the positions of the PCOs in generalproduces an exact form on the bosonic moduli space. This leads to various ambiguities inthe integral when we glue together the contributions from various patches to obtain the totalamplitude. The observation of [5–7] is that this ambiguity is in general unavoidable if weinsist on trying to reduce the computation on the integral over the bosonic moduli space,and that the computation free of ambiguity is possible if we consider the integral over thesupermoduli space as it is. Therefore, when computing general amplitudes, it is not a verygood idea to start from the integration over the bosonic moduli space of correlation functionswith PCOs and then to worry the possible ambiguities.What we are going to show in this note is that the two favorable cases mentioned above,and in general the condition (1.4), correspond to the cases where those exact forms arezero. In the language using PCOS, we find that the correlation functions are completelyindependent of their positions. In the case of embedding of the N = 0 string to N = 1string, this independence was already mentioned in the original paper [9]. The independenceon the positions of PCOs in the graviphoton amplitude is a new observation.Differently put, the two cases mentioned above and in general the case covered by (1.4)are very special in that the integral over the supermoduli space naturally reduces to anintegral over the bosonic moduli space. Therefore we use a more general framework of [5–7]in this note, explaining the correspondence with the more traditional framework of [1] inappropriate places. For example, using PCOs corresponds to taking the worldsheet gravitinoto be supported by delta functions. Our results applies to more general gauge choices of theworldsheet gravitino; the only condition is that the gravitino is zero where the physical vertexoperators sit.The rest of the article is organized as follows. In Sec. 2 we review the basics of thesuperstring perturbation theory as an integration over the moduli space of super Riemannsurfaces. In particular we recall where the difficulties from the integration of odd moduliarise. In Sec. 3 we explain when these difficulties do not arise, thanks to the appearance ofthe Poincar´e dual of the reduced space in the superspace. In Sec. 4 we apply this analysisto the embedding of the N = 0 string to N = 1 string by Berkovits and Vafa, and in Sec 5the same analysis is similarly applied to the reduction of the graviphoton amplitudes to thetopological string amplitude. The relation of the approach used in this note and the moretraditional formalism using the PCOs are reviewed and explained in Appendix A.We note that the contents of Sec. 3 and of Sec. 4 and much more were independentlyreached by Witten [12]. 2 A short review on integrals over supermoduli
In this section we collect the basics of supermanifolds and super Riemann surfaces to for-mulate superstring perturbation theory. Those readers who are familiar with the contentsof [5–7] can directly skip to the next section.
A supermanifold of dimension p | q , where p is the bosonic dimension and q is the fermionicdimension, is constructed by pasting subsets of superspace R p | q . Two patches representedby coordinates ( x i | θ µ ) and ( x i ′ | θ µ ′ ) are glued together with gluing functions: x i ′ = f i ( x | θ ) , θ µ ′ = ψ µ ( x | θ ) (2.1)where f is even and ψ is odd. If f and ψ are both real, then the resulting supermanifold M is called real. If the constraint is only f ( x | θ = 0) is real, then M is called cs manifold. If wereplace R p | q with C p | q and require f and ψ to be holomorphic , we get complex supermanifold.We can construct an ordinary manifold M red , called the reduced manifold, form super-manifold M by ignoring all fermionic things. Gluing functions of M red is simply f i ( x | θ = 0).Reduced manifolds of real or cs supermanifolds are real manifolds, and reduced manifolds ofcomplex supermanifolds are complex manifolds.If we can take coordinates such that f of (2.1) depend only on bosonic coordinates, themanifold called projected. All smooth supermanifolds are smoothly projected but in generalcomplex supermanifolds are not holomorphically projected. This prevents us from naivelydealing with odd moduli of superstring worldsheets.We have to integrate superfunctions over supermanifold to set up superstring pertur-bation theory. We define an integral top form as Ω = f ( x | θ ) Q i δ (d x i ) Q µ δ (d θ µ ) whered x i = δ (d x i ) for bosonic variables x i . The integration of an integral top form Ω over a patch U α is defined as: Z U α Ω | U α := Z U α, red Y i d x i f full ( x ) (2.2)where we expand Ω | U α asΩ | U α = f ( x ) + f µ ( x ) θ µ + · · · + f full ( x ) Y µ θ µ ! Y i δ (d x i ) Y µ δ (d θ µ ) . (2.3)With this definition, we can check the validity of various natural properties that integrationoperations should have, for instance the super-analog of Stokes’ theorem.For Ω to be globally defined, Ω | U α and Ω | U β should be glued together by gluing functiongluing U α and U β . Note that this consistency does not always mean the gluing consistency of3he locally defined top form f full Q i d x i on the ordinary manifold M red . A concrete examplecan be found in subsection 2.4. N = 1 super Riemann surfaces are complex supermanifolds of dimension 1 | η with gluing functions of a particular form given by z ′ = u ( z | η ) + θζ ( z | η ) p u ( z | η ) ,θ ′ = ζ ( z | η ) + θ p ∂ z u ( z | η ) + ζ ( z | η ) ∂ z ζ ( z | η ) . (2.4)This is the constraint so that the superconformal transformation generates the coordinatechange as defined in (2.4).The supermoduli space M is the complex supermanifold which parameterizes isomor-phism classes of super Riemann surfaces of certain genus. The dimension of M is 3 g − | g − g ≥ ζ ’s are called split. For asplit super Riemann surface, the transformation (2.4) means that θ transforms as a sectionof a spin bundle T Σ / of the reduced Riemann surface Σ red . So, the reduced space M red ofthe supermoduli M is isomorphic to the moduli space M spin of ordinary Riemann surfaceswith spin structure.Super Riemann surfaces with punctures represents superstring worldsheets with vertexoperators. There are two types of punctures, NS punctures and R punctures. The dimensionof the moduli space M n NS ,n R of Super Riemann surfaces with n NS NS punctures and n R punctures is 3 g − n NS + n R | g − n NS + n R / M R and that for the left mover M L . For type II theory, both are supermoduli spaces andfor heterotic theory one is a supermoduli space and the other is an ordinary moduli space.Then we define a integration cycle Γ ⊂ M L × M R such that Γ red equals the diagonal M red of M R, red × M L, red . Then superstring amplitudes are given by an integral of a top form onΓ. The choice of Γ is not canonical, but integration on Γ is. Superstring perturbation theory is given by an integral over the super moduli space of a topform F V ( J , δ J ) := Z D (matter, ghost) exp( − b I ) V (2.5)involving the action b I and the product of vertex operators V = Q i V i where each vertexoperator V i is an unintegrated vertex operator of conformal dimension 0 and picture number − − / b I isobtained as b I = I + 12 π Z Σ D (˜ z, z | ˜ θ, θ ) (cid:16) δ J B − δ e J e B (cid:17) (2.6)4here I is the original worldsheet action, δ J and δ e J are superfields representing differentialsof supercomplex structure of the worldsheet, and B and e B are superfields of the holomorphicand antiholomorphic superghosts.Let V consist of n R R vertex operators and n NS NS vertex operators in terms of rightmovers, and e n R R vertex operators and e n NS NS vertex operators in terms of lift movers.Then F V is a form on M R,n R ,n NS × M L, e n R , e n NS . The amplitude of the vertices V is then A V := 12 g Z Γ F V . (2.7)The factor g comes from GSO projection.For heterotic string theory, the result is similar except for that the supermoduli space M L is replaced by the bosonic moduli space. One way to calculate the amplitude (2.7) is to take an explicit coordinate on supermodulispace M g,n NS ,n R with dimension ∆ e | ∆ o where∆ e = 3 g − n NS + n R , ∆ o = 2 g − n NS + n R / . (2.8)To study odd coordinates, consider a split super Riemann surface Σ. Odd deformations fromΣ can be identified with χ ∈ H (Σ red , b R ) where b R = R ⊗ O − n NS X i =1 z i ! , R ≃ T Σ red ⊗ O − n R X i =1 x i ! . (2.9)We call the modes χ the gravitino backgrounds, as they come from the two-dimensionalsupergravity field which couples to superstring. We then choose a particular basis { χ σ } of H (Σ red , b R ) and expand χ as χ = ∆ o X σ =1 η σ χ σ . (2.10)The term in b I involving χ is now I η = ∆ o X σ =1 η σ π Z Σ red d zχ σ b S (2.11)where b S is the supercurrent twisted to live in Γ( b R − ⊗ K Σ red ): b S = f S (2.12)5here f is a locally defined function which behaves f ( z ) ≃ √ z − x i near an R vertex at x i and f ( z ) ≃ z − z i near an unintegrated NS vertex at z i . The choice of a branch of f corresponds to the choice of a square root of (2.9). In Type II theory, there is also a terminvoling both χ and e χ . The coupling between δ J and β is similarly given by I d η = ∆ o X σ =1 d η σ π Z Σ red d zχ σ b β (2.13)where b β is defined as in (2.12).Inserting these couplings into (2.5), we get F V ( m ; d η | η ; d m ) = * n Y i =1 V i exp − ∆ o X σ =1 η σ π Z Σ red d zχ σ b S − ∆ o X σ =1 d η σ π Z Σ red d zχ σ b β ! × exp − ∆ e X j =1 d m j π Z Σ red d zµ j b b ! × (antiholomorphic part) × ( χ e χ term) + m,g . (2.14)Here hi m,g denotes the correlation function under the metric background corresponding tocoordinates m of the genus- g bosonic moduli space M g, red and { µ i } is a basis of Beltramidifferentials of n NS + n R punctured Riemann surface. b b = gb is twisted to live in Γ( K Σ red ⊗O ( P z i ) ⊗ O ( P x i )) (i.e. g ∼ z − z i near a NS or R vertex at z i ). Consider two open subsets U and U of M related by a superdiffeomorphism caused by avector field θy (˜ z, z ) ∂ z . This changes the gravitino background as χ → χ ′ = χ + ˜ ∂y (2.15)This induces to the change of basis χ σ as χ σ → χ σ ′ = χ σ + ˜ ∂y σ (2.16)where P η σ y σ = y .The modes χ σ and χ σ ′ represent the same class of H (Σ red , b R ) and the coordinate η σ does not change. But the vector field θy (˜ z, z ) ∂ z causes a change on the metric at secondorder: h zz → h zz + yχ = h zz + X σ,σ ′ η σ η σ ′ y σ ′ χ σ . (2.17)The metric h determines the bosonic coordinates of the moduli, and therefore this superdif-feomorphism gives rise to a change of the coordinates, mixing odd and even moduli param-eters. 6his mixing of odd and even coordinates leads to the subtleties mentioned above. Con-sider a dimension 1 | M and an integral Z M ω (2.18)where ω is locally defined in a patch U as ω = ( γ ( t ) + γ ( t ) η η )d tδ (d η ) δ (d η ) . (2.19)If we integrate first on η and η , it reduces to −→ Z U red γ ( t )d t. (2.20)But if M is not holomorphically projected, we need a coordinate change of the form t ′ = t + a ( t ) η η , η ′ = η , η ′ = η . (2.21)In the new coordinate system ( t ′ | η ′ , η ′ ), the integrand ω is now ω = ( γ ′ ( t ′ ) + γ ′ ( t ′ ) η η )d t ′ δ (d η ′ ) δ (d η ′ ) (2.22)with γ ′ ( t ′ ) = γ ( t ′ ) − ∂ t ′ ( a ( t ′ ) γ ( t ′ )) (2.23)and the integral (2.18) reduces to −→ Z U red γ ′ ( t )d t. (2.24)Then the reductions (2.20) and (2.24) differ by an integral of an exact term ∂ t ( a ( t ) γ ( t )) d t on U red . This causes difficulties when we try to combine local contributions from patchestogether. In that case we should go to projected coordinates on M to define the integra-tion precisely, and this procedure destroys the holomorphic factorization property of theintegrand.Complex supermanifolds are not projected in general, so holomorphically factorised inte-grands on a complex supermanifold do not reduce to holomorphically factorised integrandson its reduced manifold. The arguments in the previous section also points the way out. Namely, we have an unam-biguous equality Z M ω = Z M red f (3.1)7f ω = f odddim M Y µ =1 η µ δ (d η µ ) . (3.2)More invariantly under the coordinate change, we state that the form locally defined as ̟ = Y i η i δ (d η i ) (3.3)is well-defined globally, and is the Poincar´e dual of M red ⊂ M , since we have Z M α ∧ ̟ = Z M red α | M red (3.4)for all differential forms α without δ (d η ) so that the multiplication on the left hand sidemeaningful. α | M red can be obtained ignoring all the terms containing η ’s and d η ’s from α .Let us check that the form ̟ is invariant under the coordinate change. Suppose thattwo patches are glued as η µ ′ = ψ µ ( x | η ). Then, Y µ δ (d η µ ′ ) = Y µ δ (d ψ µ ( x | η )) (3.5)= Y µ δ X i ∂ i ψ µ d x i + X ν ∂ ν ψ µ d η ν ! (3.6)= exp X i,µ ∂ i ψ µ d x i ∂ µ ! Y µ δ X ν ∂ ν ψ µ d η ν ! (3.7)= exp X i,µ ∂ i ψ µ d x i ∂ µ ! A Y µ (d η µ ) (3.8)where A µν := ∂ µ ψ ν | η =0 . (3.9)Similarly, we have Y µ ψ µ = Y µ X ν ∂ ν ψ µ | η =0 η ν + higher order in η ’s ! = Det A Y µ η µ . (3.10)8herefore we have ̟ ′ = Y µ η µ ′ δ (d η µ ′ ) (3.11)= Y µ ψ µ exp X i ∂ i ψ µ d x i ∂ µ ! δ X ν ∂ ν ψ µ d η ν ! (3.12)= Det A Y µ η µ exp ( O ( η )) 1Det A Y µ δ (d η µ ) (3.13)= Y µ η µ δ (d η µ ) (3.14)= ̟. (3.15)So, if we are to reduce superstring amplitudes to bosonic string amplitudes, we shouldcheck that the form F V in (2.14) to be integrated can be represented as ω red ∧ ̟ . We call thisphenomenon the saturation of η ’s and dη ’s. This saturation also guarantees that the shiftof the bosonic part of the supercoordinate near the degenerate super Riemann surfaces byeven nilpotent terms does not affect the computation. In particular, we can safely integratethe fermionic coordinates of NS vertices.We will explain in Appendix A that this condition, in the more traditional langauge ofPCOs, implies that the correlation function is completely independent of the positions ofthe PCOs. In this section we study how the mechanism studied in the previous section manifests itselfin the embedding N = 0 string theory to N = 1 superstring theory in [11].The N = 0 theory has the matter part X m and the ( b, c ) ghost system. We then constructan N = 1 matter system consisting of the matter system X m , the shifted ghost system ( b , c )whose spin are (3 / , − / b, c ) and ( β, γ ). The superVirasoro generators of the N = 1 matter system are S mat = b + c ( T m + ∂c b ) + 52 ∂ c , (4.1) T mat = T m − b ∂c − ∂b c + 12 ∂ ( c ∂c ) , (4.2)where T m is the stress energy tensor of X m . The central charge of T mat is 15.The shifted ghost system ( b , c ) and ( β, γ ) system have the same spin and the oppositestatistics. They are expected to cancel and the whole system goes back to the original N = 0system consisting of X m and the ( b, c ) ghost. We will see below that the integration of ( b , c )and ( β, γ ) gives ̟ = Q ηδ (d η ) which is the Poincar´e dual of the reduced moduli space inthe supermoduli space. 9he correspondence between N = 1 vertex operators and N = 0 vertex operators is asfollows. Let V i be a dimension-1 vertex operator of X m . Then c V i + θV i is a dimension-1 / N = 1 matter system ( X m , ( b , c )). Then, we can constructan N = 1 BRST invariant NS operator V i ( e z, z | θ ) = cδ ( γ )( c V i + θV i ). Let us compute aform F V defined in (2.5). Denote n is the number of vertex operators. Denote the productof the vertex operators by V = Q ni =1 V i where each V i is defined as above. Here we use NSoperators V i fixed at ( e z i , z i | θ = 0). Then, F V becomes F V ( m ; d η | η ; d m ) = * n Y i =1 V i exp − g − n X σ =1 η σ π Z Σ red d zχ σ b S ! exp − g − n X σ =1 d η σ π Z Σ red d zχ σ b β ! × exp − g − n X i =1 d m i π Z Σ red d zµ i b b !+ m,g × (antiholomorphic part) . (4.3)Here, { χ σ } is the basis of gravitino backgrounds as above.Only terms in (4.3) which have the b c ghost number 2 g − Q V i = Q cδ ( γ ) c V i has the b c ghost number − n . Therefore we need toprovide the b c ghost number 2 g − n from the expansion of exp (cid:16) − P g − nσ =1 η σ π R Σ red d zχ σ b S (cid:17) .The only possibility is g − n Y σ =1 − η σ π Z Σ red d zχ σ b b . (4.4)From a similar consideration on the bc ghost number we conclude that only the term g − n Y i =1 − d m i π Z Σ red d zµ i b b (4.5)contributes in the expansion ofexp − g − n X i =1 d m i π Z Σ red d zµ i b b ! . (4.6)Next, we explicitly calculate the integration with b and β field and confirm that theresult does not depend on choice of { χ σ } and reproduce the bosonic string amplitude. Letus expand b by modes: b b = g − n X α =1 v α b b α + X λ w λ b b λ . (4.7)where b b α are bosonic zero modes, b b λ are bosonic non-zero modes and v α and w λ are fermionicvariables. The field b is allowed to have poles at z i where vertex operators c sit. Therefore10ero modes { b α } include meromorphic ones that have poles at z i . Equivalently, { b b α } is abasis of H (Σ red , T Σ − / ⊗ O ( X i z i )) (4.8)which is dual to the space of gravitino background H (Σ red , T Σ / ⊗ O ( − X i z i )) . (4.9)Note that the superghost β is allowed to have poles at z i because of presence of vertexoperator δ ( γ ). Then, we can expand b β by the same basis as b b : b β = g − n X α =1 ν α b b α + X λ ω λ b b λ . (4.10)where ν α and ω λ are bosonic variables.Integration with zero mode factors v α and ν α in (4.3) gives the factor Z d g − v α d g − ν α exp − X σ,α η σ π Z Σ red d zχ σ v α b b α − X σ,α d η σ π Z Σ red d zχ σ ν α b b α ! = Det M Y σ δ ( η σ ) 1Det M Y s δ (d η s ) = Y σ δ ( η σ ) δ (d η σ ) . (4.11)where M σα := π R Σ red χ σ b b α . Now it is obvious that spurious singularity which occurs whenthe matrix M degenerates does not remain in the last form because of the cancellation withthe contribution of b integration.Hence, F V is proportional to ̟ = Q g − σ =1 ( η σ δ (d η σ )) and according to argument in Sec. 3,the amplitude becomes A V = 12 g Z Γ n NS ,n R , red *Y i V i Y j Z d zµ j b b + bosonic . (4.12)The final point to consider is that the integration space in (4.12) is not the integrationspace of the bosonic string, because Γ n NS ,n R , red = M spin is the moduli space of Riemannsurfaces with spin structures. This point is resolved as follows [9]. The moduli space of spinRiemann surfaces M spin has two connected components M + and M − , the moduli space ofRiemann surfaces with even and odd spin structure respectively. We can define a phase ofamplitude to each connected component separately. A Riemann surface with genus g has2 g − (2 g + 1) even spin structures and 2 g − (2 g −
1) odd structure. We should give a factor of − A V = Z M bosonic *Y i V i Y j Z d zµ j b b + bosonic . (4.13)11 Embedding topological string to superstring
Let us now move on to the study of the graviphoton amplitudes in type II string theory andits reduction to the topological string amplitude, which is a bosonic string theory as far asthe integration over the moduli space is concerned.Let us consider the compactification of Type IIA/B theory with N = 2 superconformalsymmetry with central charge c = 9. For definiteness we take Type IIA theory. We Wick-rotate the time direction and introduce complex coordinates X u = X + iX , X v = X + iX . (5.1)We use N = (1 ,
1) superfield X µ = X µ + θψ µ + e θ e ψ µ + θ e θF µ . Here and in the following µ runs over 1 , , ,
4. With this set up, the worldsheet supercurrents are given by S = iψ µ ∂X µ + G + + G − , e S = i e ψ µ e ∂X µ + e G + + e G − . (5.2)We would like to consider amplitudes among g − p µ , g − q µ , one graviton of momentum p µ , and one graviton of mo-mentum q µ . We choose the polarizations so that graviton vertices V R and the graviphotonvertices V T are V R,uvuv = c e cδ ( γ ) δ ( e γ ) D θ X u D e θ X u e i ( p u X u + p v X v ) ,V R, ¯ u ¯ v ¯ u ¯ v = c e cδ ( γ ) δ ( e γ ) D θ X ¯ v D e θ X ¯ v e i ( q ¯ u X ¯ u + q ¯ v X ¯ v ) ,V T,uv = p v c e c Θ e Θ S e S e ip u X u + ip v X v Σ e Σ ,V T, ¯ u ¯ v = q ¯ u c e c Θ e Θ S e S e iq ¯ u X ¯ u + iq ¯ v X ¯ v Σ e Σ . (5.3)Here S and S are spin fields which have charges (1 / , /
2) and ( − / , − /
2) under thebosonized currents of ψ u and ψ v , Θ is the spin field for βγ system, and Σ is the left-moving and the right-moving vertex operators of the internal system which has U (1) R charge (3 / , ∓ /
2) for typeIIA model and (3 / , /
2) for typeIIB model constructed fromthe bosonized version of the U (1) R current. Here we suppose that metric and gravitinobackgrounds are turned off near vertex operators.We would like to check that there are no subtleties due to the odd moduli integration inthe proof that the scattering amplitude of two gravitons and 2 g − g topological vacuum amplitude F top g : [10, 11] A g (cid:0) V R,uvuv V R, ¯ u ¯ v ¯ u ¯ v ( V T,uv ) g − ( V T, ¯ u ¯ v ) g − (cid:1) = V R p v p g − v q u q g − u ( g !) F top g (5.4)to the leading order in the zero momentum limit. We will see the mechanism of Sec. 3 atwork again.Let us denote the amplitude in the right hand side of (5.4) as A . It has 2 g − g − | g −
1. Notethat here we use unintegrated NS vertices. We write A as A = 12 g Z Γ F. (5.5)12he integrand F is F = *Y i V T,uv ( x i , e x i ) Y j V T, ¯ u ¯ v ( y j , e y i ) V R,uvuv ( z, e z | θ , e θ ) V R, ¯ u ¯ v ¯ u ¯ v ( w, e w | θ , e θ ) × exp − g − X σ =1 η σ π Z Σ red d zχ σ b S − g − X σ =1 d η σ π Z Σ red d zχ σ b β − g − X i =1 d m i π Z Σ red d zµ i b b ! × exp − g − X σ =1 e η σ π Z Σ red d z e χ σ be S − g − X σ =1 d e η σ π Z Σ red d z e χ σ be β − g − X i =1 d e m i π Z Σ red d z e µ i be b ! × exp − X σ,σ ′ η σ e η σ ′ π Z Σ red d z ( e ψ µ ψ µ + A ++ + A + − + A − + + A −− ) χ σ e χ σ ′ !+ m,g . (5.6) χ σ ∈ H ( σ red , b R ) are the gravitino basis. { µ i } is a basis of Beltrami differentials ofRiemann surfaces with 2 g marked punctures. A ±± is an operator of internal theory whichcouples to χ ∓ e χ ∓ to complete N = (2 ,
2) local supersymmetry. Note that A ±± have ± U (1) R symmetry because they coupleto internal gravitino.Let us first discuss the internal U (1) R charge. The field V T has (3 / , ∓ /
2) internal U (1) R charge. So we need to bring down sufficient number of operators from second, thirdand fourth line in (5.6) to saturate U (1) R charge to get a nonzero contribution. All operatorsin second to third line in (5.6) have 0 or ± U (1) R charge. So we should bring down 3 g − η ’s and 3 g − e η .Second, consider the terms involving V R . Recall again that we use unintegrated verticeshere; the degrees of freedom of the bosonic and fermionic positions are counted in theBeltrami differentials µ ’s and χ ’s. We therefore pick a specific value of the supercoordinatesof the NS vertices, which we take to be θ = θ = 0 , e θ = e θ = 0 for simplicity. Then, vertexoperators become V R,uvuv ( θ = 0) = c e cδ ( γ ) δ ( e γ ) ψ u e ψ u e i ( p u X u + p v X v ) , (5.7) V R, ¯ u ¯ v ¯ u ¯ v ( θ = 0) = c e cδ ( γ ) δ ( e γ ) ψ ¯ v e ψ ¯ v e i ( q ¯ u X ¯ u + q ¯ v X ¯ v ) . (5.8)Operators which has the charge of ψ u are the following: ψ u itself in V R , S , in V T and ψ u , ψ ¯ u in χS term. Therefore, to have a nonzero contribution, we need to use the χS term. We cantreat ψ ¯ v , e ψ u and e ψ ¯ v in a similar manner. In total, we need two η ’s and two e η ’s to saturatethese charges.Considering both the charge of U (1) R and the charges of ψ ’s, we see that the only termwhich contributes has 3 g − η ’s and 3 g − e η ’s. Therefore, it involves all the oddmoduli, and the amplitudes we are considering can be represented as an integral over theordinary bosonic moduli space, by the mechanism of Sec. 3. We can safely integrate the oddmoduli. 13wo of η ’s and two of ˜ η ’s correspond to the fermionic positions of unintegrated NSNSvertices. We integrate these odd moduli first to convert them to the integrated NSNS vertices.Then we have A = 12 g Z Γ ′ F int (5.9)where Γ ′ is a integration cycle in the supermoduli space of super Riemann surfaces with2 g − g − | g −
3. The integrand F int is F int = *Y i V T,zw ( x i ) Y j V T, ¯ u ¯ v ( y j ) Z Σ red d zV ′ R,uvuv Z Σ red d wV ′ R, ¯ u ¯ v ¯ u ¯ v × g − Y σ =1 η σ π Z Σ red d zχ σ b G − g − Y i =1 d m i π Z Σ red d zµ i b b exp − g − X σ =1 d η σ π Z Σ red d zχ σ b β ! × g − Y σ =1 e η σ π Z Σ red d z e χ σ be G ± g − Y i =1 d e m i π Z Σ red d z e µ i be b exp − g − X σ =1 d e η σ π Z Σ red d z e χ σ be β ! + (terms involving Aχ e χ ) i m,g (5.10)where V ′ R is the picture number 0 NSNS vertex: V ′ R,uvuv = (cid:16) ∂X u e ∂X u − p v ψ u ψ v e ψ u e ψ v (cid:17) e i ( p u X u + p v X v ) ,V ′ R, ¯ u ¯ v ¯ u ¯ v = (cid:16) ∂X ¯ v e ∂X ¯ v − q u ψ ¯ u ψ ¯ v e ψ ¯ u e ψ ¯ v (cid:17) e i ( q ¯ u X ¯ u + q ¯ v X ¯ v ) . (5.11)Now m, η, µ, χ and twisted fields b b, b G, b β and their antiholomorphic counterparts are appro-priate ones for Γ ′ .Then we integrate the η directions of Γ ′ , resulting in A = 12 g Z Γ red F red (5.12)where F red = *Y i V T,uv ( x i ) Y j V T, ¯ u ¯ v ( y j ) Z Σ red d zV ′ R,uvuv Z Σ red d wV ′ R, ¯ u ¯ v ¯ u ¯ v × g − Y σ =1 Z Σ red d zχ σ b G − g − Y σ =1 δ (cid:18)Z Σ red d zχ σ b β (cid:19) g − Y i =1 d m i π Z Σ red d zµ i b b × g − Y σ =1 Z Σ red d z e χ σ be G ± g − Y σ =1 δ (cid:18)Z Σ red d z e χ σ be β (cid:19) g − Y i =1 d e m i π Z Σ red d z e µ i be b + (terms involving Aχ e χ ) i m,g (5.13)14t this stage we can set χ σ to have delta function support at appropriate places, atleast on generic points of Γ red . Then we go to the familiar picture changing formalism. Asreviewed in Appendix. A, the saturation as above guarantees that the correlation functiondoes not depend at all on the positions of the PCOs. Therefore we can place the PCOsat the points most suitable for calculations. First, the terms involing A vanish if we placeholomorphic and antiholomorphic PCOs at distinct points. Then, we choose the place thePCOS as in the calculation of [10]. The rest of the computation goes unchanged comparedto [10]. To briefly summarize, we explicitly evaluate the contributions from the spacetimebosons and fermions, the ghosts ( b, c ) and ( β, γ ), and the internal U (1) R boson. Then theintegral over x i , y i and Beltrami differentials associated to them can also be performed. Weend up with F red = p v p g − v q u q g − u V R ( g !) * g − Y i =1 d m i π Z Σ red d zµ i G − g − Y i =1 d e m i π Z Σ red d z e µ i e G − + m,g (5.14)at the leading order of momenta, where G , e G are now topologically twisted. This is therelation (5.4) we wanted to show. Acknowledgments
The authors thank Edward Witten for helpful discussions. They alsothank their colleagues who spent tens of hours attending the journal club by KO on thecontents of [5–7]. YT thanks the hospitality of the Institute for Advanced Study where thismanuscript was finalized. The work of KO is partially supported by an Advanced LeadingGraduate Course for Photon Science grant. The work of YT is partially supported by WorldPremier International Research Center Initiative (WPI Initiative), MEXT, Japan throughthe Institute for the Physics and Mathematics of the Universe, the University of Tokyo. When the internal CFT is a free CFT, such as orbifolds of T , this independence from the positions ofthe PCOs implies a host of intricate identities among higher-genus theta functions. The authors have notbeen able to prove these identities by themselves. Rather, they regard these identities as dervied via theCFT methods. Formalism using PCOs
In this appendix, we review the relationship between the approach using the supermodulispace and the approach which uses the picture changing operators (PCOs).
A.1 Pointlike gravitinos and PCOs
We start from the integrand F ( m, d η | η, d m ) (2.14) of a general superstring amplitude. Letus explicitly perform the integration over η directions, which produces factors of the form Y σ δ (cid:18)Z Σ red d zχ σ b β (cid:19) δ (cid:18)Z Σ red d zχ σ b S (cid:19) δ (cid:18)Z Σ red d z e χ σ be β (cid:19) δ (cid:18)Z Σ red d z e χ σ be S (cid:19) + (contribution from χ e χ term) . (A.1)A traditional choice of the gravitino basis { χ } is to take χ σ = δ ( p σ ) , e χ σ = δ ( q σ ) for some p σ q σ in Σ red . If all p σ and q σ are different, χ e χ term does not contribute.Then, the holomorphic part of the factors (A.1) above becomes Y σ Y ( p σ ) = Y σ δ (cid:16) b β ( p σ ) (cid:17) b S ( p σ ) = Y σ δ ( β ( p σ )) S ( p σ ) (A.2)where the operator Y ( p σ ) := δ (cid:16) b β ( p σ ) (cid:17) b S ( p σ ) is the PCO. Summarizing, we have Z U F ( m, d η | η, d m ) = Z U red *Y i V i ∆ o Y σ =1 Y ( p σ ) ∆ e Y j =1 (cid:18) d m j Z Σ red d zµ j b b (cid:19)+ × (antiholomorphic part) (A.3)where U is a patch in the supermoduli space, and U red is the corresponding patch in thebosonic moduli space. In the following we do not explicitly write down the antiholomorphicpart, as it can be dealt with separately.There are two points to be kept in mind when this formula is used. The first point is thatwe can not always guarantee that χ σ are linearly independent. Namely, we assumed abovethat [ χ σ ] = [ δ ( p σ )] forms a basis of H (Σ red , b R ), where [ • ] denotes the class in H (Σ red , b R ).This is not always the case. To see this, let us write down the condition when [ δ ( p σ )] arelinearly dependent in H (Σ red , b R ), namely that the equation e ∂y = ∆ o X σ =1 e σ δ ( p σ ) (A.4)has solutions for some complex numbers e σ and y . This means that y has meromorphic andhas poles only at p σ ’s. Equivalently,dim H (Σ red , b R ⊗ O ( X σ p σ )) > . (A.5)16hen this happens the superghost correlation function has a pole, which is called the spu-rious singularity in the literature. Its appearance can be seen e.g. in (4.11). A.2 Dependence of the correlators on the positions of the PCOs
Another point to be kept in mind is more severe: the change in the positions of the PCOsgenerate exact forms on the bosonic moduli space M red , as a manifestation of the phenomenondescribed in Sec. 2.4. This is because the change in the positions of the PCOs is a changein the coordinate system of the supermoduli space. The following argument is based onthe one given in [3], and it is only a slight extension thereof. The change can be derivedby manipulating the derivative of the correlator (A.3) with the PCOs with respect to thepositions p σ directly on a purely bosonic Riemann surface. Equivalently, the same changecan be derived by studying its effect on the coordinates on the moduli space of the superRiemann surfaces. We use this second point of view below, and compute∆ Z U F ( m, d η | η, d m ) = Z U red ∆ *Y i V i Y Y ( p σ ) Y (cid:18) d m j Z Σ red d zµ j b b (cid:19)+ (A.6)under the change of the positions of the PCOs.Let us first put PCOs at { p σ } , and and consider the effect of moving one PCO at p to p + ∆ p and consider only first order in ∆ p . We should find a superconformal gaugetransformation parameter y in (2.15) producing this gauge transformation. Explicitly, weshould solve X σ ′ η ′ σ ′ δ ( z − p ′ σ ′ ) = e ∂y ( z ) + X σ η σ δ ( z − p σ ) (A.7)in y and η ′ . Here p ′ σ = p σ for σ = 1 and p ′ = p + ∆ p . This is a slight extension ofthe transformation dealt in Sec. 2.4, in that we allowed η ’s to vary. As we see below, thetransformation of η ’s are linear, and thus the change in η s does not cause any effect, as itcan be absorbed in a redefinition of η s.To solve (A.7), we define a Green’s function G ( z, w ) with the property ∂ e z G ( z, w ) = δ ( z − w ) + X σ R σ ( w ) δ ( z − p ′ σ ) . (A.8)The terms R σ ( w ) are source terms necessary to solve the Laplace equation on a closednontrivial Riemann surfaces. Then, we can solve (A.7) with y ( z ) = − Z d wG ( z, w ) χ ( w ) = − X σ G ( z, p σ ) η σ , (A.9) η ′ σ = − Z d wR σ ( w ) χ ( w ) = − X τ R σ ( p τ ) η τ . (A.10)17onsidering that y is a section of b R and (A.7),(A.9), we see that G ( z, w ) should trans-forms as a section of b R as a function of the z -plane and as a section of b R − ⊗ K Σ red as afunction of the w -plane.The Green’s function (A.8) can be written as a correlator of twisted βγ system: G ( z, w ) = 1 Z *b γ ( z ) b β ( w ) Y σ δ ( b β ( p ′ σ )) + b β b γ , (A.11) Z = *Y σ δ ( b β ( p ′ σ )) + b β b γ . (A.12)Here, twitsted fields b β and b γ are defined as b β = f β, b γ = f − γ (A.13)using the same f as in (2.12). This means that b γ ∈ Γ( b R ) and b β ∈ Γ( b R − ⊗ K Σ red ). Hence,the expression (A.11) reproduces desired transformation laws and positions and residues ofpoles of G ( z, w ). More on twisted βγ system can be found in [5].The superconformal transformation (A.7) with y ( z ) also causes the transformation of themetric as in (2.17). So, the bosonic moduli parameter m transforms as∆ m i = Z d zy ( z ) χ ( z ) b b i = − X σ =1 η η σ G ( p σ , p ) b i ( p σ ) . (A.14)Here, { b i } is a basis of (twisted) quadratic differentials which is dual to a given basis { µ i } of Beltrami differentials. We also used G ( z, p σ ) = 0.Summarizing, the superconformal transformation with parameter y ( z ) corresponds to acoordinate change of supermoduli space given by η σ → η ′ σ , m i → m ′ i = m i − ∆ m i (A.15)where ∆ m i = X σ =1 η η σ G ( p σ , p ′ ) b i ( p σ ) . (A.16)Let us write the integrand of the superstring amplitude as Z U F ( m, d η | η, d m ) = Z U Y i d m i Y σ δ (d η σ ) F ( m | η ) . (A.17)Under the changes (A.15) this integral is changed according to Y i d m i Y σ δ (d η σ ) F ( m | η ) → Y i d m ′ i Y σ δ (d η σ ) F ( m ′ | η ) − X i ∂ i (∆ mF ( m ′ | η )) ! (A.18)18o the first order of ∆ p . Therefore, the total change to the first order is∆ Z U F ( m, d η | η, d m )= − Z U red Y i d m ′ i X i ∂ i [∆ m i F ( m ′ | η )] full (A.19)= Z U red Y i d m ′ i X i ∂ i "X σ ( − σ − G ( p σ , p ′ ) b i ( p σ )( F ( m | η ) | σ ) , (A.20)= Z U red ∂ i hX σ ( − σ − b i ( p σ ) Db γ ( p σ ) b β ( p ′ ) Y τ δ ( b β ( p τ )) × Y ρ =1 ,σ b S ( p ρ ) Y i V i Y (cid:18) d m j Z Σ red d zµ j b b (cid:19)Ei (A.21)Here, in (A.19), we denoted the coefficient of Q τ η τ in A ( m | η ) as A | full , and in (A.20), thecoefficient of Q τ =1 ,σ η τ in A ( m | η ) is denoted by as A | σ , and to go from (A.20) to (A.21), weused the fact that there is a factor of Z given in (A.12) in the definition of F which cancels Z in the denominator of the expression for G ( z, w ) in (A.11).Comparing with (A.6) and using the definition of the PCO (A.2), we finally find∆ *Y i V i Y Y ( p σ ) Y (cid:18) d m j Z Σ red d zµ j b b (cid:19)+ = ∂ i hX σ ( − σ − b i ( p σ ) Db γ ( p σ ) b β ( p ′ ) δ ( b β ( p )) δ ( b β ( p σ )) × Y ρ =1 ,σ Y ( p ρ ) Y i V i Y (cid:18) d m j Z Σ red d zµ j b b (cid:19)Ei . (A.22)The right hand side is an exact form on the patch of the bosonic moduli space. This can alsobe obtained by a schematic manipulation as follows [1, 2]: To move a PCO Y ( p ) to anotherposition q , we first go to the large Hilbert space and move the BRST operator: h Y ( p ) · · ·i = h [ Q BRST , ξ ( p )] ξ ( q ) · · ·i (A.23)= h ξ ( p )[ Q BRST , ξ ( q )] · · ·i + h ξ ( p ) ξ ( q )[ Q BRST , · · · ] i (A.24)= h Y ( q ) · · ·i + h ξ ( p ) ξ ( q )[ Q BRST , · · · ] i . (A.25)The second term in the last line contains terms where Q BRST acts on the b ghost, thusgenerating exact terms on the moduli space of Riemann surfaces.At this point, it is obvious that saturation of η ’s guarantees that the correlator is inde-pendent of the positions of PCO’s. When F ( m, d η | η, d m ) is proportional to ̟ , as discussedin Sec. 3, F ( m | η ) as defined in (A.17) only has terms with all factors of η , and F ( m | η ) | σ van-ishes by definition. 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