Boundary contributions to three loop superstring amplitudes
aa r X i v : . [ h e p - t h ] S e p Boundary contributions to three loop superstring amplitudes
Kowshik Bettadapura , Hai Lin , Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
Abstract
In type II superstring theory, the vacuum amplitude at a given loop order g can receive contributions from the boundary of the compactified, genus g su-permoduli space of curves M g . These contributions capture the long distance orinfrared behaviour of the amplitude. The boundary parametrises degenerationsof genus g super Riemann surfaces. A holomorphic projection of the supermodulispace onto its reduced space would then provide a way to integrate the holomor-phic, superstring measure and thereby give the superstring vacuum amplitude at g -loop order. However, such a projection does not generally exist over the bulkof the supermoduli spaces in higher genera. Nevertheless, certain boundary di-visors in ∂ M g may holomorphically map onto a bosonic space upon compositionwith universal morphisms, thereby enabling an integration of the holomorphic,superstring measure here. Making use of ansatz factorisations of the superstringmeasure near the boundary, our analysis shows that the boundary contributions tothe three loop vacuum amplitude will vanish in closed oriented type II superstringtheory with unbroken spacetime supersymmetry. Introduction
A super Riemann surface of genus g is a (1 | g , Riemann surface and its module of odddifferentials defines a spin structure on the surface. In contrast to punctures on Riemannsurfaces however, there can be two types of punctures on a super Riemann surfaceoriginating from external states in superstring theory. These are Neveu-Schwarz (NS)type punctures and the Ramond (R) type punctures.The moduli space of super Riemann surfaces is referred to generally as supermodulispace . It is a subjest of much importance in several areas of mathematics and physics.It provides a unified framework in which to study problems in algebraic geometry, themathematics of supersymmetry and superconformal quantum field theory. Methodsfrom supergeometry and superanalysis are essential for studying supersymmetric stringtheory from the viewpoint of supermoduli space. For an early introduction to themathematics of supergeometry and superanalysis, see [1, 2, 3] and references therein.In this paper we are interested in superstring amplitudes and the boundary compo-nents of supermoduli spaces. More precisely, we aim to understand contributions to thesuperstring amplitude at g loop order arising from the boundary of the compactified,genus g supermoduli space M g . As an illustration of our ideas we specialise to genus g = 3 and analyse the boundary contributions to the three loop vacuum amplitude inType II superstring theory.More generally, the supermoduli space M g,n,n ′ is the moduli space of genus g su-per Riemann surfaces with n Neveu-Schwarz punctures and n ′ Ramond punctures. Inparticular, the dimensionality of the supermoduli space is increased by (1 |
1) for eachNeveu-Schwarz puncture and by (1 | ) for each Ramond puncture. Two Ramond punc-tures contribute to one fermionic modulus. These moduli in superstring perturbationtheory play the role of the Schwinger parameters in quantum field theory. In the contextof a supersymmetric theory, fermionic moduli play the role of supersymmetric partnersto the bosonic moduli.The Mumford isomorphisms in algebraic geometry are a collection of isomorphismsbetween certain line bundles over the Deligne-Mumford compactification of the modulispace of Riemann surfaces. In bosonic string theory, defined by the path integral with thePolyakov action functional, these isomorphisms can be used to construct the holomor-phic string measure [4, 5]. Due to the existence of tachyons, bosonic string theory is notentirely a physical theory, in contrast to its supersymmetric extension—superstring the-ory. In superstring theory however, the non-compact superstring configuration space isno longer the moduli space of Riemann surfaces but rather its supersymmetric analogue,the supermoduli space. Accordingly, the Mumford isomorphisms can be generalised toisomorphisms between certain line bundles on supermoduli space [6], leading thereby toa construction of the holomorphic, superstring measure [7]. For a construction of thismeasure in the presence of NS- and R-punctures, see [8] and [9].1he superstring measure on the supermoduli space can also be derived from world-sheet superconformal field theory [7, 10, 11, 12, 13, 14, 15]. The integration of thismeasure over the supermoduli space then gives the superstring amplitude. A chiralsplitting procedure is important [10, 16, 14] in the computation of the genus two super-string measure [17]. In the chiral splitting procedure, one introduces loop momenta towrite conformal correlators via an integral over loop momenta whose integrand is theproduct of the left and right chiral conformal blocks. Each chiral block is analytic inthe moduli of the surface as well as in the inserted vertex points.Integration of the genus g , bosonic string measure over the Deligne-Mumford com-pactifcation of the bosonic moduli space, i.e. the moduli space of Riemann surfaces, givesthe g loop bosonic string amplitude. Similarly and as mentioned above, integration ofthe genus g , superstring measure gives a contribution to the superstring amplitude at g loop order. The space over which one integrates is the analogue of the compactifi-cation of bosonic moduli space, being a compactification of supermoduli space. Sucha compactification was sketched by Deligne in a letter to Manin circa 1987, and de-scribed recently and in more detail by Witten in [18] and Donagi and Witten in [19].Expressions for the superstring amplitude in genus g = 0 , g = 2 by D’Hoker and Phong [17]. However, as observed byWitten [21], these expressions ought to receive potentially non-vanishing contributionsfrom the boundary divisors in the compactification of supermoduli space. At two looporder, Witten nevertheless notes that these boundary contributions will vanish. In thispaper we note that the boundary contributions of vacuum amplitude at three loop orderwill also vanish.From more mathematical perspectives, one of the main results by Donagi and Witten[19, 22] concerns the question of holomorphically projecting the genus g supermodulispace M g onto its reduced space. A holomorphic projection π : M g → SM g would al-low for computing the superstring amplitude by firstly integrating along the odd fibersas stipulated by Berezin [1], and then integrating over the spin moduli space SM g .Since the spin moduli space discretely covers the moduli space of Riemann surfaces M g , measures on SM g can be reduced to measures on M g by summing over the spinstructures—a procedure known as GSO projection. This method of reducing the super-string measure to a measure on M g was used by Green and Schwarz [20] and by D’Hokerand Phong [17] in their derivation of the superstring amplitude to loop orders zero, oneand two. Donagi and Witten found however that M g cannot be projected onto SM g for any genus g ≥
5, thereby placing a mathematical obstruction to applying knownmethods to calculate superstring amplitudes to arbitrary loop order. Note, this is incontrast to bosonic string theory where the bosonic string amplitudes are in principleknown to any loop order.Supermoduli space parametrises smooth, super Riemann surfaces. In its compactifi-cation, boundary divisors parametrise super Riemann surfaces which can degenerate intotwo kinds, separating degenerations and non-separating degenerations. These boundary2ivisors can be identified with punctured supermoduli spaces of generally lower generathrough a process known as clutching, as studied in a more classical setting in [23]. Theboundary contributions to superstring amplitudes involve integrating the superstringmeasure over these boundary divisors. In this paper we look at these boundary con-tributions along boundary divisors in the genus g = 3 supermoduli space. As observedearlier, there do not exist holomorphic projections of supermoduli in genus g ≥
5. Ingenus g = 3 there does exist a projection, but it is not holomorphic—it is singular alongthe hyperelliptic locus [21]. As such, although one can use super period matrix [21, 24]as a basis, it is unclear as to how to apply D’Hoker and Phong’s integration procedureover the genus g = 3 supermoduli space. As mentioned above, we have largely consid-ered boundary components of the supermoduli space, which involve supermoduli spacesof lower genera.The organisation of this paper is as follows. In Section 2, we describe the boundarycomponents of supermoduli space. This section is divided into three subsections whichemphasise perspectives from algebraic geometry, from gluing in local, geometric models,and from superstring worldsheet theory. Section 3 is devoted to superstring measuresand contributions to the amplitude from the boundary of supermoduli space. Thissection is divided into five subsections offering perspectives on superstring measures andamplitudes from supergeometry and sheaf theory in algebraic geometry. We analysethe genus three case as a detailed example. In Section 4 we discuss our results anddraw conclusions. In the interests of being self-contained, we give a brief overview ofthe compactification of the moduli space of Riemann surfaces with and without spinstructures in Appendix A. In this section we analyse the boundary of supermoduli space and related aspects. Theboundary parametrises super Riemann surfaces with prescribed degenerations. Thatis, super Riemann surfaces which develop nodes of Neveu-Schwarz (NS) or Ramond(R) type. As described in [18], there are two distinct types of degenerations of a su-per Riemann surface, referred to as separating and non-separating degenerations. Theconstruction of the boundary of supermoduli space parallels that of the boundary ofthe Deligne-Mumford compactification of the moduli space of Riemann surfaces in [23]and [25]. In Section 2.1 we analyse the boundary components of supermoduli spaceparametrising degenerating super Riemann surfaces. We emphasise the role of clutch-ing morphisms, among other things. Section 2.2 reviews gluing formulae for these de-generating surfaces at Neveu-Schwarz and Ramond nodes. In Section 2.3 we describethe relation between the superstring amplitudes and the boundary of the supermodulispaces. 3 .1 Boundary components via clutching
Like the objects it parametrises, the moduli space of genus g super Riemann surfaces M g is a superspace. Its reduced space SM g parametrises Riemann surfaces with spinstructure. As such, its compactification might resemble that of the spin moduli spacedescribed by Cornalba [26] and which in turn resembles the Deligne-Mumford compact-ification [25]. And indeed, the compactification of M g is similar to that of M g . To beself-contained, we give a brief overview of the compactifications of the ordinary bosonicmoduli space M g and spin moduli space SM g in Appendix A.Denote by M g,n,n ′ the moduli space of super Riemann surfaces of genus g with n -many NS-punctures and n ′ -many R-punctures. We use a prime in denoting the Rpuncture number so as to distinguish them from the NS puncture number. The reducedspace of M g,n,n ′ is SM g,n + n ′ . If n = n ′ = 0, we will just write M g rather than M g, , ′ forbrevity. Similarly, if either n = 0 or n ′ = 0, the respective subscript will be omitted. Aswith the case of spin moduli space, the parity of spin structures induces a decompositionof the supermoduli space: M g,n,n ′ = M g,n,n ′ ;+ ∪ M g,n,n ′ ; − , where + or − denotes even orodd for the parity of the spin structure. It is M g,n,n ′ ;+ which is of primary interest insuperstring theory. But as illustrated in the boundary components of spin moduli spacein Appendix A, the boundary of ∂ M g,n,n ′ ;+ will inevitably involve odd spin structures.Now in contrast to Neveu-Schwarz punctures, there must always be an even number ofRamond punctures, i.e., n ′ is an even integer. In genus g ≥
2, appropriately applyingthe Riemann-Roch theorem reveals the dimension:dim M g,n,n ′ = 3 g − n + n ′ (cid:12)(cid:12) g − n + 12 n ′ . (2.1)In analogy with stable curves, compactifying M g,n,n ′ involves allowing super Riemannsurfaces to develop nodes of Neveu-Schwarz or Ramond type. The boundary components ∂ M g,n,n ′ ⊂ M g,n,n ′ ought then be similar to those of M g,n from (A.1). This is indeedthe case along the Neveu-Schwarz nodes. As such, components of ∂ M g,n,n ′ are given bythe analogue of α - and β -type clutchings in (A.1), (A.2) and (A.3) for super Riemannsurfaces along NS nodes. This leads now to the following clutching morphisms for theboundary component of the even part of supermoduli space: M g ,n +1 ,n ′ ; ± × M g ,n +1 ,n ′ ; ± α ( ± , ± ) −→ M g,n,n ′ ;+ and M g − ,n +2 ,n ′ ;+ β + −→ M g,n,n ′ ;+ (2.2)where g + g = g , n + n = n and n ′ + n ′ = n ′ . There are similar clutching morphismsto (2.2) describing the boundary of the odd part of supermoduli space. Note thataccordingly we have the inclusions,im α ⊂ ∂ M g,n,n ′ ⊂ M g,n,n ′ , im β ⊂ ∂ M g,n,n ′ ⊂ M g,n,n ′ , (2.3)where α and β are as in (2.2). We sometimes denote α = α g ,g and β = β g − withthe subscript indicating the genus; and sometimes we omit the subscript for brevity.4f D NS ; sep. and D NS ; nsep. in ∂ M g,n,n ′ denote the divisors parametrising separating andnon-separating degenerations along NS-nodes, then: D NS ; sep. ; ± ∼ = M g ,n +1 ,n ′ ; ± × M g ,n +1 ,n ′ ; ± and D NS ; nsep. ;+ ∼ = M g − ,n +2 ,n ′ ;+ . (2.4)Note D NS ; sep. = D NS ; sep. ;+ ∪D NS ; sep. ; − and similarly for D NS ; nsep. . These divisors form theboundary components of M g,n.n ′ corresponding to degenerations along NS nodes. Thedivisors parametrising degenerations along Ramond nodes cannot be so elegantly de-scribed however. As discussed by Witten [18], the divisors D R ; sep. and D R ; nsep. in ∂ M g,n,n ′ parametrising separating and non-separating degenerations along Ramond nodes arefibered over the expected boundary components with fermionic fiber. More generally, if X is a supermanifold with boundary ∂ X , the boundary ∂ X ⊂ X will have codimension-(1 | M g,n,n ′ − (dim M g ,n ,n ′ + dim M g ,n ,n ′ )= n ′ − ( n ′ + n ′ ) + 3 (cid:12)(cid:12)
12 ( n ′ − ( n ′ + n ′ )) + 2 (2.5)where we have taken g = g + g and n = n + n . Since the number of Ramond puncturesmust be even, and since n ′ < n ′ + n ′ , the dimension formula in (2.5) only makessense when n ′ + 2 = n ′ + n ′ . Evidently, the divisor D R ; sep. ⊂ ∂ M g,n,n ′ parametrisingseparating degeneration along Ramond punctures is a (0 | M g ,n ,n ′ × M g ,n ,n ′ . A similar analysis in the non-separating case D R ; nsep. ⊂ ∂ M g,n,n ′ reveals it will fiber over M g − ,n,n ′ +2 with (0 | D R ; sep. , D R ; nsep. ⊂ ∂ M g,n.n ′ parametrising separating and non-separatingRamond degenerations can be realised as fibrations: C | ⊂ / / D R ; sep. ; ± π R ; sep. ; ± (cid:15) (cid:15) M g ,n ,n ′ ; ± × M g ,n ,n ′ ; ± and C | ⊂ / / D R ; nsep. ;+ π R ; nsep. ;+ (cid:15) (cid:15) M g − ,n,n ′ +2;+ (2.6)where g + g = g , n + n = n and n ′ + n ′ = n ′ + 2. Note the contrast with NS-casein (2.4). As we will see in Section 2.2, this contrast will be apparent in the gluinglaws along the nodes in the following sense: there are no free, odd parameters in thegluing laws along NS nodes. As in the Neveu-Schwarz case, the divisors parametis-ing Ramond degenerations decompose according to the parity of spin structures, i.e., D R ; sep. = D R ; sep. ;+ ∪ D R ; sep. ; − , and similarly for D R ; nsep. We can now write the boundary of supermoduli space as follows: ∂ M g,n,n ′ = ∂ M g,n,n ′ ; NS ∪ ∂ M g,n,n ′ ; R . (2.7)The compactified supermoduli space is now the union of the bulk with the boundary, M g,n,n ′ = M g,n,n ′ ∪ ∂ M g,n,n ′ . (2.8)5nother description of the boundary is as a union over irreducible components ∂ M g,n,n ′ = S j ∆ j , where ∆ j denotes an irreducible component. Then to the embedding∆ j ⊂ M g,n,n ′ there is a bundle of normal sections to the embedding over ∆ j , denoted N ∆ j . Its sheaf of sections, denoted ˆ N ∆ j , fits into a short exact sequence0 → T ∆ j → T M g,n,n ′ | ∆ j → ˆ N ∆ j → , (2.9)or dually 0 → ˆ N ∗ ∆ j → T ∗ M g,n,n ′ | ∆ j → T ∗ ∆ j → . (2.10)We denote by N ∆ j the fiber of the normal bundle N ∆ j to the embedding ∆ j ⊂ M g,n,n ′ .This gives a fibration of spaces, N ∆ j → N ∆ j → ∆ j . (2.11)As mentioned earlier, the sheaf of sections of N ∆ j over ∆ j is ˆ N ∆ j .As in the case of Riemann surfaces with marked points, we can form a forgetful mapon super Riemann surfaces, M g,n +1 ,n ′ p −→ M g,n,n ′ , (2.12)given by forgetting the ( n + 1)-th NS puncture and stabilizing the resulting puncturedsuper Riemann surface. The pull-back along the forgetful map is associated to theinsertion of an vertex operator in superconformal field theory. We also denote p = p g with the subscript meaning the genus. Sometimes we omit the subscript for brevity.Composite forgetful maps gives the morphism M g,n +2 ,n ′ p −→ M g,n,n ′ .The following are diagrams of morphisms relevant for subsequent constructions inthis paper: M g ,n +1 ,n ′ × M g ,n +1 ,n ′ α −−−−→ M g,n,n ′ y p y p M g ,n ,n ′ × M g ,n ,n ′ (2.13)and M g − ,n +2 ,n ′ α −−−−→ M g,n,n ′ y p M g − ,n,n ′ (2.14)The clutching morphisms are defined on boundary components of supermoduli spaceswhich parametrise degenerating super Riemann surfaces. These degenerations can berealised in a geometric model involving gluing maps in Section 2.2. They are importantfor Section 3.3.In this section we have described the compactification of supermoduli space. InSection 3.2 we consider a ‘partial compactification’ of supermoduli space. This is in orderto ensure that a certain, holomorphic map defined on the bulk will remain holomorphicwhen extended to the (partial) compactification.6 .2 Degenerations and gluing maps in a geometric model The degeneration of super Riemann surfaces parametrised by the boundary of supermod-uli spaces can be realised in a geometric model via gluing. These models are necessarilylocal however, in the sense that the gluing map is a local map. It is only defined nearthe node along which the super Riemann surface degenerates.As described in [7], we can glue super Riemann surfaces S ℓ and S r with local coordi-nates x | θ and y | ψ , respectively. We glue them so that the gluing happens at the points a | α ∈ S ℓ and b | β ∈ S r ,( x − a − αθ )( y − b − βψ ) = − ε = q NS , ( y − b − βψ )( θ − α ) = ε ( ψ − β ) , ( x − a − αθ )( ψ − β ) = − ε ( θ − α ) , ( θ − α )( ψ − β ) = 0 . (2.15)In the language of theoretical physics, the superconformal structures are defined by thevector fields D θ = ∂ θ + θ∂ x and D ψ = ∂ ψ + ψ∂ y on the surfaces S ℓ and S r respectively.The gluing formula maps the superconformal coordinates of the left component S ℓ tothe superconformal coordinates of the right component S r . The above (2.15) is gluingalong the NS punctures and then smoothed out. The α, β are fermionic parameters.The Ramond punctures define divisors on the super Riemann surface [7], of dimen-sion 0 |
1. We can also glue along these divisors in a manner analogous to but moresubtly than in (2.15). In the case of a separating Ramond degeneration, the Ramondpunctures in S ℓ and S r are glued and smoothed out. In local coordinates x | θ on S l and y | ψ on S r the superconformal structures are defined by the vector fields ∂ θ + xθ∂ x and ∂ ψ + yψ∂ y respectively. The divisors corresponding to the Ramond punctures aredefined respectively by x = 0 and y = 0; and the gluing is [7] xy = q R ,θ = ζ ± √− ψ. (2.16)where q R is a bosonic gluing parameter. The free fermionic parameter ζ parametrizesthe fiber of a component of the fibration D sep. ; R , c.f., (2.6). Evidently, each Ramonddivisor has fermionic fiber C | . We would like to calculate amplitudes for the scattering of n + n ′ superstring states.These can be calculated by the correlation functions h V . . . V n + n ′ i of vertex operators V , . . . , V n + n ′ in worldsheet superconformal field theory. We use the RNS formalism withmanifest worldsheet supersymmetry to quantise the theory. The superstring worldsheetis manifestly a super Riemann surface and so the amplitude can also be calculated by7ntegrating over supermoduli space. To illustrate, consider a form ˆ F V ,...,V n + n ′ on thesupermoduli spaceˆ F V ,...,V n + n ′ = Z D ( X, B, C, ˜ B, ˜ C ) exp( − ˆ I ) n + n ′ Y i =1 V i ( p i ) , (2.17)where X denotes worldsheet matter fields which are also the spacetime coordinates, C, ˜ C denote worldsheet ghost fields, B, ˜ B denote worldsheet antighost fields, p i denote points,and ˆ I is the action of the worldsheet theory after gauge-fixing. In the simplest formalismwe use NS vertex operators of picture number −
1, and Ramond vertex operators ofpicture number − / g total contribution to the scattering amplitude is ˆ A = h V . . . V n + n ′ i . Weconsider ˆ A to be the summation of contributions from the bulk and from the boundaryof supermoduli space ˆ A bulk = ˆ A − A and ˆ A bndy = A respectively. The form restrictedto the boundary is F V ,...,V n + n ′ = ˆ F V ,...,V n + n ′ (cid:12)(cid:12) ∂ M g,n,n ′ . (2.18)The contribution to the amplitude, from the boundary of the supermoduli space, wherethe super Riemann surfaces degenerate, is given by A = Z ∂ M g,n,n ′ Z N F V ,...,V n + n ′ . (2.19)Here N is the fiber of the normal bundle to the boundary divisor, discussed in (2.11). In this section we analyse contributions to amplitudes from the boundary of supermod-uli spaces. We begin in Section 3.1 by presenting an overview of the Berezinian ofbundles over supermoduli space. We present Lemma 3.1 concerning a particular class ofBerezinians. These are relevant in the formulation of the super Mumford isomorphismsand forms which define the superstring measures. In Section 3.2 we investigate contribu-tions from the boundary of supermoduli space for a general genus. This involves firstlyconstructing a holomorphic map from a partial compactification of supermoduli spacein genus two; and secondly, using this map as a building block to extract the aforemen-tioned boundary contributions. In Section 3.3 we use the geometric model of gluing mapsto illustrate the pole behaviour of the super Mumford form near the boundary divisors.In Section 3.4 we overview the notion of integration on supermanifolds more generally,with the aim to apply these notions to compute superstring amplitudes through inte-gration over supermoduli spaces. We present a useful integration formula (3.53) whichwill be used. Section 3.5 serves as an example of these formalisms for the three loopvacuum amplitude. 8 .1 Super Mumford isomorphisms and forms
A supermanifold X is a space which is modelled on the data of a manifold X and avector bundle T ∗ X, − on X , thought of as the module of ‘odd’ differentials or ‘fermionicparameters’. The subscript + or − denote even or odd respectively, in this section.We can also think of T ∗ X, − as a locally free sheaf on X . The space X is itself referredto as the reduced space of X . The prototype supermanifold associated to ( X, T ∗ X, − ) isthe split model, which is the locally ringed space ( X, ∧ • T ∗ X, − ). Its dimension is definedby a pair ( p | q ) where p = dim X and q = rank T ∗ X, − . More generally, a supermanifold modelled on ( X, T ∗ X, − ) is a locally ringed space X = ( X, O X ) where O X is a sheaf of (local)supercommutative algebras on X , referred to as the structure sheaf. It is, additionally,required to be locally isomorphic to ∧ • T ∗ X, − as sheaves over X . The dimension of X coincides with the dimension of the split model. If X is isomorphic to the split model, itis split; otherwise, it is non-split. A more relevant and weaker condition for the purposesof physics is a (holomorphic) projection or fibration of X over its reduced space X . Aswe will discuss in sections to follow, the existence of a projection allows for the reductionof measures on superspace to measures on the reduced space where classical methods ofintegration can be applied. Note, if X is split, then it is projected; but not necessarilyconversely.Since the structure sheaf O X of a supermanifold X = ( X, O X ) is a sheaf of super-commutative algebras, it is globally Z -graded. Hence we can write O X = O X , + ⊕ O X , − as O X , + -modules. The tangent sheaf can be graded compatibly and so we have T X = T X , + ⊕ T X , − . (3.1)If J = O X , − · O X ⊂ O X denotes the fermionic ideal, then the structure sheaf of thereduced space X is O X = O X / J . If X is modelled on ( X, T ∗ X, − ) then T X , ± ∼ = T X, ± mod J ,where T X, + = T X is the tangent sheaf of X and T X, − are the ‘odd’ tangent vectors.Hence, if X is a supermanifold modelled on ( X, T ∗ X, − ) with tangent sheaf as in (3.1), itsBerezinian will then be given byBer X = Ber T ∗ X ∼ = ω X det T ∗ X, − = ω X ⊗ det T X, − mod J (3.2)where ω X = det T ∗ X, + is the canonical bundle of X . Here, forming the denominatormeans tensoring by its dual.A super Riemann surface of genus g is a (1 | S modelledon a Riemann surface C and a spin structure T ∗ C, − as its module of odd differentials. Spinstructures are also referred to as theta characteristics [26] in the language of algebraicgeometry. Recall that a line bundle on a Riemann surface is an ordinary spin structureif its quadratic tensor power is isomorphic to the canonical bundle. That the module ofodd differentials must be a spin structure follows from a more general characterisation:a super Riemann surface S is a supermanifold with C as its reduced space and a choice9f nowhere integrable distribution D ⊂ T S , where D is generated by the superconformalvector fields on the super Riemann surface. In denoting the superconformal vector fieldby D θ = ∂ θ + θw ( x ) ∂ x , see that D θ = w ( x ) ∂ x . In particular, the function w ( x ) has zerosalong the Ramond punctures where D θ = 0. We denote the divisors for Neveu-Schwarzor Ramond punctures on the super Riemann surface as P i or F i respectively in ourconvention, and their sum by P = P ni =1 P i , F = P n ′ i =1 F i . On a super Riemann surfacethen with n Neveu-Schwarz punctures and n ′ Ramond punctures, the distribution D sitsin a short exact sequence0 → D → T S → D ⊗ ⊗ O ( P + F ) → . (3.3)One might alternatively and equivalently denote super Riemann surfaces by the pair( S , D ) and define a generalised spin structure by D s def = D ∗ | C , following [18]. Here D is achoice of distribution on S , and D s is a generalised spin structure on the reduced space C . If S is a (1 | C , then everydistribution D ⊂ T S satisfying (3.3) is in bijective correspondence with generalised spinstructures D s on C .Consequently, viewing M g,n,n ′ as a supermanifold and using the description of tan-gent spaces of supermanifolds (modulo the fermionic ideal), at any isomorphism class[( S , D )] ∈ M g,n,n ′ we have: T M g,n,n ′ , + | [( S , D )] = H ( C, T C ⊗ O ( −P − F )) ,T M g,n,n ′ , − | [( S , D )] = H ( C, D s ∗ ) (3.4)where D s is the spin structure on C which is uniquely associated to the distribution D and D s ∗ is its dual.In (3.2) we have a general expression for the Berezinian of a supermanifold. When X = M g,n,n ′ is a supermoduli space, its tangent bundle is described in (3.4). Then bySerre duality we find, at a super Riemann surface class [( S , D )] with underlying Riemannsurface C , its cotangent space is given by: T ∗ M g,n,n ′ , + | [( S , D )] = H ( C, ω ⊗ C ⊗O ( P + F )) , T ∗ M g,n,n ′ , − | [( S , D )] = H ( C, ω C ⊗D s ) (3.5)where ω C is the canonical bundle on the curve C . Using (3.2) now, the fiber of theBerezinian of M g,n,n ′ at a point [( S , D )] on supermoduli space is:Ber M g,n,n ′ | [( S , D )] ∼ = H ( C, ω ⊗ C ⊗ O ( P + F )) H ( C, ω C ⊗ D s ) , (3.6)modulo the fermionic ideal in O M g,n,n ′ .The classical Mumford isomorphism between certain line bundles over the mod-uli space of Riemann surfaces can be generalised to isomorphisms between certain10erezinians over the supermoduli space M g . With L g a line bundle with fiber L g | [ C ] =det H ( C, ω C ), the bosonic Mumford isomorphism is L ng ∼ = L ⊗ (6 n − n +1) g . Over superRiemann surfaces we can define a line bundle or, more generally, a sheaf L / g with fiber L / g | [( S , D ) = det H ( C, ω C ⊗ D s ). In supergeometry, the Berezinian plays the role of thedeterminants and so taking Berezinians and tensor powers leads to the following. Lemma 3.1.
Over a genus g super Riemann surface class [( S , D )] with underlyingRiemann surface C , we can define a family of sheaves L n,mg with fiber L n,mg | [( S , D )] = Ber (cid:8) H ( C, ω ⊗ nC ) ⊕ H ( C, ( ω C ⊗ D s ) ⊗ m ) (cid:9) . (3.7) Then L n,mg ∼ = L ng ⊗ (cid:16) L m g (cid:17) ∗ . (3.8) Proof.
Since ( D s ) ⊗ ∼ = ω C , we can set D s = ω / C and thereby identify H ( C, ( ω C ⊗D s ) ⊗ m ) with L m g . Hence we have that L n,mg ∼ = L ng ⊗ ( L m/ g ) ∗ . (cid:3) We see that (3.6) is a particular case of (3.7) for ( n, m ) = (2 , M g = L , g . The Mumford isomorphism realises the line bundles L ng as a certain tensor powersof L g = L g . With 3 m/ m ′ + 1 / m ′ the integral part, we obtain fromthe classical Mumford isomorphism, L n,mg ∼ = L ng ⊗ (cid:16) L m ′ + g (cid:17) ∗ ∼ = L ⊗ (6 n − n +1) g ⊗ (cid:16) L ⊗ (6 m ′ − / g (cid:17) ∗ . (3.9)For ( n, m ) = (2 , M g ∼ = L ⊗ g ⊗ (cid:16) L ⊗ g (cid:17) ∗ ∼ = ( L / g ) . (3.10)This is the Mumford isomorphism for the Berezinian of supermoduli space, i.e., thesuper Mumford isomorphism. Now in (3.10) we see that the right-hand side is a tensorproduct of powers of the line bundle L g = L g and ( L g ) ∗ = L − g .There is a variant of this description. In [8] and [6], the authors use the Mumfordisomorphism over supermoduli space in a slightly different form to that in (3.10). Notefrom (3.4) that (cid:0) Ber T M g (cid:1) ∗ ∼ = L g ⊗ (cid:16) L g (cid:17) ∗ ∼ = L / g . (3.11)Hence we find Ber M g ∼ = ( L / g ) ∼ = ( (cid:0) Ber T M g (cid:1) ∗ ) = (cid:0) Ber T M g (cid:1) − . (3.12)11he isomorphism Ber M g ∼ = (cid:0) Ber T M g (cid:1) − in (3.12) is used by Witten [8] for superstringperturbation theory. We have seen that this also follows from our Lemma 3.1.To understand the relation to the super Mumford form, firstly observe that thesuper Mumford isomorphism in (3.12) is equivalent to requiring Ber T ∗ M g ⊗ (Ber T M g ) be holomorphically trivial, meaning Ber T ∗ M g ⊗ (Ber T M g ) ∼ = O M g . The genus g superMumford form, denoted Ψ g , is then a global section of this trivial bundle, i.e., thatΨ g ∈ Γ( M g , Ber T ∗ M g ⊗ (Ber T M g ) ). These isomorphisms and forms can be generalisedto the case where the super Riemann surfaces have n Neveu-Schwarz and n ′ Ramondpunctures, leading to super Mumford forms Ψ g,n,n ′ over M g,n,n ′ . The vacuum superstring amplitude at g loop order is an integral of the superstring mea-sure, which ought to be computed not over M g but its compactification M g . The total, g loop superstring amplitude therefore receives contributions from the bulk M g and fromthe boundary ∂ M g . Witten [21] looks at the boundary in genus g = 2 and observesthat contributions to D’Hoker and Phong’s derivation of the superstring amplitude totwo-loop order will be vanishing. This is consistent with the vanishing of the two-loopvacuum amplitude obtained by D’Hoker and Phong. In this paper we look at boundarycontributions to the three-loop and higher loop amplitudes. As discussed in previoussections, the boundary of supermoduli space has codimension (1 |
0) and parametrisessuper Riemann surfaces of lower genus. In particular while M g,n,n ′ , for a given ( g, n, n ′ )may have complications related to projectability of its bulk M g,n,n ′ , some of its lowergenus boundary components may nevertheless fiber over their reduced space with odddimensional fibers or holomorphically map to a bosonic space. Now for dimensionalreasons, M g is split for genus g = 0 ,
1, and hence projected, i.e., can be holomorphicallyfibered over its reduced space. In contrast, Donagi and Witten [19] showed that: M g,n,n ′ (and so also its compactification M g,n,n ′ ) will be non -projected for g − ≥ n + n ′ ≥ . (3.13)Note in particular that M , will be non-projected. Surprisingly however, M in fact is projected as ilustrated by D’Hoker and Phong in their computation of the superstringvacuum amplitude at two loop order.The projection M → SM → M constructed by D’Hoker and Phong usesthe identification of g = 2 super Riemann surfaces S with their period matrices Ω( S );and a formula relating the period matrix Ω( S ) to the period matrix of the underlyingRiemann surface C , denoted Ω( C ). In a particular gauge, termed split gauge , thisformula identifies Ω( S ) with Ω( C ), thereby leading to the holomorphic projection S 7→ Ω( S ) ≡ Ω( C ) C . Since M , and, by extension, M , are non-projected by (3.13),12n analogous procedure to that performed by D’Hoker and Phong will not result ina holomorphic projection. It results instead in a meromorphic projection M , →SM , . It is singular along the divisor parametrising NS degenerations. However, incomposing the projection π : M → SM with the forgetful map M , p → M ,we obtain a holomorphic map q , : M , p −→ M π −→ SM . (3.14)In this way, measures on M , can be reduced to measures on SM and, uponsumming over the even spin structures (GSO projection), to measures on M . Nowmore generally, if µ is an integration measure on M , , then along q , we have bythe pushforward formula q , ∗ (( q ∗ , f ) µ ) = f q , ∗ µ (3.15)for any function f on SM . Extending to the compactification leads to an integrationrelation: Z M , ;+ ( q ∗ , f ) µ = Z SM ;+ f q , ∗ µ (3.16)where now f must be compactly supported. Note that since the fiber of q , is (1 | q , ∗ will not coincide with thefamiliar Berezin-integration. Indeed, Berezin integration will reduce measures on M , to SM , along the meromorphic projection π , : M , → SM , . As a result,the Berezin integration of µ along π , will introduce singularities in the pushed-forwardmeasure π , ∗ µ .The integration formula in (3.16) involves the compactifications of supermoduli spaceand spin moduli space. Witten in [8, Sec. 5], observes however that the D’Hoker-Phongprojection π : M → SM does not extend to a holomorphic projection of thecompactification M . Indeed, along the divisor D NS ; sep. ; − parametrising separatingNS degenerations of type ( − , − ), π fails to both (1) be holomorphic along D NS ; sep. ; − ;and (2) to project D NS ; sep. ; − onto its reduced space. As a result, the extension to theboundary is generally meromorphic, so there exists a commutative diagram: M π (cid:15) (cid:15) ⊂ / / M π (cid:15) (cid:15) SM ⊂ / / SM (3.17)where π is the holomorphic D’Hoker-Phong projection, and π is a meromorphicextension to the compactification. There are however boundary components along which π will extend holomorphically. One such boundary component is the divisor D NS ; sep. ;+ parametrising separating NS degenerations of type (+ , +). It was noted by Witten in138, Sec. 5], that π will both (1) be holomorphic along D NS ; sep. ;+ ; and (2) project D NS ; sep. ;+ onto its reduced space. The relation in (3.16) will therefore be valid near thedivisor D NS ; sep. ;+ .Now recall that the vacuum amplitude at g -loop order is obtained by integrationover M g . A partial g -loop boundary contribution is then a contribution from the factor M , in the boundary. As explained above however, the holomorphic, D’Hoker-Phongprojection π : M → SM will extend generally to a meromorphic projectionupon full compactification; and as noted in the comments succeeding (3.17), π willbe holomorphic along the NS separating divisor D NS ; sep. ;+ . Hence we can look at the‘partial’ compactification M g ; NS ; sep. := M g ∪ D NS ; sep. ⊂ M g , (3.18)where super Riemann surfaces are only allowed to develop NS-nodes. For the even partwe have, M NS ; sep. ;+ := M ∪ D NS ; sep. ;+ ⊂ M . (3.19)The projection map M NS ; sep. ;+ π −→ SM , is holomorphic.More generally now, for any superspace Y with integration measure µ Y and holo-morphic map ρ : Y → M , (3.20)we can integrate ρ ∗ µ Y along the partial compactification M NS ; sep. ;+ as in (3.16). Theintegral of superstring measures along this partial compactification will be referred toas a partial g -loop contribution by using the genus two supermoduli space as a buildingblock, or factor.The boundary of M g , denoted ∂ M g , has codimension (1 | ν g to the embedding ∂ M g ⊂ M g has rank (1 |
0) and can locally be parametrisedby one even variable. The conormal bundle sequence to this embedding is0 −→ ν ∗ g −→ Ω M g | ∂ M g −→ Ω ∂ M g −→ . (3.21)Taking the Berezinian of (3.21) and using that ν ∗ g has rank-(1 |
0) therefore gives theisomorphism Ber M g | ∂ M g ∼ = ν ∗ g ⊗ Ber ∂ M g . (3.22)The isomorphism (3.22) relates measures on M g with measures on the boundary ∂ M g .With the characterisation of the boundary components in (2.4) and (2.6), the isomor-phism in (3.22) leads to ansatz factorisations for the superstring measure near specifiedboundary components, e.g., Ber ∂ M g | ∆ g ,g − g ∼ = Ber M g , ⊗ Ber M g − g , .The configuration space for the superstring vacuum states at g -loop order is M g .Along a NS separating divisor D NS ; sep. ⊂ ∂ M g with generic component in (2.4), observethat the number of punctures of each factor will always satisfy the inequality (3.13) when14 ≥
2. Hence D NS ; sep. cannot be projected for any g ≥
2. As in the case of stable curveshowever, the forgetful map from punctured supermoduli space M g, p → M g realises M g, as the universal family of genus g , super Riemann surfaces over M g . And so,forgetting the puncture yields a morphism D NS,sep. ∼ = M g , × M g , → M g × M g ,where g + g = g . If either of the factors M g or M g are projected, then measures on D NS,sep. can be pushed-forward and integrated over the projected factor M g or M g .In this way, components of the integration measure over D NS,sep. can be calculated byintegrating along the composite morphism D NS,sep. ∼ = M g , × M g , → M g or D NS,sep. ∼ = M g , × M g , → M g . (3.23)Now recall the morphism q , from (3.14). Extending it to the compactification resultsin a map, ¯ q , : M , p −→ M π −→ SM . (3.24)Specialising q , to the partial compactification along (+ , +)-separating NS nodes in(3.19) then gives a holomorphic map. In genus g ≥ g = 2 so that g = g − M , × M g − , ± α −−→ M g ; ± y p M y π SM (3.25)Note that the above diagram is compatible with both even and odd components of M g and M g − , respectively. In this way we obtain a holomorphic map D NS ; sep. → M .This is the special case of Y = D NS ; sep. in (3.20). In genus g ≥ g -loop vacuum contribution from the boundary component D NS ; sep. . In thecase of a non-separating, NS degeneration D NS ; nsep. ⊂ ∂ M g , recall from (2.4) that D NS ; nsep. ∼ = M g − , . The inequality in (3.13) holds for genus g ≥
4; while in genus g = 3, (3.13) does not hold. Nevertheless, in genus g = 3, projecting out the twoNS-punctures yields: D NS ; nsep. ;+ ∼ = −→ M , p −→ M . (3.26)As in the general case in (3.25) then, we can obtain a partial, three-loop boundarycontribution from the non-separating divisor D NS ; nsep. ;+ to the three-loop superstringvacuum amplitude.Recall from (2.6) that the divisors parametrising degenerations along Ramond punc-tures are fibered over supermoduli spaces with (0 | π R ; sep. ;+ and π R ; nsep. ;+ in (2.6) respectively. This allows for reducingthe integration measure to measures on supermoduli spaces. An analysis similar to thecase of NS-degenerations can be undertaken to evaluate partial three-loop contributionsalong the divisors parametrising degenerations along Ramond punctures.The below diagram is another example for a higher genus, namely g = 4: M , × M , α −−→ M y p × p M × M y π × π SM × SM (3.27)We will illustrate the example of g = 3 in Section 3.5. In some special cases in thecontext of g = 3 ,
4, both factors could be projectable as in above diagram (3.27).
In analogy with the Mumford relations on the moduli space of Riemann surfaces, thereare analogous relations between divisors on the supermoduli space, as illustrated in(3.12). Generalising these relations to the punctured case leads to a trivialising sectionof a certain tensor product involving canonical bundles, Ψ g,n,n ′ on M g,n,n ′ , referred to asthe super Mumford form. This global section Ψ g,n,n ′ defines a holomorphic measure on M g,n,n ′ and, in perturbative superstring theory, is taken to be the g loop, superstringmeasure with n + n ′ external states. Its integral over M g,n,n ′ gives the g loop scatteringamplitude. There are a number of issues surrounding the computation of this amplitudehowever, arising primarily from the complexity in the geometry of the supermoduli spaceitself.For instance, Ψ g, corresponds to vacuum amplitudes where there are no externalparticles; and Ψ g, corresponds to tadpole amplitudes or one-point amplitudes. Theform Ψ g, corresponds to two-point amplitudes or propagators of a string state. Thetadpole graphs are those with only one external state or external particle. The tadpoleamplitudes describe the amplitudes for a particle to disappear into the vacuum. Inunitary superstring theory, the tadpole amplitude for a massless stable particle is zero.An attractive feature of looking at the boundary of supermoduli space is that, asis clear from (2.4), (2.6), its components parametrise super Riemann surfaces of lowergenera. Furthermore, near the boundary, the superstring measure itself factorises intolower genus components, c.f., the discussions around (3.22). In genus g = g + g , we16ave the following α - and β -type clutchings such as M g ,n +1 ,n ′ × M g , ,n +1 ,n ′ α −→ M g,n,n ′ , M g − ,n +2 ,n ′ β −→ M g,n,n ′ . (3.28)The images of α and β form components of the boundary ∂ M g,n,n ′ which we denote ∆ α and ∆ β respectively. We use the geometric model of gluing maps to analyse boundarycontributions, via observations by Witten [7, 8]. Along a separating or non-separatingdegeneration respectively, with g = g + g , the super Mumford form satisfies the ansatzfactorisationΨ g,n,n ′ | ∆ α ∼ Ψ g ,n +1 ,n ′ F ( ε )Ψ g ,n +1 ,n ′ , Ψ g,n,n ′ | ∆ β ∼ Ψ g − ,n +2 ,n ′ G ( ε ) , (3.29)for some possible singular forms F ( ε ) and G ( ε ) depending on the degeneration parameter ε . The tilde sign ∼ means up to a multiplicative constant, which is a normalisationconstant in the definition of Ψ. By (3.29) then, the super Mumford form is expected tosatisfy relevant asymptotics near the boundary divisors. Integrating these asymptoticexpressions for Ψ g,n,n ′ along the boundary reveals how the g -loop amplitude will receivecontributions from the boundary of M g,n,n ′ .Away from the boundary divisor, the node is smoothed out into a thin cylinder. Thecloser to the boundary divisor in the supermoduli space, the thinner the cylinder is.The thin cylinder means a long distance propagation of a closed string state in stringtheory. From the worldsheet superconformal field theory point of view, gluing along theNS punctures as in (2.15) is equivalent to insertion of the form V ℓ ( a | α ) ⊗ V r ( b | β ) ε m d ε ,where ε is the gluing parameter and V ℓ and V r are operators inserted on the two sidesof the cylinder at points a | α and b | β . The conformal dimension of the operator is 1 + m for m ≥ − . The forms F ( ε ) and G ( ε ) are defined on the cylinder described by the degenerationparameter ε . A form with a general order m is given by the expression[d a | d α ] ε m d ε [d b | d β ] . (3.30)In the gluing relations (2.15), notice that ε is related to the square-root of the NS gluingparameter q NS . To investigate this relation further, we can consider a more generalchange of variable from ε to a new gluing parameter q as follows,( − q ) / = ε + ε p Cαβ (3.31)or equivalently q = − ε ( ε + 2 Cε p αβ ) with p a general real number. Integrating over theodd moduli α and β while fixing q then yields, Z [d a | d α ] ε m d ε [d b | d β ] ∼ Z [d α ]((d aCq ( m + p − d q d b ) αβ + . . . )[d β ] ∼ Z d a Cq ( m + p − d q d b . (3.32)17ere d a , d q , d b are independent integration variables. Accordingly, near a boundarycomponent ∆ j the super Mumford form factorises as follows,Ψ g,n,n ′ | ∆ j ∼ Ψ ∆ j d qq (2 − m − p ) (3.33)where Ψ ∆ j ∈ Γ(Ber T ∗ ∆ j ⊗ (Ber T ∆ j ) ) and Ψ g,n,n ′ ∈ Γ(Ber T ∗ M g,n,n ′ ⊗ (Ber T M g,n,n ′ ) )has a pole of a certain order on the boundary ∆ j ⊂ M g,n,n ′ . This formula (3.33) isderived from the gluing map of the geometric model by analytic methods, and is alsoa realisation of the factorisation in (3.22). The integration on the reduced space of thecylinder is Z d qq (2 − m − p ) ¯ q (2 − m − p ) (3.34)which would have a pole for m + p <
2. This physically means a propagator of a closedstring state on a cylinder. In fact, the Neveu-Schwarz (or Ramond) node is the limitwhen the circle winding the cylinder, for a propagating Neveu-Schwarz state (or Ramondstate), is shrinking to zero size.In the case where the boundary divisor parametrises separating NS degenerations of(+ , +)-type, Witten [7, 18] derived,Ψ g ; ± ∼ Ψ g ;+ [d a | d α ] d εε [d b | d β ]Ψ g ; ± . (3.35)This corresponds to m = − , p = 0. Note that there is always a Ψ g ;+ factor with +spin structure in the above factorisation. At a ( − , − ) degeneration [7, 18],Ψ g ;+ ∼ Ψ g ; − ( V ) [d a | d α ]d ε ε [d b | d β ] Ψ g − ( V ) , (3.36)where Ψ g i ; − ( V ) is computed by inserting the superconformal primary operator V ofdimension 5, and m = 8 , p = −
2. Note that this term scales as a positive powerof ε . After change of variable, this term scales as a positive power of q . Hence theintegration of this term (3.36) is vanishing in the limit of ε → q →
0, and so thecontribution to the g -loop amplitude from this boundary component is vanishing.For non-separating degeneration of NS type we have the factorisation,Ψ g ∼ Ψ g − [d a | d α ] d εε [d b | d β ] . (3.37)Changing variable from ε to q via ( − q ) / = ε + αβ [7] and integrating over α and β with fixed q gives, Z [d α ] ε − d ε [d β ] ∼ Z [d α ]( q − d q αβ + . . . )[d β ] = Z q − d q, (3.38)18nd hence: π ∗ Ψ g | im α g ,g ∼ Ψ d ag d qq Ψ d bg . (3.39) π ∗ Ψ g | im β g − ∼ Ψ d a d bg − d qq . (3.40)Here π ∗ denotes integrating out two fermionic moduli e.g. d α d β . Furthermore, we haveused forgetful morphsims in defining the forms appearing on the right hand side above,e.g., Ψ g , = p ∗ (Ψ g ). The form Ψ d ag is then the contraction of Ψ g along d a . Eq. (3.39),like (3.32), contains factors like d a d q d b with independent variables for integration.From the worldsheet superconformal field theory point of view [7, 18], gluing alongthe Ramond punctures as in (2.16) is equivalent to insertions of the form e ζG [d ζ ] d q R q R , (3.41)where q R and ζ are bosonic and fermionic gluing parameters and G is the zero-modeof the worldsheet supercurrent. The factor e ζG is due to the coupling of the worldsheetgravitino with the worldsheet supercurrent in the action. The d ζ is integrated over C | .From the point of view of the boundary divisor, the C | is the extra fermionic fiber ofthe boundary divisor with Ramond degeneration in the supermoduli space.For non-separating degeneration of Ramond type,Ψ g | im β g − ∼ X α i ,α j Ψ g − , ′ (Ξ α i , Ξ α j ) d q R q R e ζG [d ζ ] . (3.42)Here Ψ g − ′ (Ξ α , Ξ α ) is the super Mumford form for a genus g − n ′ = 2 Ramond punctures and with the superconformal primary operatorsΞ α , Ξ α inserted at these punctures [8]. Recall that in the notation we use primes todenote the Ramond puncture number.For separating degenerations of Ramond type near the boundary of the supermodulispace,Ψ g, ′ (Ξ α , Ξ α ) | im α g ,g ∼ X α i ,α j Ψ g , ′ (Ξ α , Ξ α i ) d q R q R e ζG [d ζ ]Ψ g , ′ (Ξ α , Ξ α j ) , (3.43)where g + g = g . Note that (3.43) is for the case when there are two external masslessstring states on the left-hand side of (3.43).The factorisation is valid near the degeneration, when | ε | or | q R | are small respec-tively, and the degeneration is the limit when ε → q R → .4 Integration in supergeometry and supermoduli spaces To calculate the superstring amplitude, integration on the supermoduli space is needed.This would involve firstly integrating out the odd (fermionic) coordinates, and thenintegrating the resulting measure over a classical moduli space. In this section, we firstdescribe integration of holomorphic measures. We then describe integration over smoothsupermanifolds and over complex supermanifolds. Then we present a useful integrationformula (3.53) which will be used in Section 3.5. Finally, we describe integral forms whichare also relevant to the question of integrating along the boundary of the supermodulispace.On smooth, real, orientable manifolds M there is, up to a positive constant, a naturaland unique volume measure ν M . This ν M is a global section of the line bundle of volumeforms det T ∗ M = Ω dim M ( M ) , (3.44)where Ω ( M ) = Γ( M, T ∗ M ) are the global sections of the cotangent vector bundleover M . Now we describe integration of holomorphic measures. Complex manifolds X are smooth manifolds equipped with a choice of integrable complex structure. Any n -dimensional complex manifold X will have an underlying 2 n -dimensional real manifoldwhich we denote by X ∞ . With J an integrable, complex structure we can identify X = ( X ∞ , J ). Volume forms on X can be integrated over M = X ∞ as a smoothreal manifold. We want to describe holomorphic volume measures however which comefrom the complex manifold X . This requires understanding the decomposition of thedifferential forms on X ∞ . Let T X ∞∗ denote the cotangent bundle. With respect to thecomplex structure J we have a decomposition into holomorphic and anti-holomorphicforms T X ∞∗ ∼ = T , X ∞∗ ⊕ T , X ∞∗ and hence a tensor product factorisation of thevolume forms on X ∞ . Now with T ∗ X the holomorphic cotangent bundle of X there isa natural inclusion T ∗ X ⊂ T , X ∞∗ ; and similarly an inclusion of the anti-holomorphiccotangent bundle T ∗ X ⊂ T , X ∞∗ . The inclusions are at the level of the sections ofvector bundles. Denote det X = det T ∗ X and det X = det T ∗ X . Using (3.44) we havetherefore an inclusion into the volume forms on X ∞ ,det X ⊗ det X ⊂ det T , X ∞∗ ⊗ det T , X ∞∗ ∼ = det X ∞ . (3.45)Sections of det X are referred to as holomorphic volume forms on X ; and similarlysections of det X are anti-holomorphic volume forms. Complex conjugation z z induces a conjugation on holomorphic forms. In particular, to any holomorphic volumeform ω ∈ det X we have the conjugate-squaringdet X −→ det X ⊗ det X, ω ω ⊗ ω. (3.46)Composing (3.45) with (3.46) gives a mapping between volume forms det X → det X ∞ .And so, with this mapping, we can integrate holomorphic functions against holomorphic20olume forms on X by simply integrating the resulting ( n, n )-form over the underlyingreal manifold X ∞ . Explicitly, for a holomorphic volume form ω on a complex manifold X , we have Z X ω def = Z X ∞ ω ⊗ ω. (3.47)Before turning to integration in supergeometry, we discuss a notion in algebraictopology serving to motivate subsequent notions in supergeometry. A vector bundle E over a manifold M is fibered over M with linear fibers. With π : E → M denoting thefibration, compactly supported forms on E can be formally reduced to forms on M viathe integration-along-fiber map π ∗ : Ω jcpct. ( E ) → Ω j − rank E ( M ). As explained by Bottand Tu [27], if x denote local coordinates on the base M and y coordinates on the fiber,then ( x, y ) will be local coordinates on E and π ∗ is defined on Ω jcpct. ( E ) by: π ∗ : ( π ∗ f )d x · · · d x m d y · · · d y m ′ (cid:26) f ( x ) d x · · · d x j − rank E if m ′ = rank E . (3.48)Intuitively, that π ∗ (d y · · · d y m ′ ) = 1 if m ′ = rank E and is zero otherwise. Berezin [1]defined integration over supermanifolds analogously to the integration-along-fiber mapabove. Crucially, this definition only makes sense if the supermanifold can be fiberedover its reduced space with purely odd fibers—a property known as ‘projected’. Thatany supermanifold can be smoothly projected over its reduced space is a consequenceof Batchelor’s splitting theorem [28]. Holomorphically however, there are generally ob-structions to fibering (or, projecting) complex supermanifolds over their reduced spaces.Hence, holomorphic measures over complex supermanifolds cannot generally be inte-grated in the way outlined by Berezin. In the case where complex supermanifolds X are non-projected , i.e., cannot be holomorphically fibered over their reduced space, it is anopen question as to how to integrate holomorphic measures over X . Donagi and Witten[19, 22] found that M g cannot be globally holomorphically projected onto its reducedspace for any genus greater or equal to five.To continue our discussion of the integration on supermanifolds now, there are twokinds of objects which can be integrated over supermanifolds. They are (1) Berezinianvolume forms; and (2) integral forms. The former are similar to volume forms onmanifolds as discussed above; and the latter are similar to distributions. We will firstlyconsider Berezinian volume forms.There is no unique ‘top form’ on a supermanifold since the differential of odd, orGrassmann, variables are no longer nilpotent. For example, if θ is odd, then d θ ∧ d θ = 0.One can nevertheless form the module of volume forms on a supermanifold analogous tothe determinant line bundle from (3.44). To any super vector space V = V ⊕ Π W , whereΠ W is the vector space of Grassmann variables with fermionic statistics, we can form itsBerezinian Ber V , which is a (1 | X its cotangent bundle T ∗ X is is a bundle of super vector spaces. It makes sense to21herefore set Ber X = Ber T ∗ X . To see how to integrate these volume forms, let | X | bethe reduced space of X . It is smooth manifold and embeds naturally inside X . Witha projection map π : X → | X | , the supermanifold X can be realised as fibered over itsreduced space with odd or ‘fermionic fiber’. Generalising the classical integration-along-fiber construction in differential topology, we can use π to integrate out the fermionicfibers to recover thereby a volume form on | X | . Denoting by π ∗ the integration-along-fiber map, we have therefore a morphism of sheaves π ∗ : Ber X → det | X | . With π thenwe can define, for any σ X ∈ Γ( X , Ber X ): Z X σ X def = Z | X | π ∗ σ X . (3.49)Now suppose X is endowed with a covering U = ( U α ) where each U α is isomorphicto ( | U α | , C ∞ ( | U α | ) ⊗ ∧ • R q ) where | U α | is the reduced space of U α . Denote the odddimension of X by q . Let F ∈ O X ( X ) be a global, smooth function. Then over U α we have F | U α ∈ O X ( U α ) ∼ = C ∞ ( | U α | ) ⊗ ∧ • R q . With the projection map π : X → | X | ,we can write F | U α = ( π ∗ g α ) ⊗ Θ α for some g α ∈ C ∞ ( | U α | ) and a Grassmann constantΘ α ∈ ∧ • R q . Note that this constant can be absorbed into the Berezinian volume form σ X ,α = σ X | U α . Finally now, in order to ensure the ultimate integral is well defined, choosea partition of unity ρ | U | subordinate to | U | and set f α = ρ α g α . For each index α , ρ α iscompactly supported in | U α | . Then over U α we have by (3.49), Z U α F | U α σ α = Z | X | f α π ∗ (Θ α σ X ,α ) . (3.50)We can integrate against any volume form σ ∈ Γ( X , Ber X ), Z X F σ X = X α Z U α F α σ X ,α . (3.51)By (3.49), the integration above only depends on a choice of projection π : X → | X | .We now describe integration over complex supermanifolds. Firstly, there are a num-ber of ways to define a complex supermanifold. For our purposes, a complex super-manifold Y is a supermanifold where: (1) the reduced space | Y | of Y is a complexmanifold and; (2) the restriction of the tangent bundle T Y to | Y | is holomorphic. Thetangent bundle of Y is Z -graded, so T Y ∼ = T + Y ⊕ T − Y . Restricting T Y to | Y | gives T Y | | Y | ∼ = T | Y | ⊕ N | Y | , where N | Y | → | Y | is a vector bundle. If | Y | is a complexmanifold, then T | Y | will automatically be holomorphic. Condition (2) then amounts torequiring N | Y | also be holomorphic. As in the case of complex manifolds, for complexsupermanifolds there will be an underlying smooth supermanifold which we denote by Y ∞ . Our conventions here are such that the complex structure is only defined by ref-erence to the even coordinates. And so, if ( z | θ ) denote local coordinates on Y , their22onjugation is ] ( z | θ ) = ( e z | θ ). As explained by Witten in [31], the conjugate e z coincideswith the familiar complex conjugate z on the reduced space | Y ∞ | . That is, along theembedding | Y ∞ | ⊂ Y ∞ we have ] ( z |
0) = ( z | e Y only differs from Y in that the reduced space is con-jugate, i.e, | Y | = | Y | . Where the odd tangent bundle is concerned however, we have T − e Y = T − Y .Now let π : Y → | Y | be a holomorphic projection. On Berezinian volume forms itdefines the mapping, ( f + gθ )[d x | d θ ] π ∗ g dx , for x denoting a complex, even variable x and θ the odd variable. For multiple even variables x , ..., x m and odd variables θ , ..., θ n , θ · · · θ n [d x · · · d x m | d θ · · · d θ n ] π ∗ d x · · · d x m . (3.52)Then as in the smooth case, π realises Y as being holomorphically fibered over itsreduced space with fermionic fibers. With Ber Y the space of holomorphic volumeforms on Y , and det | Y | the holomorphic volume forms on the reduced space | Y | , theintegration-along-fiber map gives a relation π ∗ : Ber Y → det | Y | . Recall that by ourconventions here, we only conjugate the even parameters. As such, and since the evenand odd parameters are globally distinguished on Y , there is no need to implementan operation as in (3.46) for volume forms on Y directly. We can instead defer thisoperation to the reduced space | Y | . Therefore, for a volume form Ψ ∈ Γ( Y , Ber Y ) anda holomorphic projection π : Y → | Y | , we can define: Z Y Ψ def = Z | Y | π ∗ Ψ ⊗ π ∗ Ψ . (3.53)Note that the right-hand side above does not make any reference to the underlying,smooth supermanifold Y ∞ . The procedure for calculating π ∗ Ψ however is similar tothat for volume forms on smooth supermanifolds X since π here is holomorphic. Thiscan be viewed as being in analogy with the integration of holomorphic volume forms in(3.47). In Section 3.5, we use the definition (3.53) extensively in forming our integrations.The projection map π : X → | X | fibering a smooth supermanifold over its reducedspace always exists, albeit non-canonically so. This is in contrast to the complex casewhere holomorphic projections π : Y → | Y | need not exist. The formula (3.53) issuitable only in the case where Y is such that π exists as a holomorphic map. Asmentioned earlier, when Y = M g is the supermoduli space of genus g curves, Donagi andWitten in [19] illustrated precisely this: that a holomorphic projection π : M g → | M g | does not exist when g ≥ integral forms .The integration of integral forms on supermanfolds was discussed in [29, 30, 31]. Aparticularly appealing feature of the codimension-one integral forms lies in their relationto a generalised Stokes’ Theorem. 23uppose X is a supermanifold with boundary ∂ X and de Rham differential d. Stokes’Theorem asserts that any codimension-one, compactly supported integral form ν satisfies Z X d ν = Z ∂ X ν. (3.54)Specialising to supermoduli space then, for any codimension-one integral form ν on M g,n,n ′ , we have by (3.54) that R M g g ,n n ′ ,n n ′ d ν = R ∂ M g g ,n n ′ ,n n ′ ν . By (2.3),the images of α and β form components of the boundary ∂ M g,n,n ′ . Therefore, Z M g,n,n ′ d ν = Z ∂ M g,n,n ′ ν = Z im α ν + Z im β ν. (3.55)In this way we see how codimension-one, integral forms will receive contributions fromboundary divisors. Integral forms can also be useful in describing other observables onthe boundary of the supermoduli space, such as anomalies. As an illustration of the formalisms in the previous sections, we analyse the three loopvacuum amplitudes in detail in this section. We use the factorisation of super Mumfordforms Ψ near the boundary of the supermoduli space to analyse the contribution tothe vacuum amplitude at genus three, from the boundary of the supermoduli space,and consider the cases of NS and Ramond nodes at the degeneration. The supermodulispace of the super Riemann surfaces is denoted by M g , while the spin moduli space ofthe Riemann surfaces is denoted by SM g , and the moduli space of Riemann surfaces isdenoted by M g .In this section, g = g + g = 3. Hence the clutching maps describing the degenera-tions are: M , × M , α , −→ M , . (3.56) M , × M , α , −→ M , . (3.57) M , β −→ M , . (3.58)Now recall that in genus g = 2, D’Hoker and Phong constructed a holomorphic pro-jection π , + : M , + → SM , + . This projection can be extended to a meromorphicmapping π , : M , → SM , which will be meromorphic on the compactification.To retain holomorphy however we can, as in (3.24), specialise the morphism ¯ q , : M , p −→ M π −→ SM to the partial compactification formed by allowing (+ , +)24S nodes. With the clutching maps above, we obtain a diagram M , × M , α −−→ M y p M × M , y π × π , SM × SM , (3.59)The D’Hoker-Phong projection π , + : M , + → SM , + is defined by sending a genus g = 2 super Riemann surface with prescribed period matrix to a genus g = 2 Riemannsurface with the same period matrix. This mapping generalises to define mappings¯ q , × π , in the diagram above and a projection π : M → SM . However, wedo not consider π here since it is meromorphic. As described in Section 3.3, nearthe boundary of supermoduli space the super Mumford form admits a factorisation asfollows: Ψ | im α , ∼ Ψ , F ( ε )Ψ , . (3.60)Ψ | im α , ∼ Ψ , F ( ε )Ψ , . (3.61)Ψ | im β ∼ Ψ , G ( ε ) . (3.62)Here, the ε is the degeneration parameter near the boundary, and Ψ , = p ∗ (Ψ ) for p the forgetful morphism. Note that while π ∗ (Ψ ) will be singular, π , ∗ (Ψ , ) will not besingular and π ∗ (Ψ ) will be non-singular along the (+ , +) separating NS divisor. Asa result, factors such as q , ∗ (Ψ , ) can be calculated via the D’Hoker-Phong method.In terms of the gluing parameters ε and local coordinates near the punctures a | α and b | β , we have more explicitly:Ψ | im α , ∼ Ψ [d a | d α ] d εε [d b | d β ]Ψ . (3.63)Ψ | im α , ∼ Ψ [d a | d α ] d εε [d b | d β ]Ψ . (3.64)Ψ | im β ∼ Ψ [d a | d α ] d εε [d b | d β ] . (3.65)Changing variable from ε to q as (3.38) and integrating over α and β with fixed q thengives: Ψ | im α , ∼ Ψ , d qq Ψ , . (3.66)Ψ | im α , ∼ Ψ , d qq Ψ , . (3.67)Ψ | im β ∼ Ψ , d qq . (3.68)25ote, we have used forgetful morphisms in defining the above forms, e.g., Ψ , = p ∗ Ψ .For non-separating degeneration of Ramond type,Ψ | im β ∼ X α , α Ψ , ′ (Ξ α , Ξ α ) e ζG [d ζ ] d q R q R (3.69)where Ψ , ′ (Ξ α , Ξ α ) is the super Mumford form for a genus 2 super Riemann surfacewith n ′ = 2 Ramond punctures and with superconformal primary operators Ξ α , Ξ α inserted at these punctures. The terms q R and ζ are bosonic and fermionic gluingparameters.The bosonic gluing parameters ε or q R can be viewed as the bosonic coordinate ofthe fiber of the normal bundle of the boundary component, in the separating or non-separating cases respectively, c.f., (2.11).In the case of an odd spin structure, the superstring vacuum amplitude is zero, sinceone needs the insertion of operators to absorb the ten fermionic zero-modes of the RNSfermions and for the vacuum amplitude there is no such operator insertions on the genusthree surface. Hence for the three loop vacuum amplitude, it suffices to consider genusthree surfaces with even spin structures.The contribution to the superstring amplitude from the boundary of the supermodulispace is the integration as in (2.19), A := Z ∂ M g Z N F g . (3.70)We shall denote by N the reduced space of the fiber N of the normal bundle to theboundary divisor.For general g and g with g + g = g = 3 then, we can use the factorisation of thesuper Mumford form from (3.66)–(3.68) to get, A = Z M g , × M g , Z N Ψ g , d qq Ψ g , . (3.71)It is known for g ≤
2, massless tadpole graphs, i.e. one-point functions, all vanish intype II superstring theory with unbroken spacetime supersymmetry [20, 17, 7]. That is,e.g., Z M g , Ψ g , = 0 . (3.72)Therefore, since the factor in (3.71) contains a tadpole graph, the amplitude, i.e. thefull integral (3.71) will vanish. Now in the expression (3.71), recall that Ψ g i , = p ∗ (Ψ g i ),where p : M g i, → M g i is the forgetful morphism. Pushing the measure forward under p then gives R M gi, p ∗ (Ψ g i ) = R M gi Ψ g i , c.f., (3.16). The amplitude in (3.71) for separating26egenerations therefore reduces to, A = Z M g × M g Z N Ψ g d qq Ψ g . (3.73)For non-separating degenerations of the NS type we have, A = Z M g − Z N Ψ g − d qq , (3.74)where we have used R M g − , Ψ g − , = R M g − , p ∗ p ∗ (Ψ g − ).For non-separating degenerations of Ramond type, A = Z M g − , ′ Z N Z C | X α ,α Ψ g − , ′ (Ξ α , Ξ α ) e ζG [ dζ ] d q R q R . (3.75)We have the factorisation by the factor R N R C | e ζG [d ζ ] d q R q R . This factor can be holo-morphically projected to its reduced space. The integration of q R will be as follows.With an infrared regulator ǫ , R N d q R q R ¯ q R ∼ R ǫ ≤| q R | d q R q R ¯ q R ∼ ln ǫ could have had a ln ǫ in-frared divergence, however, the other prefactor would be vanishing due to summationsof superconformal operators in the presence of unbroken spacetime supersymmetry.When g , g −
1, or g ≤
2, the projections π , π , and π , to the reduced spaces(which we denote generally by π ) are holomorphic, as we discussed in Section 3.2.Note that, for instance, π , being holomorphic implies that π , is also holomorphic.Hence we can use an alternative method by holomorphic projection of one of the factorsinvolving the lower genus supermoduli space and use the integration formula (3.53), aswe describe as follows. Here N is the reduced space of the fiber N of the normal bundleof the boundary divisor. Because one of the components of the Riemann surface is atadpole graph, and the factor of the tadpole graph vanishes, we have e.g., Z M g π ∗ Ψ g π ∗ Ψ g Z N d qq ¯ q = 0 . (3.76)With an infrared regulator ǫ , R N d qq ¯ q ∼ R ǫ ≤| q | d qq ¯ q ∼ ǫ could have had a ǫ infrareddivergence, but due to the vanishing of the prefactor which is a vanishing tadpole graphas in Eq. (3.72), the full integral (3.76) will be vanishing. For the clutching of thesecond type, with NS degeneration, A = Z M g − Z N π ∗ Ψ g − π ∗ Ψ g − d qq ¯ q . (3.77)D’Hoker and Phong have shown that the two-loop two-point function for massless NSsector vanish, implying that R M g − π ∗ Ψ g − π ∗ Ψ g − = 0. Hence the full integral (3.77) isvanishing. 27e also mention that in the case g = 2, g = 1, we can project both factors. Thenby the integration formula (3.53), A = Z M g ×M g Z N π ∗ Ψ g π ∗ Ψ g d qq ¯ q π ∗ Ψ g π ∗ Ψ g , (3.78)which also shows the vanishing.There are two methods of computations above. We could first factorise and thenproject on the reduced space of one of the factors. For some graphs, we could also firstfactorise and then project both factors. The two methods are different ways of goingalong the arrows in the diagram (3.59) as we illustrated.Our analysis shows that the boundary contribution to the three-loop vacuum am-plitude, from the boundary of the supermoduli space will vanish in closed oriented typeII superstring theory with unbroken spacetime supersymmetry. Furthermore, we knowthat the vacuum amplitudes at genus zero, one and two are also vanishing [20, 17, 7], inthe closed oriented type II superstring theory, with unbroken spacetime supersymmetry.Here, what we have analysed is the boundary contribution to the superstring amplitude,not the bulk contribution.There is an analysis of the superstring amplitude at three loop order from the bulkof the bosonic moduli space using modular forms [32, 33, 34, 35, 36]. These resultsare compatible with our analysis. However the approach there is not derived fromsupermoduli space or from manifest supersymmetry. It is therefore not obvious how todirectly relate their bosonic ansatz with the approach here through supermoduli space.By similar calculations as presented in this section, we can deduce the followingremark. Remark 3.1.
In closed oriented type II superstring theory in spacetime backgroundswith unbroken supersymmetry, the contribution to massless tadpole graphs, i.e. one-point functions, from the boundary of the supermoduli space, is vanishing at three-looporder.
The idea behind the above remark is, since the Ramond puncture number are alwayseven, one only need to add one additional NS puncture on one component of the superRiemann surface with lower genus. Again one uses the factorisations. One could alsouse the forgetful morphism associated to that NS puncture, which would be equivalentto the formalism of integrated NS vertex operator.
One of the main goals in this paper is to obtain an understanding of scattering ampli-tudes in perturbative superstring theory. We focussed in particular on contributions tothe superstring amplitude from the boundary of supermoduli space, which in turn can28e described by clutching morphisms on supermoduli spaces of generally lower genera.The physical interpretation of these clutching morphisms is that they are related totaking an infrared or large distance limit of superstring amplitudes. Fundamentally,the superstring amplitudes can be calculated by integrating the superstring measure,which itself defines a measure on supermoduli space. Following earlier work in [6, 8], wediscussed how this measure could be constructed from the generalisation to supermodulispace of the classical Mumford isomorphisms on the moduli space of Riemann surfaces.The superstring measure, or super Mumford form, is a measure defined on super-moduli space. It is holomorphic on the bulk but, in the compactification, acquires polesnear some components of the boundary. We can understand this pole behaviour bylooking at how the super Mumford form would factorise into forms over lower generasupermoduli spaces along specified boundary components.In genus two, and so at two loop order, D’Hoker and Phong calculated the superstringamplitude by integrating the superstring measure over the genus two supermoduli space.This integration involved a projection of supermoduli space onto its reduced, bosonicspace, where more classical integration methods could then be applied. In genus three,i.e., at three loop order, it was not clear how to apply D’Hoker and Phong’s method ofcalculation since it is not known whether there exists a holomorphic projection of thegenus three supermoduli space onto its reduced, bosonic space. We observe however thatsince the boundary in genus three parametrises super Riemann surfaces of lower genera,some of these components may at least admit a holomorphic map to a bosonic space.Considering only those boundary components which admit such a map led to the notionof a partial compactification. Following observations by Witten in [8] we note that thegenus two boundary component parametrising (++) Neveu-Schwarz degenerations of agenus three super Riemann surface admits precisely such a map.To state our result more clearly, our analysis shows that the boundary contribu-tion to the three loop vacuum amplitude, from the boundary of the supermoduli space,will vanish in closed, oriented, Type II superstring theory with unbroken spacetime su-persymmetry. It also implies that upon compactification to four spacetime dimensionspreserving the supersymmetry, the boundary contribution to the cosmological constantat three loop level is also zero in Type II superstring with unbroken spacetime super-symmetry.Furthermore, our observations are compatible with the results on superstring am-plitudes at three loop order in [32, 33, 34, 35, 36] which are obtained using ansatz formodular forms on bosonic moduli space. It is an open and interesting problem as to howto understand the bulk, or interior, contribution to the three loop vacuum amplitudefrom the viewpoint of supermoduli space. Moreover, it would also be desirable to un-derstand the relation between the approach from supermoduli space and from modularforms in the calculation of superstring amplitudes.In theories with unbroken supersymmetry the vacuum energy is vanishing since con-tributions from bosons and fermions cancel each other exactly. It would also be good to29nderstand the vanishing of boundary contributions from the point of view of possiblenonrenormalisation theorems [37, 38] at three loop, as well as from the point of viewof, and relations to, other superstring formalisms such as the Green-Schwarz formalismand the pure spinor formalism.The boundary contributions to the superstring amplitude can also be viewed ascorrection terms to calculations of superstring amplitudes over the bulk. Correctionterms in genus three may then be relevant for heterotic string theory. Calculations inheterotic string theory are more subtle however, as they involve embedded integrationcycles in products of moduli spaces M g ; L × M g ; R parametrising left and right movers[39, 40]. Boundary contributions, or correction terms, may also be relevant in thecalculation of amplitudes for open strings in type II string theory, where there is only onefactor of the chiral measure. In the α ′ → U (1) symmetry [42, 43, 44]. As a consequence, itgenerates a nonzero mass term at one-loop for tree-level massless scalars charged un-der the anomalous U (1). It also induces a nonzero dilaton tadpole and nonzero vacuumenergy [40, 39] at two-loop. There is tree-level spacetime supersymmetry but it is break-ing at one-loop. The aspect of nonzero tadpole amplitudes and associated spacetimesupersymmetry breaking by string loop effects is a fundamental idea in superstring the-ory. A similar degeneration of super Riemann surfaces where there are infinitely thincylinders can also occur, e.g., as in [44], and so are related to the boundary of super-moduli spaces. As such, these models are particularly relevant for approaches involvingsupermoduli space and its boundary.In this paper we have constrained our analysis to Type II superstring theory. Theanalysis for heterotic SO (32) string theory is more involved [39, 40]. In that case, forcertain orbifold backgrounds with tree-level supersymmetry, spacetime supersymmetryis broken at one-loop and there is non-zero two-loop vacuum amplitude [40, 39]. Thisis mainly due to the breaking of supersymmetry by string loop effects. One maindifference is that in the type II theory with unbroken spacetime supersymmetry, wehave zero tadpole amplitudes for stable massless particles.Factorisations of two loop superstring amplitudes also occur in [45] in the type IIcase, which is useful for checking S-duality of type II superstring theory. The techniquesof [45] are also useful in computing the normalisation factor for the vacuum energy.The S-duality covariance and factorisation constraints would be a good consistencycheck for computing the amplitudes and the effective action in Type II superstring and30upergravity theory. Acknowledgments
The work was supported in part by Yau Mathematical Sciences Center and TsinghuaUniversity, and by grant TH-533310008 of Tsinghua University (to H.L.).
A Compactification of moduli space and spin mod-uli space
In this appendix, in order to be self-contained and for the convenience of the reader, webriefly overview the compactification of the moduli space of ordinary Riemann surfacesand the compactification of the spin moduli space of spin Rieman surfaces. Note thatwe use the term Riemann surface interchangeably with (complex) curve.The moduli space M g parametrises families of smooth curves X over a base B . Thenon-compactness of M g follows from the property that such families a non-completebase B cannot generally be extended to smooth families of curves over the completion b B , even up to finitely many base-changes. Remarkably however, if one allows familiesof curves to have at-worst nodal singularities, then any family of smooth curves X → B can be extended to a family of stable curves b X → b B with at-worst nodal singularities. Astable curve C has a nodal singularity at a point p ∈ C if, in any local, affine coordinatesystem ( x, y ) at p that xy = 0 at p . The boundary divisor of the compactification ∂ M g ⊂M g consists of degenerations of two distinct kinds: separating and non-separating . If D sep. ⊂ ∂ M g parametrises separating degenerations, then a generic component of D sep. is isomorphic to the product M g , × M g , where g + g = g . If D nsep. ⊂ ∂ M g parametrises non-separating degenerations, then D nsep. ∼ = M g − , . More generally, with M g,n the Deligne-Mumford compactification of n -pointed, stable, genus g curves, itsboundary ∂ M g,n will also parametrise separating and non-separating degenerations.With D sep. ⊂ ∂ M g,n parametrising the separating degenerations, a generic component isisomorphic to the product M g ,n +1 ×M g ,n +1 where n + n = n . With D nsep. ⊂ ∂ M g,n parametrising the non-separating degenerations, D nsep. ∼ = M g − ,n +2 . The morphisms onboundary components induced by the inclusion ∂ M g,n ⊂ M g,n , M g ,n +1 × M g ,n +1 α −→ M g,n and M g − ,n +2 β −→ M g,n (A.1)are referred to as α - and β - clutchings respectively.A spin curve is a smooth curve C equipped with a choice of spin structure L C , whichis a line bundle over C satisfying L ⊗ C ∼ = ω C for ω C the canonical bundle. The notionof stable, pointed curves C can be generalised to stable pointed spin curves ( C, L C ).31ere, L C restricts to a spin structure L C sm. on the smooth locus of the stable curve C sm. ⊂ C , and vanishes along the nodes of C . Blowing up along the nodes { p i } ∈ C results in a smooth curve π : e C → C . Off the exceptional divisor, e C \ π − ( { p i } ) isisomorphic to the smooth locus C sm. . As such, the spin structure L C sm. on C sm. pullsback to e C \ π − ( { p i } ). The exceptional divisor E = π − ( { p i } ) ⊂ e C is isomorphic toa product of P C ’s and so the spin structure L C sm. can be continued along E by gluingin the standard spin structure O P C ( −
1) on P C . Cornalba’s compactification SM g,n isholomorphically fibered over M g,n with boundary components comprising clutchings, inanalogy with (A.1). In contrast to M g however, the spin structures L C are themselvesendowed with parity: L C is even or odd if h ( C, L C ) ≡ h ( C, L C ) ≡ SM g = SM + g ∪ SM − g . Inthe compactification of SM g , one needs therefore to account for the parity of the spinstructure in the separating degenerations at a node. The analogue of the α -clutchingsin (A.1) making up the boundary components ∂ SM g,n ⊂ SM + g,n are therefore: SM + g ,n +1 × SM + g ,n +1 α ++ −→ SM + g,n and SM − g ,n +1 × SM − g ,n +1 α −− −→ SM + g,n (A.2)where g + g = g and n + n = n . Similarly, the β -clutching is given by: SM + g − , β + −→ SM + g,n . (A.3)Since clutchings are natural in families, both the α - and β -clutchings on the spin modulispace and the moduli space respectively are compatible with the holomorphic fibration SM g → M g . References [1] F. A. 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