The Super Period Matrix with Ramond Punctures in the supergravity formulation

In a very recent preprint, Witten showed how to construct a $g|r \, \times \, g|r$ super period matrix for super Riemann surfaces of genus $g$ with $2r$ Ramond punctures, which is symmetric in the ${\bf Z}_2$ graded sense. He also showed how it can be applied to analyze supersymmetry breaking in string compactifications which are supersymmetric at tree-level. Witten's construction is in the purely holomorphic formulation of super Riemann surfaces. In this paper, a construction is given in the formulation of two-dimensional supergravity. The variations of the super period matrix with respect to supermoduli deformations are also given, as well as an explicit illustration of how the super period matrix with two Ramond punctures would emerge from a degeneration of the super period matrix without punctures in higher genus.


Introduction
Such formulas for super Riemann surfaces with Neveu-Schwarz punctures were instrumental in the derivation of the measure [3,15] and scattering amplitudes [5,16] for Neveu-Schwarz states. Their counterparts for surfaces with Ramond punctures can be expected to play a similar role in future investigations of Ramond states scattering.
The paper is organized as follows. In §2, we begin by reviewing the supergravity approach to super Riemann surfaces. We explain the turning on of the gravitino field χ and, for the purpose of subsequent publications on the evaluation of Ramond amplitudes, the simultaneous turning on of a Beltrami differential µ, the result of which is to parametrize the fiber of super Riemann surfaces with Ramond punctures over a given reduced surface Σ red with Ramond punctures. We also develop the basic function theory which is needed in the presence of Ramond punctures, namely explicit formulas for the holomorphic sections of the bundle R −1 and construction of an anti-symmetric Szegö kernel. In §3, we construct a basis of both even and odd superholomorphic forms of U(1) weight 1/2 using the function theory developed in §2. The key issue in the construction of a super period matrix is the prescription of a canonical normalization that would single out a basis, the periods of which would then define the entries of the super period matrix. This is one of the problems solved by Witten in [14], and a brief summary of his prescription is given in §3.3. Witten's prescription is then implemented in the supergravity formalism in §3. 4. The resulting periods are worked out in §3.5-7. In §3.8, we illustrate, in a basic case for Ramond punctures, the relation between superholomorphic forms in the supergravity formalism and holomorphic forms on the underlying reduced surface. Such a relation played an essential role in the calculations of [15]. In §3.9, we discuss briefly an additional set of periods, already known to Witten [17] but not discussed in [14], which are the ones obtained by considering the line integrals of the normalized basis of superholomorphic 1/2 forms between pairs of Ramond punctures. In §4, we derive the first order variations of all the periods of the superholomorphic 1/2 forms under a deformation of super complex structures. These formulas are needed in the forthcoming work on the derivation of scattering amplitudes for Ramond states. Finally, in §5, we show how the super period matrix with 2 Ramond punctures in genus g arises in the non-separating degeneration limit of the super period matrix of a super Riemann surface in genus g + 1 without R-punctures. Appendices A and B contain various technical definitions and detailed derivations.

Supergravity formulation with Ramond punctures
The supergravity Ramond-Neveu-Schwarz formulation of superstring perturbation theory is based on two-dimensional supergravity, which governs the field theory on the worldsheets of the strings. The fields consist of the metric g and the gravitino field χ ± , as well as of the fields x µ , ψ µ ± which describe respectively the bosonic and fermionic string degrees of freedom with µ = 1, · · · , D in a D-dimensional space-time. The fields χ ± , ψ µ ± are worldsheet spinors, and the index ± labels the two chiralities of the spinors. On a surface of genus g, the spinor fields χ ± , ψ µ ± may carry any one of the 2 2g inequivalent spin structures.
In the chiral splitting approach [18,19], the starting point for the construction of amplitudes for closed oriented superstrings is the functional integral over the fields g, χ ± , x µ , ψ µ ± on worldsheets of genus g. In the critical dimension D = 10 of space-time, gauge symmetries lead to the familiar ghost fields and reduce the integration over g and χ at any given loop order g to a finite-dimensional integral over even and odd moduli of the surface. The Chiral Splitting Theorem [19] guarantees that this functional integral, at fixed spin structure, fixed internal loop momenta, and fixed metric g and gravitino χ ± is the absolute value squared of a chiral amplitude. The chiral amplitude of positive chirality, for example, depends only on the fields χ + and on holomorphic deformations of the metric, which we parametrize by the Beltrami differential µ. Type II amplitudes are obtained by pairing independent chiral amplitudes of opposite chirality, while Heterotic string amplitudes are obtained by pairing a chiral amplitude of positive chirality with a suitable bosonic amplitude, in both cases supplemented with a suitable GSO summation over spin structures.
The chiral splitting theorem produces a set of effective field theory rules for the construction of chiral amplitudes at fixed spin structure and fixed internal loop momenta directly in terms of chiral fields x µ + , ψ µ + and ghosts b, c, β, γ as a function of the chiral data χ + and µ only, without appealing to the complex conjugate fields. These effective rules may be reformulated in superspace on the worldsheet, thus promoting the worldsheet into a super Riemann surface. Their moduli space may be viewed as the equivalence classes (χ + , µ) under the action of diffeomorphisms and local supersymmetry transformations. This effective supergravity formulation is now closely related with the purely holomorphic approach to super Riemann surfaces and their moduli space, as formulated in section 3.5.2 of [10], and both approaches will produce the same chiral amplitudes.
The supergravity approach leads to a convenient construction of chiral superstring amplitudes in which the χ and µ fields are paired respectively against the worldsheet supercurrent and stress tensor. It was used at two-loop order in the calculation of the measure [3,15,20] and scattering amplitudes [5] of NS states (see also [13] for a review). The holomorphic formulation of super Riemann surfaces and their moduli space was used in [21] to give a completely novel and very elegant derivation of the superstring measure at genus 2.

Super Riemann surfaces with Ramond punctures
In the present paper, we shall be interested in using the supergravity and chiral splitting approach to describe super Riemann surfaces with Ramond punctures (or R-punctures for short). The starting point will be a reduced Riemann surface Σ red corresponding to a super Riemann surface Σ for which all odd coordinates (on the surface and on super moduli space) have been set to zero. The space of super Riemann surfaces may then be described in terms of the function theory on the ordinary Riemann surface Σ red by turning on the worldsheet gravitino field χ along with deformations µ of the metric.
In superstring theory, R-punctures are distinguishable since they will support fermion vertex operators with distinguishable quantum numbers. An R-puncture may be formulated in an infinite number of possible pictures [1]. As stressed in [10], on a super Riemann surface Σ of arbitrary genus, an R-puncture in the -1/2 picture corresponds to a Ramond divisor on Σ (namely a submanifold of Σ of dimension 0|1), while an R-puncture in the -3/2 picture corresponds to a specific point in the Ramond divisor. Lower picture numbers −1/2 − k with k ≥ 2 are allowed as well, but will not be considered here.
On a compact super Riemann surface, the number of R-punctures is an even number 2r where r is a positive integer. Since an R-puncture is associated with a quadratic branch point it is natural to partition the set of all R-punctures into pairs (p η , q η ) for η = 1, · · · , r, each pair representing a quadratic branch cut. It is natural and it will be useful to define the following sums of Ramond divisors, The partitioning of R-punctures into pairs also has a natural origin when surfaces with R-punctures are obtained by non-separating degeneration from surfaces of higher genus, including from surfaces of genus g + r without R-punctures, as will be discussed in §5 below.
Since the reduced surface Σ red is the starting point for the construction of a super Riemann surface Σ in the supergravity approach, we shall give a description of the deformation fields µ and χ and associated symmetries from the point of view of the reduced surface Σ red . The generators v of infinitesimal diffeomorphisms in the presence of a Ramond divisor F, must leave F invariant and therefore be smooth sections of the line bundle K −1 ⊗ O(−F) over Σ red . Throughout, K will denote the canonical bundle over Σ red . Similarly, the generators ξ of infinitesimal local supersymmetries must be sections of a line bundle R over Σ red which may be defined by the isomorphism, along with a generalized spin structure needed to specify the square root uniquely.
The transformation properties under diffeomorphisms v and local supersymmetries ξ of the fields µ and χ (henceforth, we shall denote the positive chirality component χ + simply by χ) on Σ red dictate the bundles over Σ red of which µ and χ should be sections, We recall that the action of an infinitesimal diffeomorphism v on µ is given by δ v µ = ∂zv, while the action of a local supersymmetry transformation ξ on µ and χ is given by, In either picture -1/2 or -3/2, the divisor F is transformed into itself. The difference between the two pictures is that the generators of local supersymmetries in the −1/2 picture are allowed to take arbitrary values on F so that supersymmetry acts by an automorphism on F, while they are required to vanish on F in the −3/2 picture, so that the individual points in F are left invariant.

Function theory on the reduced surface Σ red
Our goal is to construct moduli for super Riemann surfaces Σ of genus g with 2r R-punctures. For this, we need to develop some basic function theory on the underlying reduced surface Σ red , which is a Riemann surface of genus g with 2r R-punctures obtained by setting to zero all anti-commuting parameters in Σ. We refer the reader to Appendix A for a brief review of standard function theory on compact surfaces without punctures. We shall concentrate here in the body of the text on the new features that arise in the presence of R-punctures. The surface Σ red has genus g and 2r R-punctures partitioned into pairs (p η , q η ) with η = 1, · · · , r. As usual (see Appendix A), we fix a canonical homology basis (A I , B I ) of H 1 (Σ red , Z). The canonical bundle K of Σ red has dim H 0 (Σ red , K) = g holomorphic sections. A canonical basis ω I of these holomorphic sections is determined by the requirement, (2.5) The g × g period matrix Ω IJ of Σ red is defined by, The matrix Ω is symmetric with positive definite imaginary part. Along with the positions of the 2r R-punctures (p η , q η ), it provides the moduli for the surface Σ red . The Jacobian is defined by J(Σ red ) = C g /(Z g + ΩZ g ).
To complete the basic function theory on Σ red in the presence of R-punctures, we shall also need the Abel map and the prime form, which are multiple-valued on the surface Σ red . In order to define them properly, we cut the surface Σ red along the homology cycles A I , B I for I = 1, · · · , g and parametrize the surface by a simply connected domain Σ cut (see Fig. 1 for g = 2). We choose a reference point z 0 from which the Abel map may be defined by the integral z z 0 ω I as a single-valued function throughout Σ cut . The prime form E(z, w) may be similarly defined, as is done explicitly in Appendix A. Part of the function theory on Σ red that we shall need in the presence of R-punctures will be holomorphic and meromorphic sections of the bundles R −1 and K ⊗ R. Generically, on the moduli space for Σ red with 2r Ramond punctures, we have, It is shown in [14] that these relations hold throughout moduli space when r ≥ g. To obtain the corresponding holomorphic and meromorphic sections in terms of ϑ-functions, it will be necessary to consider the Abel map evaluated on Ramond divisors, such as, for example, The factors of 1 2 arise in taking the square root of R −2 . Using a description of the surface Σ red in terms of the simply connected cut domain Σ cut , the range of the punctures p η and q η is restricted to Σ cut . To describe the bundle R −1 completely and uniquely, including the specification of its generalized spin structure, the Abel map needs to supplemented by one of the 2 2g half-integer characteristics, which we shall denote by δ below. It is in this sense that δ, for a given choice of the cut domain Σ cut , labels all the generalized spin structures.
Explicit expressions for the r holomorphic sections of R −1 will be obtained in the remaining parts of this section. Since the case of two R-punctures is illuminating by itself, we shall treat it first, before moving on to the case of general r ≥ 1.

Spinors and propagators for 2 Ramond punctures
The explicit formulas for the holomorphic and meromorphic sections of R −1 are most transparent in the simplest case of two R-punctures, denoted p and q, with F = p + q, and we shall discuss this case first. The holomorphic section h(z) for generic moduli is given by, (2.9) The expression (2.9) should be understood as giving the section h(z) on Σ red \ F. The expression for h(z) in a coordinate chart centered at either one of the punctures p or q is obtained by multiplying (2.9) by the suitable transition function for the bundle R −1 . In particular, in such a chart, h(z) is indeed given by a holomorphic function near p or q.
The expression (2.9) shows explicitly that h(z) is non-trivial only for generic moduli, such that ϑ[δ](z − 1 2 p − 1 2 q) does not vanish identically in z, or equivalently, that δ − 1 2 p − 1 2 q not be a special divisor. It is also instructive to determine how h(z) transforms as p is moved around an arbitrary full period 2ε. Because of the presence of p/2 in the argument of the ϑ-function in (2.9), the monodromy of the Abel map as p is moved around an arbitrary full period 2ε will produce a shift by an arbitrary half-period ε in the argument of the ϑ-function. This will map the holomorphic section h with generalized spin structure δ to a holomorphic section h with generalized spin structure δ + ε thus producing the transitive action on all spin structures, even and odd, as reviewed in §4.2.4 of [10].
The other ingredient that we need is a worldsheet fermion propagator, or Szegö kernel, S(z, w) for spinors on a surface with R-punctures. It is a meromorphic section of R −1 z ⊗ R −1 w , and is characterized by the following conditions ∂zS(z, w) = 2πδ(z, w), (2.10) together with the anti-symmetry requirement S(z, w) = −S(w, z). (2.11) Note that the condition (2.10) alone cannot completely specify S(z, w), because of the existence of the holomorphic spinor h(z). To construct S(z, w), we begin by considering the propagator S pq (z, w), also valued in R −1 z ⊗ R −1 w and defined explicitly by The propagator S pq (z, w) satisfies the condition (2.10), together with The propagator S pq (z, w) is not anti-symmetric in z and w, and we have instead As a result, the unique Szegö kernel S(z, w) which is anti-symmetric under the interchange of its arguments is defined by Equivalently, this unique Szegö kernel may be obtained by anti-symmetrizing S pq (z, w), Its only pole is at z = w, with unit residue so that S(z, w) = 1/(z − w) + O(z − w).

Field theoretic interpretation
The preceding function theory on Σ red can be expressed in terms of fermion fields ψ andψ valued in R −1 and spin fields Σ ± . Let the OPE relations of these fields be given bȳ The OPEs ψ(z)ψ(w) andψ(z)ψ(w) are O(z −w) while the OPEs ψ(z)Σ + (w) andψ(z)Σ − (w) are of order O(z − w) 1 2 , neither of which will be needed in the sequel. Then we have The OPE relations guarantee that the relations (2.13) hold automatically. The field theoretic formulation of the anti-symmetric S is obtained by combining (2.16) with (2.18).

The general case of 2r Ramond punctures
We extend all the preceding concepts to the general case of 2r R-punctures. Recall that the 2r punctures may be grouped into pairs (p η , q η ) with η = 1, · · · , r.

Holomorphic sections
For generic moduli, the space H 0 (Σ red , R −1 ) has dimension r. An explicit basis for its sections is given by the following formula, valid on Σ red \ F, Its field theoretic interpretation is The normalization is arranged so that the monodromy of h p η (z), as the R-punctures p η , q η are moved around Σ red , is just factors of ±1 or ±i. The normalizations of the OPE relation and of the behavior of h p η (z) in z at the point p η implied by (2.19) precisely match, and we have (z − p η ) 1 2 h p η (z) → δ ηη as z → p η . It is natural and useful to introduce also a conjugate basis of holomorphic sections of R −1 by effecting the interchange of p η ↔ q η for all η. We shall denote these sections by h q η (z). There exists a linear relation between the h p η and h q η , by an invertible matrix Λ, In terms of Λ, we may now collect all the limits we shall need, We shall show below by indirect arguments that the matrix Λ satisfies Λ t Λ = −I, so that the change of basis h p ↔ h q is indeed guaranteed to be invertible.

The Szegö kernel
Next, we construct Szegö kernels satisfying the same conditions (2.10) and (2.11) as in the simplest case of 2 R-punctures. Again, the condition (2.10) is unaffected by the addition of an arbitrary linear combination of h p η (z)h p γ (w) to S(z, w). As in the case of 2 R-punctures, we begin by considering the following Szegö kernel, By construction, we have, S pq (p η , w) = S pq (z, q η ) = 0 η = 1, · · · , r (2.24) On Σ red \ F, the Szegö kernel is given by the following explicit formula, Since the Szegö kernel S pq (z, w) is not anti-symmetric in z, w, we consider its symmetric part S pq (z, w) + S pq (w, z). This sum is a holomorphic section of R −1 z ⊗ R −1 w and may be written in two equivalent ways, either in the basis generated by h p η , or in the basis generated by h q η , By construction, the matrices H p ηη and H q ηη are symmetric in η, η . They may be determined using the relations of (2.24), namely S pq (p η , w) = 0 on the first line, and S pq (z, q η ) = 0 on the second line, along with the following limits which may be read off from the explicit expressions given for S pq , h p η , and h q η , Applying these limits to both sides of (2.26) and using (2.27), we obtain, a result which is in agreement with the case r = 1 computed previously. Using the second equality in (2.26), along with the equations of (2.28) and the linear relation between h p η and h q η in (2.21), we deduce the previously announced result, Returning to the construction of an anti-symmetric Szegö kernel, we introduce the following family S M of candidate Szegö kernels, indexed by an r × r matrix M of parameters, We have the following reflection symmetry relation, where M t is the transpose of M . A Szegö kernel satisfying (2.10) and (2.11) can then be obtained by setting M to be any anti-symmetric matrix. Up to this point, the antisymmetric matrix M is arbitrary. But we shall see later that a unique choice is dictated by the requirement that the ensuing super period matrix be symmetric in the Z 2 -graded sense.

Superholomorphic Forms and their Periods
The key ingredients in the construction of the period matrix for an ordinary Riemann surface were the holomorphic 1-forms ω I which are holomorphic sections of the canonical bundle K. Analogously, in the supergravity formulation, the key ingredient for the super period matrix will be the superholomorphic 1/2 forms [18,19]. A formal approach to the super period matrix based more on algebraic geometry may be found in [22]. A useful review of the subject may be found in [10]. In this section, we shall give a description and explicit construction of these forms for super Riemann surfaces of general genus g and with an arbitrary number 2r of R-punctures. We shall then use those forms to derive even and odd periods.

Differential equations for superholomorphic 1/2 forms
We shall be interested in superholomorphic formsω of U (1) weight 1/2 and their periods for a given super Riemann surface defined in the supergravity formulation by the equivalence class (g, χ). The condition for superholomorphicity reads D −ω = 0, where D − is the analog of the Cauchy-Riemann operator for the super Riemann surface Σ (see for example [18]). We introduce a system of local complex coordinates (z,z) associated with the complex structure given by the metric g, and an odd coordinate θ which should be viewed as a section of R. The formsω may be decomposedω(z, θ) =ω + (z) + θω z (z) in componentsω + and ω z which are sections of R −1 and K respectively. The condition for superholomorphicity D −ω = 0 is then given by the following component equations, 1 As a result, the 1-form dzω z − 1 2 dz χω + is closed, and we can define the period ofω around a closed cycle C by We shall see that in the presence of R-punctures, superholomorphic forms of weight 1/2 exist for both even or odd grading. The componentsω z in the odd forms are even, and their periods along closed contours C are even moduli. The componentsω z in the even forms are odd, and their periods along closed contours C are odd moduli.
Considering now the generalized situation in which we view a super Riemann surface as a deformation, parametrized by (χ, µ) of a reduced Riemann surface Σ red with metric g, we may write down also the equations for superholomorphicity D −ω = 0 of a formω in this context. We now introduce a system of local complex coordinates (z,z) associated with the metric g of Σ red , and carry out the deformation of complex structures by µ on equations (3.1) by replacing ∂z → ∂z + µ∂ z on scalars, with analogous modifications for spinors and tensors. The D −ω = 0 equations then read as follows, and the periods of a superholomorphic formω receive a µ-dependent correction, The equations of (3.3) are invariant under conformal transformations withω + ,ω z , χ, and µ transforming respectively with weights (1/2, 0), (1, 0), (−1/2, 1), and (−1, 1). They are also invariant under local supersymmetry transformations with parameter ξ, provided χ and µ are transformed as in (2.4) and the components ofω transform as follows,

A basis of superholomorphic forms of U(1) weight 1/2
Using the function theory on Σ red described in §2, we can easily write down a basis (ω M I |ω M η ) of superholomorphic forms of U(1) weight 1/2. The upper index M refers to the choice of propagators S M (z, w) in the expressions below. First, since χ and µ are nilpotent, it is advantageous to rewrite the equation (3.3) for superholomorphicity as an integral equation which can be solved after a finite number of iterations, The contributions + (z) and z (z) are respectively any holomorphic half-form and any holomorphic one-form with respect to the complex structure for Σ red . In the above integral equations, we have further used the inverse of ∂z on one-forms and half-forms. The inverse of the operator ∂z on 1-forms is proportional to ω(z, w) = ∂ z ∂ w ln E(z, w). The inverse of the operator ∂z on sections of R −1 is given by the Szegö kernel S M (z, w) in the presence of R-punctures described in §2. Different choices of matrices M in the Szegö kernel S M (z, w) will result in different choices of basis vectors for the same space of superholomorphic forms.
For the odd superholomorphic 1/2 forms, we choose z = ω I and + = 0, where ω I is the basis of holomorphic forms for Σ red , with the normalization (2.5), and we obtain the following basis of g odd superholomorphic 1/2 forms, The superscript M shows the dependence of the forms on the as yet unspecified matrix M .
For the even superholomorphic 1/2 forms, we choose z = 0 and + = h N η , where h N η is an as yet unspecified linear combination of the holomorphic section h p η of R −1 given in §2. This linear combination may be expressed in terms of an as yet unspecified r × r matrix N , The resulting even superholomorphic 1/2 forms will be denotedρ M N η thereby exhibiting their dependence on the matrices M, N . They are given bŷ We have obtained in this way a g|r-dimensional basis of superholomorphic forms (ω M I |ρ M N η ), with I = 1, · · · , g and η = 1, · · · , r, generated from their values at χ = 0, µ = 0 by a finite number of iterations when χ and µ are generated by a finite number of odd moduli.
The evaluation of the periods of the basis (ω M I |ρ M N η ) is a bit subtle, because the expressions in (3.7) involve integrals that are only conditionally convergent. We shall discuss their evaluation in detail below in §3.7, but for the moment, we shall quote the result that the basis does have the expected A-periods, has not yet been fixed, but will determine the matrix N uniquely.

Witten's basis and Super Period Matrix
A key insight of [14] is that the residual arbitrariness in the basis of superholomorphic forms requires a proper normalization, defined by introducing so-called "fermionic periods". Once this proper normalization is satisfied, the resulting matrix of periods will have all the desirable properties, including symmetry with respect to the natural Z 2 grading.
The fermionic periods of a superholomorphic form can be constructed as follows. Let ω(z, θ) =ω + (z) + θω z (z) denote any even or odd superholomorphic form of U (1) weight 1/2. The definite integral ofω between two points (z, θ z ) and (w, θ w ) in Σ is given by (see e.g. eq. (5.13) of [19]). It is natural to introduce an associated 1-form following [10], The form κ is closed since its components satisfy (3.3). Next, use local complex coordinates (z|θ) near the R-puncture p η for which z = 0 at p η , and in which the superconformal structure is defined by the distribution generated by The odd period w p η (ω) of the closed form κ which is related to the superholomorphic 1/2 formω at the puncture p η is defined in [14] by The fermionic period w p η (ω) must be independent of θ since dκ = 0. The purpose of the normalization factor N is to assure symmetry of the super period matrix, in the Z 2 graded sense. Similarly, one defines the fermionic period w q η at the puncture q η . The fermionic periods w A η (ω) and w B η (ω) (denoted w η (κ) andw η (κ) in [14]) can now be defined by Witten's normalization prescription for a basis of closed holomorphic 1-forms (σ I |ν η ) with I = 1, · · · , g and η = 1, · · · , r for a super Riemann surface with genus g and 2r punctures is then given as follows, Witten's g|r × g|r super period matrix can now be defined by 2 with the entries given by It was shown in [14] that the normalization conditions (3.16) can be implemented in the complement of a divisor in the reduced space of M g,0,2r and that, using the Riemann relations, they imply symmetry in the Z 2 graded sense of the matrixΩ so that its components satisfŷ Ω IJ =Ω JI ,Ω Iη =Ω ηI , andΩ ηζ = −Ω ζη .

Witten's basis in the supergravity formalism
We shall now show how to implement Witten's normalization explicitly in terms of the superholomorphic 1/2 forms constructed in §3.2. The issue is to understand and evaluate the change of bases, Recall that the basis (ω M I |ρ M N η ) depends on the choice of Szegö kernel S M (z, w) for r > 1, a choice which is parametrized by the matrix M , as well as on the choice of the basis h N η as a function of h p η of (3.8), a choice which is parametrized by the matrix N . It will turn out that the matching of the basis (ω M I ) ↔ (σ I ) of odd forms corresponds to a unique choice of the anti-symmetric matrix M in S M (z, w). Once M has been determined, the matching of the basis (ρ M N η ) ↔ (ν η ) of even forms is obtained by enforcing the fermionic A-periods of ρ M N η and will uniquely determine the matrix N .

The case of 2 Ramond punctures
In this case, there is a unique Szegö kernel S(z, w) which is anti-symmetric, so the basis (ω I |ρ 0 ) obtained in §3.2 is canonical, up to an overall normalization ofρ 0 . We shall indeed show that it essentially coincides with Witten's basis (σ I |ν 0 ) where N is the normalization given in (3.14). Here and below when discussing the r = 1 case, we shall use the notation 0 for the index η in order to clearly distinguish it from the index I which takes values I = 1, · · · , g.
The 1-form ν 0 corresponds to a superholomorphic 1/2 form proportional toρ 0 , since both are required to have vanishing A I -periods. The constant of proportionality may be computed from the explicit expression for the components ofρ 0 in (3.9), the normalization of h in (2.9), and the normalization w A (ν 0 ) = 1.
To evaluate w A (ρ 0 ), we must evaluate w p (ρ 0 ) and w q (ρ 0 ). Near p we use coordinates (z|θ) in which D * θ takes the canonical form (3.13). The formulas for h(z) and S pq (z, w) derived in (2.9) and (2.12) respectively are expressed instead in terms of coordinates (z|θ) in which z − p = 0 at p, and in which the distribution takes the form D * θ = ∂θ +θ∂ z away from F. The map between the coordinate systems is provided byθ p = (z − p) 1 2 θ p . Similarly near q, the coordinates map viaθ q = (z − q) 1 2 θ q . This map allows us to work out the odd periods w p (ρ 0 ), since near p the corresponding 1-form is given by We then have To compute these limits, we evaluate the contribution from h(z) as well as the contribution from the Szegö kernel S(z, w) of (2.15). The first, near p and near q, is given by, The factor √ −1 arises from the square root involving the prime forms as z → q in (2.9). The signs have been chosen for consistency with the sign conventions of (3.16).
It remains to evaluate the limits for the Szegö kernel S(z, w). This is straightforward with the help of the vanishing conditions (2.13), namely S pq (p, w) = S pq (z, q) = 0, and the relation (2.15). We find, Substituting these results into (3.9) for η = 0, we find, As a result, we may identify the one form √ 2 N ν 0 with the superholomorphic 1/2 formρ 0 . Next, we compare σ I andω I . We note that their bosonic periods on A I -cycles coincide. Hence, the normalizations of σ I andω I differ at most by the addition a multiple ofρ 0 tô ω I . Since allω I with these normalizations differ by multiples ofρ 0 , we shall evaluate the fermionic periods ofω I . To do so, we now need the limits, It will be convenient to relate these expressions to the bosonic B-periods of ν 0 , namely, which may be evaluated with the help of the formula (A.10) for the B I -period of the Abelian differential of the second kind ω(z, w). By substituting the limits for the Szegö kernel of (3.24) into (3.7). we find, As a result, the fermionic periods are given as follows, Proper Z 2 graded symmetry of the super period, which includes the relationΩ I0 =Ω 0I requires us to identify the √ −1 in the definition of N with the factor of i in the first line integral of (A.10). The basis of superholomorphic formsω I obeys Witten's normalization conditions, and therefore must agree with the basis of one forms σ I .

The general case of 2r Ramond punctures
For general numbers 2r of R-punctures, the odd and even superholomorphic 1/2 forms are given respectively in (3.7) and (3.9). They have been normalized only partially, and still involve a dependence on the matrices M and N , whose values have not yet been prescribed. A change in the assignment of M will induce a change of basis onρ M N η without admixture of the formsω I , while onω I the change will act by adding a linear combination ofρ M N η . A change in N will only affectρ M N η by a change of basis. We will show below that, generically, there exists a unique preferred choice of M and N , such that The change of basis is invertible if and only if the matrix (I + √ −1 Λ) is invertible, where the matrix Λ was defined in (2.21). Thus, in the supergravity formalism, the divisor in the moduli space M g,0,2r outside of which Witten's normalization can be implemented is described by the equation To prove the above statements, we consider first the normalization of the formsω M I . These forms have already been normalized on bosonic A-periods. The fermionic periods ofω M I involve the Szegö kernel S M used to construct these forms in (3.7). We shall now show that, generically, there exists a unique choice of anti-symmetric matrix M which guarantees the vanishing of the fermionic A-periods ofω M I . To this end, we compute the limits of the Szegö kernel for arbitrary M , using the definition (2.30) and the relations (2.24), and we find, We now form the combinations of these limits that enter into the fermionic periods w A η and w B η , with respective signs + For the w A η -period the sign is + While not manifest from the expression (3.34), the matrix M 0 is seen to be anti-symmetric with the help of the relation Λ t Λ = −I. In summary, we have w A (ω M 0 I ) = 0, so that the identification σ I ↔ω M 0 I is complete. It remains to normalize the fermionic A-periods ofρ M 0 N ζ . To this end, we again use (3.33) for the w A η period with the sign + √ −1, and we find Generically, Witten's normalization of (3.16) may be achieved by setting N = N 0 where In summary, we have w A η (ρ M 0 N 0 ζ ) = δ ηζ , so that the identification ν η ↔ρ M 0 N 0 η is complete. Henceforth, we shall make the choices M = M 0 and N = N 0 and drop the corresponding superscripts on the differentials and the superholomorphic 1/2 forms, and simply set The choices for both M 0 and N 0 are regular provided the matrix (I + √ −1 Λ) is invertible everywhere in the reduced moduli space with R-punctures.
Recall that the forms (ω I |ρ η ) were given explicitly by the expressions (3.7) and (3.9), but these expressions involve conditionally convergent integrals, and the evaluation of their periods has to be carried out with some care. The following formula for the interchange of conditionally convergent integrals was established in [19], eq. (5.21). Let ψ = dz ψz be an arbitrary form of type (0, 1), then we have, For the integrals of the components of the formω I of interest here, the first term on the above right hand side compensates precisely the second term on the right hand side of the expression (3.4) defining periods of superholomorphic forms. Using this result, we find that the integrals of the formsω I along any closed cycle C is given by, The three integrals we shall need here may be evaluated with the help of (A.10), Here, ω yx (u) is the third kind Abelian differential with simple poles in u at y and x with respective residues +1 and −1 and vanishing A-periods (see Appendix A). Using these careful definitions of the conditionally convergent integrals needed to evaluate the bosonic periods, we readily confirm the normalizations ofω I andρ η on the bosonic A-periods. The bosonic B-periods may be evaluated in the same way, and the results will be exhibited in §3.7 below.
3.6 Calculation of the fermionic periods of (ω I |ρ η ) We begin by recalling the necessary ingredients to compute the fermionic periods involving the Szegö kernel S(z, w) which was defined in terms of the matrix M = M 0 in (3.37). Given that M 0 satisfies (3.34), formula (3.33) simplifies to give, where h η and N were defined respectively in (3.8) and (3.14). Using the result on the first line, we readily confirm the normalizations of the fermionic A-periods ofω I andρ η . Using the result on the second line for the calculation of the fermionic w B η periods ofω I , we find, Comparison with the expression forΩ Iη , we find the following equality, which is a manifestation of the Riemann bilinear relations for super Riemann surfaces, as has been explained by Witten. The equality is easy to verify to first order in χ and zero-th order in µ, and may be proven in general by iterated solution of the integral equations for the components ofω I andρ η . The fermionic B-periods w B η (ρ ζ ) may be evaluated by the same methods, and will be exhibited in §3.7 below.

The super period matrix in the supergravity formalism
In view of the identification (σ I |ν η ) ↔ (ω I |ρ η ), the entries of the super period matrix are given by the periods of the latter bŷ As was shown by Witten in [14], the super period matrix is symmetric in the Z 2 -graded sense, so that its components satisfy the relationsΩ JI =Ω IJ ,Ω Iη =Ω ηI , andΩ ηζ = −Ω ζη . Collecting the results derived above, we find the following expressions for its components, Recall that the matrix M 0 is given explicitly by (3.34) in terms of the matrix Λ, which is itself defined in (2.21), so everything in the above formulas is explicit. The validity of the Riemann relations may also be verified on these expressions using the iterated solutions of the integral equations forω I andρ η of (3.7) and (3.9) respectively. For example, if we assume that µ = O(χ 2 ), then the lowest order non-trivial corrections are given as follows, The Z 2 -graded symmetry is now manifest. The expression for the pseudo-classical blockΩ IJ is formally identical to its expression in the absence of R-punctures upon substitution of the Szegö kernel with R-punctures.

Complex structure on Σ red and superholomorphicity
The parametrization of super Riemann surfaces Σ used in §2 and §3 starts from the reduced surface Σ red supplemented with a gravitino field χ and a Beltrami differential µ. The period matrix Ω IJ of Σ red and the pseudoclassical blockΩ IJ of the super period matrix do not, in general, agree. In fact, for genus g ≥ 4, and arbitrary gravitino field χ, there will, in general, be no choice of µ that can achieveΩ IJ = Ω IJ since the ordinary period matrix Ω IJ must satisfy the Schottky relations but, in general,Ω IJ will not. For g ≤ 3, however, there are no Schottky relations, and for arbitrary Ω IJ and χ, it will always be possible to find a Beltrami differential µ such thatΩ IJ = Ω IJ . In fact, the condition to be enforced on µ may be read off from (3.45), and is given by, for all I, J = 1, · · · , g. In turn, this condition implies that the (0, 1) form µω Iw + 1 2 χω I+ must be in the range of the ∂w operator acting on functions, so that we have, for some g-component function λ I . The functions λ I may be reconstructed explicitly from this equation, up to an additive constant, and we have, The addition to ω uv of a linear combination of holomorphic forms ω I is immaterial in view of (3.47). This property guarantees that λ I (u) are well-defined and single-valued on Σ red . The function λ I provides a powerful connection relating the superholomorphic structure given by the 1/2 formsω I of the super Riemann surface Σ to the complex structure given by the 1-forms ω I of the reduced surface Σ red . To see this, we first establish the following relation, by inspection of (3.7),ω The correspondence may be expressed directly in superspace language, and we have, It is now immediate from these expressions that the period matricesΩ IJ and Ω IJ coincide, since the differentials on which they are built differ just by a total derivative. Relations of the form (3.50) were instrumental in the study of NS scattering amplitudes [23]. They can be expected to play a similar role in the study of Ramond scattering amplitudes.

Additional Periods
In addition to the periods of super holomorphic 1/2 forms reviewed earlier in this section, it is natural to consider further periods obtained as line integrals of superholomorphic 1/2 forms between pairs of R-punctures. 3 In particular, such periods emerge automatically upon the non-separating degeneration of a super Riemann surface when the degenerating cycle carries a spin structure associated with a Ramond state, as will be shown in §5. The additional periods may be either even or odd, and their description will depend upon the picture chosen to describe Ramond states. In picture -1/2, R-punctures are characterized by a Ramond divisor whose fiber is of dimension 0|1 at each puncture. Supersymmetry transformations act by automorphism on the fiber but leave the divisor invariant.
The additional periods may have practical applications to parametrize the moduli space of super Riemann surfaces with a small number of R-punctures. The reason is that the superperiod matrix (Ω IJ ,Ω Iη ,Ω ηζ ) does not, for low r, have a sufficient number of independent parameters to account for all the even and odd moduli of M g,0,2r . For example, the number of odd entriesΩ Iη equals gr while the number of odd moduli in picture -1/2 is 2g − 2 + r. For g ≥ 2, the super period marix provides enough odd moduli only when r ≥ 2. Similarly, the entries ofΩ ηζ provide enough even moduli to account for the 2r R-punctures only when r ≥ 5. Thus, for surfaces of low r, it may be convenient to have additional periods available.
We shall now provide a brief account of the additional periods for general genus g but restrict to the simplest case r = 1 which is perhaps the most important one. We denote the two R-punctures by p, q. To facilitate the description of the additional periods, it will be convenient to specify not just the divisors p, q, but also auxiliary points θ p and θ q in each one of the 0|1 fibers. In the -1/2 picture, the variables θ p , θ q transform under supersymmetry by automorphism of the fiber, and should therefore not be thought of as moduli.
The additional even periodsV I and odd periodV 0 are given as follows, They may be evaluated in components using (3.7) and (3.9) giving the even periods, and the odd period, Here, ω pq (z) is the third kind Abelian differential with vanishing A-periods, as reviewed in (B.6). It is manifest that the additional periodsV I andV 0 have non-trivial monodromy as p and q are taken around non-trivial homology cycles on the surface, and we have, The dependence on the fermionic periods in (3.53) and (3.54) may be rendered explicit using the results of (3.26) and (3.25). The additional even periods become, while the additional odd period becomes, Here, we have found it natural to introduce the notations, Under the supersymmetry transformations of (2.4) on χ and µ, the additional periodsV I ,V 0 are invariant provided the points θ p , θ q in the 0|1-dimensional fibers transform as follows, We note that the additional periods are invariant under diffeomorphisms that leave the R-punctures invariant.
In genus 2 with two R-punctures, it is useful to consider also deformations µ which fix at the same time the additional even periodsV I introduced in (3.52).

Variations of Periods
In the absence of R-punctures the genus g = 2 super period matrix provides a holomorphic projection of the moduli space M 2,0,0 of genus 2 super Riemann surfaces onto the moduli space M 2 of ordinary Riemann surfaces, and has provided a valuable tool for the construction of the superstring measure and scattering amplitudes. In the presence of two R-punctures, Witten has shown [14] that the genus 2 super period matrix provides a natural meromorphic projection of supermoduli space M 2,0,2 onto M 2 , which may be used to derive powerful results on the fate of the vacuum energy in general heterotic superstring compactifications which enjoy N = 1 tree-level supersymmetry. Some of these results appear to generalize to genus 3 as well [14].
There is good reason to expect that the super period matrix with R-punctures will play a valuable role in the evaluation of a measure and scattering amplitudes with R-punctures at genus 2 as well. For low values of r, additional periods may be required, as explained in §3.9. Once these periods have been chosen to provide a parametrization by coordinates collectively denoted m A of the moduli space M 2,0,2r of super Riemann surfaces of genus 2 with 2r punctures, then a key ingredient in the construction of the measure and scattering amplitudes is the super Beltrami differential H A =θ∂(−µ − θχ)/∂m A associated with the choice of coordinates m A , and the space of superholomorphic 3/2 differentials Φ B dual to H A . In this section we shall evaluate H A and Φ B . As applications may exist to genus 2 and genus 3 in superstring theory for an arbitrary number of R-punctures, we shall calculate the variational formulas of Witten's super period matrix for general g and r, and supplement this with the variational formulas for the additional periods for the case r = 1.

Summary of variational formulas
The derivations of the variational formulas are rather technical, so we shall begin by giving an overview of the results, and relay derivations to later subsections. We shall consider the variation generated by deformations δχ of χ and δµ of µ of the various entries of the super period matrix. For the case r = 1 we shall also evaluate the variations of the additional periods. The most succinct formulation of the variational formulas is in superspace language. Introducing the superspace Beltrami differential, the deformations of the components of the super period matrix are found to be as follows, where the super holomorphic 3/2 forms are given by, Their superholomorphicity, namely D − Φ IJ = D − Φ ηI = D − Φ ηζ = 0, follows from the superholomorphicity ofω I andρ η . The structure of these variations in terms of superholomorphic 3/2 forms is analogous to the structure found without R-punctures in [15] for genus 2.
The variations of the additional periods may be computed as well. Here, we shall do so only for the case r = 1, which is perhaps of greatest physical relevance. One obtains, where Ψ pq I and Ψ pq 0 are now super meromorphic 3/2 forms which have poles at p and q. They may be expressed in terms of super holomorphic and meromorphic 1/2 forms, whereω pq is the superholomorphic 1/2 form with simple poles at p and q with residues +1 and −1, and vanishing A-periods. Its explicit form is given in Appendix B. Note that we have allowed for variations of θ A and θ B , and thus of the points in the Ramond divisors.

Varying superholomorphic 1/2 forms
Deformations of χ and µ produce variations δω I and δρ η of the superholomorphic 1/2 formŝ ω I andρ η . To compute these variations in terms of existing data, it is convenient to start from the normalization conditions, and differential equations which these variations must satisfy. The normalization conditions express the fact that we require the normalizations of the bosonic and fermionic A-periods to remain unchanged, The components of the variation δω = δω + + θδω z of any one of the superholomorphic 1/2 formsω =ω + + θω z satisfy a set of differential equations derived by variation of (3.3), where the sources are given by, This system of differential equations may be solved for the deformations δω + and δω z in terms of three kernels G 0 (z, w), G + (z, w), and G 1 (z, w) which are meromorphic sections of the bundles K z ⊗ K w , R −1 z ⊗ K w , and R −1 z ⊗ R −1 w respectively, and will be more completely defined and calculated below and in Appendix B. The integral equation corresponding to (4.7) is given by, Expressing the combinations τ + ww and τw in terms of δχ and δµ provides us with the desired variational formulas. The forms δ + (z) and δ z (z) are the components of an arbitrary super holomorphic 1/2 form, to be fixed by normalization. In Appendix B, a suitable set of normalization conditions will be imposed on the kernels G 0 , G + , and G 1 , in such a manner that one can set δ z = δ + = 0.

Varying the super period matrix
Combining these results yields the general formulas for the deformations of B-periods, Expressing τ in terms of the variations δχ and δµ, and recasting the result in superspace with the help of (4.1) and the superspace expressions for super holomorphic 1/2 forms, we see that the combinationŝ will yield the superholomorphic 3/2 forms listed in the results of (4.2) and (4.3), upon settinĝ ω equal to eitherω I orρ η .

Varying additional periods
We shall now prove the variational formulas for the additional periods in the case r = 1. They are given by the line integral (3.11) between points θ p and θ q in the Ramond divisors at p, q applied to the 1/2 formsω I andρ 0 . These periods take the form, The deformations ofV I andV 0 receive contributions from the variation ofω due to the deformations of χ and µ, as well as from the variation of the points θ p , θ q in the Ramond divisor, and we have, Making use of the integrals of the kernels in (B.9), we find, where the formω in τ equalsω I , and where the formω in τ now equalsρ 0 / √ 2N. Expressed in superspace with the help of ω pq (z, θ) =ω pq + (z) + θω pq z (z), these results produce (4.4) and (4.5).

The Super Period Matrix from Degenerations
In this section, we examine the limit of the super period matrix of a super Riemann surface of genus g + 1, in the non-separating limit with Ramond monodromy around the degenerating cycle. The limiting surface is then a super Riemann surface of genus g with 2 R-punctures. Our goal is to show that the limit of the super period matrix in genus g + 1 consists indeed of the bosonic components of the limiting surface of genus g with 2 R-punctures. The case of a limiting surface with 2r R-punctures for r > 1 can then be considered as a degeneration of a surface with 2(r − 1) R-punctures. A simultaneous degeneration of r cycles in a surface with no punctures would not be generic.

5.1
The super period matrix in genus g + 1 We consider the following one-parameter family of super Riemann surface (g, χ) in genus g + 1 with no puncture. Let Σ be a Riemann surface of genus g, and construct the family of degenerating surfaces Σ ε of genus g + 1, as ε → 0, by the classical construction described in Fay [24] for ordinary Riemann surfaces. That is, we choose neighborhoods U p and U q around two points p and q, which can be identified with the unit disk through coordinates z p and z q respectively. The family of surfaces Σ ε is constructed by where the equivalence relation ∼ means that we identify the annuli {|ε| < |z p | < 1} and {|ε| < |z q | < 1} around p and q through the relation Note that for any fixed ρ with 0 < ρ < 1, and all 0 < ε < ρ, the surfaces Σ ε all contain the open set Σ \ (ρU p ∪ ρU q ).
For super Riemann surfaces Σ, it was shown in section 6 of [10] that, for Ramond degenerations, the identification of the even coordinates in (5.2) needs to be supplemented with an identification of the entire Ramond divisors F p and F q at p and q. In terms of coordinates θ p and θ q for these divisors, the identification will specify the combination θ q ± √ −1 θ p , where the two sign choices produce the two possible relations between the spin structure of Σ ε and the generalized spin structure of Σ (see section 6.2 of [10]).
We fix an even spin structure δ on Σ ε and consider a non-separating degeneration of type R to a surface Σ with R-punctures p, q. Let χ be a gravitino field on Σ, and let χ ε be gravitino fields on Σ ε which converge to χ on compact subsets of Σ \ {p, q}, and which remain uniformly bounded in a neighborhood of p and q. We fix a canonical homology basis (A I , B I ) for Σ ε with B g+1 the non-separating cycle. Let ω ε I be the corresponding basis of holomorphic forms dual to A I , and let Ω ε IJ the corresponding period matrix The super period matrixΩ ε IJ , with I, J = 1, · · · , g + 1 was derived in [19]. Letω ε I = ω ε I+ + θω ε Iz be the superholomorphic 1/2 forms for (g, χ) given by the integral equations, 4 where ω ε (z, w) = ∂ z ∂ w ln E ε (z, w) and S ε [δ](z, w) are respectively the prime form and the Szegö kernel for the surface Σ ε . Then the super period matrix is given bŷ Equations (5.4) forω I can be solved iteratively by expanding in powers of the nilpotent field χ. It suffices then to compute the degeneration limits of ω ε I , Ω ε IJ , E ε (z, w), and S ε [δ](z, w).

5.2
The degeneration limits of ω ε I , Ω ε IJ , and E ε (z, w) These degeneration limits are well-known, and we shall just quote the results from [24]. Let z ∈ Σ \ (U p ∪ U q ). Then to order O(ε), Here ω I are the holomorphic differentials of the surface Σ, and ω qp (z) is the meromorphic differential with simple poles at p and q on Σ. Whenever convenient, we shall use the shorthand h = g + 1. The period matrix on Σ ε is then given by, where Ω IJ is the period matrix of Σ and c depends only on Ω, and not on p, q. Similar formulas hold for the prime form, and its derivatives, for z, w ∈ Σ \ (U p ∪ U q ),

The degeneration limit of S ε [δ](z, w)
Evaluating the non-separating degeneration limit of the Szegö kernel is the main step in calculating the limit of the super period matrix. We decompose each even spin structures δ on the surface Σ ε in accord with the non-separating degeneration along the cycle B h = B g+1 , Here, κ parametrizes the generalized spin structure induced by δ on Σ, and δ h , δ h ∈ {0, 1 2 }. The Szegö kernel on Σ ε is given by, on the surface Σ ε . We already know that E ε (z, w) = E(z, w) + O(ε). The degenerations of the ϑ-functions are obtained as follows.

Degeneration of ϑ[δ](0, Ω ε )
We compute the degeneration of ϑ[δ](0, Ω ε ) using the series representation of (A.1) with the sum running over n I with I = 1, · · · , g + 1 = h. Since Ω ε hh → +i∞, the leading contributions in the series arise as follows. When δ h = 0 the degeneration is of the Neveu-Schwarz type, only the term n h = 0 contributes, and the ϑ-constant converges to, The characteristic κ speficies the spin structure induced by δ on the surface Σ. When δ h = 1 2 , the degeneration is of the Ramond type, both the terms n h = 0, −1 contribute, and the ϑ-constant converges to, The characteristic κ together with the positions of the points p and q in the 2 2g -fold cover of Σ, specifies the generalized spin structure induced by δ on Σ. Formula (5.12) is valid as long as this leading contribution is non-vanishing, namely κ + (p − q)/2 ∈ Θ, where Θ is the ϑ-divisor. In terms of ε, this leading term is of order ε 1 8 and corrections to it will be suppressed by integer powers of ε.

Degeneration of
We compute the degeneration of ϑ[δ](z − w, Ω ε ) using the series representation of (A.1) as well. To leading order, the Abelian integrals involving z, w are given by, where the holomorphic covering function φ of (5.13) is given by, Decomposing the summation in the series for the ϑ-function in accord with the degeneration along the cycle B h , we have, As the limits when ε → 0 of the quantities u ε , v ε I are finite, but Ω hh → +i∞, the leading contribution to the limit will result from the following terms in the summation over m. When δ h = 0, it is just m = 0 which contributes, and we obtain, When δ h = 1 2 , then m = 0, −1 contribute, and to leading order, we find, where we have used exp(iπu ε ) = φ pq (w, z) in view of (5.13).

Degeneration of the Szegö kernel
Assembling all ingredients, we find that for NS non-separating degenerations with δ h = 0, the Szegö kernel with even spin structure δ on Σ ε simply tends to the Szegö kernel with even spin structure κ on Σ, For Ramond degenerations with δ h = 1 2 , the Sezgö kernel with even spin structure δ on Σ ε tends to the following limit, This relation is obtained by assembling the limit of the Szegö kernel in (5.10) from the limits of its factors in (5.17), (5.12), and (5.8, and identification with the Szegö kernel S pq (z, w) given in (2.12). With the help of the definition (2.15) this result in turn is seen to coincide with the anti-symmetric Szegö kernel in the presence of two Ramond punctures p, q, (5.20)

The limit of the super period matrix
We shall now determine the limit ofΩ ε IJ , with I, J = 1, · · · , g + 1 = h, as ε → 0. We begin by recalling some basic relations between the moduli spaces M g,n,2r of super Riemann surfaces of genus g, with n Neveu-Schwarz punctures, and 2r Ramond punctures. For the case of interest, namely g ≥ 2, their dimensions are 3g − 3 + n + 2r|2g − 2 + n + r.
In the supergravity formulation of super Riemann surfaces, the 2g odd moduli of M g+1,0,0 are encoded in the field χ ε . As a result, the expansion in powers of χ ε of the even quantities such as the differentialsω ε Iz in (5.4) and the super period matrixΩ ε IJ of (5.5) vanish beyond O((χ ε ) 2g ), while the odd quantities such asω ε I+ in (5.4) vanish beyond O((χ ε ) 2g−1 ). Upon a non-separating degeneration of type NS or R of a surface in M g+1,0,0 , one even modulus becomes the degeneration parameter ε, while two even moduli become the bosonic coordinates of the punctures. For the NS degeneration, two odd moduli of χ ε become the odd coordinates of the NS punctures.
For the R degeneration, the fate of the odd moduli is more delicate, as the odd dimension of M g+1,0,0 is one higher than the odd dimension of its naive limit M g,0,2 . This subtlety was carefully explained and resolved in section 6.2 of [10]. The Ramond degeneration divisor D equals not M g,0,2 but rather a C 0|1 line bundle over M g,0,2 with additive structure group C 0|1 . In the supergravity formulation, this means that the 2g odd moduli in M g+1,0,0 which span χ ε will redistribute into the 2g − 1 odd moduli of M g,0,2 which span χ along with one odd coordinate α which parametrizes the C 0|1 fiber of the bundle D → M g,0,2 . The parameter α enters into the prescription for gluing the Ramond divisors F p and F q with coordinates θ p and θ q respectively through the relation, In the supergravity formulation of super Riemann surfaces, χ will depend on α, and α will transform under supersymmetry by a shift.
We now return to the degeneration limits of the super period matrix for M g+1,0,0 and even spin structure. For non-separating degenerations of type NS, the limit found in (5.18) for the Szegö kernel on Σ ε gives the Szegö kernel on Σ, and thus reduces the componentŝ Ω ε IJ for I, J = 1, · · · , g of the super period matrix on Σ ε with even spin structure δ to the componentsΩ IJ of the super period matrix for the corresponding surface Σ with even spin structure κ, related by (5.9). The componentsΩ ε Ih reduce to line integrals between the two NS punctures, andΩ hh diverges as ln ε, as was the case for bosonic surfaces in (5.7).
For non-separating degenerations of type R, we focus attention first on the g × g block of the super period matrix with componentsΩ ε IJ with I, J = 1, · · · , g. All the ingredients that enter into the expressions for these components in (5.5), namely the holomorphic forms ω ε I (z), gravitino field χ ε (z), Szegö kernel S ε [δ](z, w), and prime form E ε (z, w) converge to their respective counterparts ω I (z), χ(z), S(z, w), and E(z, w) on the super Riemann surface Σ with R-punctures p and q when z, w are outside the funnel region {|ε| < |z p | < ρ}. As a result, the components of the bockΩ ε IJ with I, J = 1, · · · , g converge to the pseudo-classical block with componentsΩ IJ , of the surface Σ with R-punctures p, q, lim ε→0Ω ε IJ =Ω IJ I, J = 1 · · · , g (5.22) SinceΩ IJ is invariant under any supersymmetry transformation, including those which do not vanish at p and q, it is clear that any dependence in χ upon the gluing parameter α will cancel out in the limitΩ IJ , which thus effectively only depends upon 2g − 1 odd moduli. For non-separating degenerations of type R, the limit ofΩ ε hI is obtained from (5.5) using the limit (5.6). Naively, this would lead to the following expression forΩ hI , This result is naive because the funnel will now contribute due to the presence of the poles at the punctures p, q caused by the differential ω pq . Considering the behavior under local supersymmetries, we see that (5.23) is invariant under all supersymmetries which vanishes at p, q, but not under supersymmetries which do not vanish at p, q. For the latter, the contribution of the funnel is non-zero, but we shall not attempt here to evaluate it. Since we have earlier identified the line integralsV I as additional periods which are invariant under all supersymmetries, it is compelling to conjecture that the correct limit, taking into account the contributions from the funnel, will give, lim ε→0Ω ε Ih =V I (5.24) whereV I are the additional bosonic periods defined by the line integrals from p to q in (3.52). Also, the componentΩ ε hh diverges like ln ε as was the case for bosonic surfaces in (5.7). Finally, note that the limits (5.22) and (5.23) demonstrate that the full super period matrix at genus g with two R-punctures may be obtained from the non-separating degeneration of the super period matrix at genus g + 1 without R-punctures. This is manifest for the quasi-classical blockΩ IJ of (5.22), while the odd components Ω 0I may be obtained from (5.23) by taking the derivative with respect to θ B in the explicit expression forV I obtained in (3.56), where θ B parametrizes the single odd modulus of the surface of genus g + 1 which has become pure gauge upon the non-separating degeneration.
A ϑ-functions, Abelian differentials, and prime form In this appendix, we review Jacobi ϑ-functions, the prime form, and Abelian differentials and their basic properties on a compact Riemann surface genus g (see [24] and for example [25,18] for applications to string perturbation theory).

A.2 Function theory on a compact Riemann surface
The basic objects of function theory on a compact Riemann surface Σ 0 (we shall reserve the customary notation Σ for super Riemann surfaces) of genus g are the Abelian differentials and the prime form. To define them, we choose a canonical homology basis in H 1 (Σ 0 , Z) given by cycles A I , B I with canonical intersection form #(A I , A J ) = #(B I , B J ) = 0 and #(A I , B J ) = δ IJ for I, J = 1, · · · , g. Modular transformations on the basis A I , B I are defined to leave the intersections invariant, and form the group Sp(2g, Z). The holomorphic Abelian differentials ω I for I = 1, · · · , g are holomorphic sections of the canonical buncle K on Σ 0 which are normalized on A-cycles, and whose integrals on B-cycles produce the period matrix Ω IJ of the surface Σ 0 , The Jacobian is defined by J(Σ 0 ) = C g /(Z g + ΩZ g ). Given a base point z 0 on Σ, the Abel map is defined for d points z i and multiplicities n i with i = 1, · · · , d by For integer multiplicities n i , the Abel map is single-valued on J(Σ 0 ). To define the bundle of spinors K 1 2 requires specifying a spin structure. On a surface of genus g there are 2 2g spin structures, of which 2 g−1 (2 g +1) are even, while 2 g−1 (2 g −1) are odd, depending on whether the number of holomorphic sections of K 1 2 is even or odd. Having fixed a reference spin structure, all other spin structures may be labelled by a half-characteristic κ which is even or odd in accord with the parity of the spin structure. Generically, there is one holomorphic section h κ (z) of K The Abelian differentials ω(z, w) of the second kind, and ω xy (z) of the third kind, with unit residues and vanishing A-periods, may be expressed in terms of the prime form by, ω(z, w) = ∂ z ∂ w ln E(z, w) ω xy (z) = ∂ z ln E(z, x) E(z, y) (A.9) Their B-periods are given as follows, Abelian differentials of second and third kind obeying other normalizations may be obtained by the addition of Abelian differential ω I of the first kind to ω(z, w) and ω xy (z) respectively.

A.3 Deformation theory on a compact surface
We shall need the following simple formulation of the deformation of a complex structure by a Beltrami differential µ on a purely bosonic Riemann surface Σ 0 (such as for example the reduced surface Σ red ): a complex structure is specified by the class of its holomorphic functions. If the holomorphic functions f (z) for a given complex structure are specified by ∂zf = 0, and if µ = µz z is a Beltrami differential, then the holomorphic functions of the deformed complex structure are defined by the equation This means that the deformation is implemented by the basic shift for derivatives ∂z → ∂z + µ∂ z (A.12) on scalars, with similar shifts on tensors of arbitrary weights. Corrections of the form µμ may be ignored since all conformal anomalies cancel in critical string theory, while corrections involving higher powers of µ alone are absent. An alternative formulation is by introducing coordinates w which are holomorphic with respect to the new complex structure, that is, which satisfy the equation (A.11). The change z → w is a quasi-conformal transformation, and function theory can then be formulated in terms of w and ∂w.
In concrete calculations of scattering amplitudes it will turn out to be much more convenient to keep the coordinates z on the super Riemann surface Σ fixed and associated with the complex structure of Σ red .
The variation δ ww φ of any object φ under the variation of the complex structure of the surface Σ 0 by a Beltrami differential µ is defined as follows, δφ = 1 2π Σ 0 d 2 w µw w δ ww φ µw w = 1 2 g ww δg ww (A.13) The variations of the basic functional objects on the surface Σ 0 that will be needed here are given as follows, δ ww Ω IJ = 2πiω I (w)ω J (w) δ ww ω I (x) = ω I (w)∂ x ∂ w ln E(x, w) δ ww ln E(x, y) = − 1 2 ∂ w ln E(w, x) − ∂ w E(w, y) 2 δ ww S(x, y) = 1 2 ∂ w S(x, w)S(w, y) − 1 2 S(x, w)∂ w S(w, y) (A.14) On the last line, S(x, y) stands for the Szegö kernel on a surface Σ 0 without R-punctures.

B Calculation of Variational Formulas B.2 Normalization of A-periods
For the bosonic A-periods, the solution of (B.3) leads to, Pairwise elimination allows us to solve these equations schematically as follows, G + = −(1 + Sχωχ) −1 Sχω G 0 = (1 + ωχSχ) −1 ω G t + = +(1 + ωχSχ) −1 ωχS G 1 = (1 + Sχωχ) −1 S (B.14) We use the basic transposition rules ω t = ω and S t = −S. To show that the two expressions on the left are consistent with one another, it suffices to expand both in powers ofχ, to use the transposition rules (Sχωχ) t =χωχS and (ωχSχ) t =χSχω, and to rearrange the results. It is manifest that these two lines are transposes of one another.