aa r X i v : . [ h e p - t h ] M a y SUPERSTRING SCATTERING AMPLITUDES INHIGHER GENUS
SAMUEL GRUSHEVSKY
Abstract.
In this paper we continue the program pioneered byD’Hoker and Phong, and recently advanced by Cacciatori, DallaPiazza, and van Geemen, of finding the chiral superstring measureby constructing modular forms satisfying certain factorization con-straints. We give new expressions for their proposed ans¨atze ingenera 2 and 3, respectively, which admit a straightforward gener-alization. We then propose an ansatz in genus 4 and verify that itsatisfies the factorization constraints and gives a vanishing cosmo-logical constant. We further conjecture a possible formula for thesuperstring amplitudes in any genus, subject to the condition thatcertain modular forms admit holomorphic roots. Introduction
The problem of finding an explicit expression for the string measureto an arbitrary loop (genus) order is one of the major open problems instring perturbation theory. For the bosonic string, chiral expressionsin any genus in terms of theta functions and additional points on theworldsheet have been proposed by Manin in [18], Beilinson and Maninin [1], and Verlinde and Verlinde in [26]. Non-chiral expressions interms of the Weil-Petersson measure and Selberg zeta functions havealso been proposed by D’Hoker and Phong in [7]. The problem is muchmore difficult for the superstring and the heterotic string. Although theone-loop scattering amplitudes had been derived by Green and Schwarzin [5] for the superstring and by Gross, Harvey, Martinec, and Rohm in[6] for the heterotic string, the case of genus g ≥ g ≥ Research is supported in part by National Science Foundation under the grantDMS-05-55867. compute the genus 2 superstring measure from first principles in a se-ries of papers [8, 9, 10, 11] and to verify that the corresponding result θ [∆] Ξ [∆] times the bosonic measure produced vanishing cosmologi-cal constant, 2- and 3-point scattering amplitudes, and other expectedphysical properties [14, 15]. With the new insight from Ξ [∆](Ω), theystarted a modern program of identifying the higher genera superstringmeasure from factorization constraints and syzygy/asyzygy conditions[12, 13]. In [13] they also proposed an ansatz for the superstring mea-sure in genus 3, including a θ [∆] factor, subject to the condition ofcertain linear combinations of modular forms having a square root.However, to date such a linear combination has not been found, andmay not exist (see below).In [2] Cacciatori and Dalla Piazza used the combinatorics of theaction of the symplectic group on the set of theta characteristics ingenus 2 to identify D’Hoker and Phong’s modular form Ξ [∆](Ω) fromcertain invariance properties. Recently Cacciatori, Dalla Piazza, andvan Geemen [3] proposed an ansatz for the chiral superstring measurein genus 3, by constructing an appropriate modular form Ξ [∆], whichhas a θ [∆] rather than the θ [∆] factor, satisfying the factorizationconstraints on the locus of products of abelian varieties of lower genera.They also promise to determine in a forthcoming paper the dimensionof the appropriate space of modular forms, and to show that their formis the unique one satisfying the factorization constraints (and thus therewould be no solution in the form suggested in [13]). They also say thattheir constraints appear to have a solution in genus 4 as well.In this paper we rewrite the ansatz of D’Hoker and Phong in genus2, and of Cacciatori, Dalla Piazza, and van Geemen in genus 3 in termsof modular forms associated to isotropic spaces of theta characteristics,which have been studied since the times of Krazer [17] and in particularused by Salvati Manni [23]. This allows us to propose a straightforwardgeneralization of the chiral superstring measure to higher genera, whichfor genus 4 is an appropriate holomorphic modular form satisfying thenecessary factorization constraints and producing vanishing cosmolog-ical constant. For higher genera we conjecture a possible ansatz, satis-fying the factorization constraints, contingent on certain monomials intheta constants admitting holomorphic roots.An expression for higher genus superstring amplitudes was also pro-posed by Matone and Volpato [19]. Their formulas depend on thechoice of points on the worldsheet and do not seem to give an explicitmodular form. It would be interesting to understand the relation ofour work to theirs. UPERSTRING SCATTERING AMPLITUDES IN HIGHER GENUS 3
The structure of this work is as follows: in section 2 we fix notationsand introduce basic notions of modular forms. In section 3 we reviewthe orbits of the action of the symplectic group on sets of theta char-acteristics. In section 4 we reinterpret the modular form G defined in[3] in terms of syzygies. Though strictly speaking this computation isnot needed to define our modular forms and construct an ansatz, thisis our motivation for considering, in section 5, modular forms corre-sponding to vectors subspaces of the space of theta characteristics andreviewing what is known about them. In section 6 we prove the crucialtheorem 15 describing the restrictions of these modular forms to lociof products. In section 7 we obtain a new expression for the ans¨atzein genera 2 and 3 in terms of our modular forms, and also verify thatin genus 4 there is a unique modular form that is a linear combinationof ours that has correct factorization properties, thus giving an ansatzin genus 4. In section 8 we describe a possible generalization to ar-bitrary genus, proving in theorem 22 that it satisfies the factorizationconstraints (this also gives another proof of this for the ansatz in genus4), and describe further tests and questions that could be used to studythe validity and uniqueness of the ansatz.2. Notations and definitions
Definition 1.
We denote by A g the moduli space of complex princi-pally polarized abelian varieties of dimension g . We denote by H g theSiegel upper half-space of symmetric complex matrices with positive-definite imaginary part, called period matrices. The right action of thesymplectic group Sp(2 g, Z ) on H g is given by (cid:18) A BC D (cid:19) ◦ τ := ( Cτ + D ) − ( Aτ + B )where we think of elements of Sp(2 g, Z ) as of consisting of four g × g blocks, and they preserve the symplectic form given in the block formas (cid:18) − (cid:19) . We then have A g = Sp(2 g, Z ) \H g . Definition 2.
Given a period matrix τ ∈ H g we denote the abelianvariety corresponding to [ τ ] ∈ A g by A τ := C g / ( Z g + τ Z g ). The thetafunction is a function of τ ∈ H g and z ∈ C g given by θ ( τ, z ) := X n ∈ Z g exp( πi ( n t τ n + 2 n t z )) . We denote by Θ τ the line bundle on A τ of which the theta function isa section SAMUEL GRUSHEVSKY
Definition 3.
Given a point of order two on A τ , which can be uniquelyrepresented as τε + δ for ε, δ ∈ Z g (where Z denotes the abelian group Z / Z = { , } and we use the additive notations throughout the text),the associated theta function with characteristic is θ (cid:20) εδ (cid:21) ( τ, z ) := X n ∈ Z g exp( πi (( n + ε ) t τ ( n + ε ) + 2( n + ε ) t ( z + δ )) . As a function of z , θ (cid:20) εδ (cid:21) is odd or even depending on whether the scalarproduct ε · δ ∈ Z is equal to 1 or 0, respectively. Theta constants arerestrictions of theta functions to z = 0, and all odd theta constantsvanish identically in τ . Definition 4.
A modular form of weight k with respect to a subgroupΓ ⊂ Sp(2 g, Z ) is a function f : H g → C g such that f ( γ ◦ τ ) = det( Cτ + D ) k f ( τ ) ∀ γ ∈ Γ , ∀ τ ∈ H g . We define the level subgroups of Sp(2 g, Z ) as follows:Γ g ( n ) := (cid:26) M = (cid:18) A BC D (cid:19) ∈ Γ g | M ≡ (cid:18) (cid:19) mod n (cid:27) Γ g ( n, n ) := (cid:8) M ∈ Γ g ( n ) | diag( A t B ) ≡ diag( C t D ) ≡ n (cid:9) . These are normal subgroups of Sp(2 g, Z ) for n >
1; however, Γ g (1 , Remark 5.
Theta constants with characteristics are not algebraicallyindependent, and satisfy a host of algebraic identities. Their squarescan be expressed algebraically in terms of a smaller set of modularforms, called theta constants of the second order, by using Riemann’sbilinear addition theorem — see [16] for details. Theta constants of thesecond order are algebraically independent for g ≤
2. The only identityamong them for g = 3 was known classically at least since the time ofSchottky, and is discussed in [4], while the ideal of relations amongthem for g > The action of
Sp(2 g, Z ) on theta characteristics Proposition 6 (see [16]) . Theta constants with characteristics aremodular forms of weight one half with respect to Γ g (4 , . Moreover,the full symplectic group acts on theta constants with characteristics asfollows: θ (cid:20) M (cid:18) εδ (cid:19)(cid:21) ( M · τ ) = φ ( ε, δ, M, τ, z ) det( Cτ + D ) θ (cid:20) εδ (cid:21) ( τ ) , UPERSTRING SCATTERING AMPLITUDES IN HIGHER GENUS 5 where φ is some explicit eighth root of unity, and the action on thecharacteristic is (1) M (cid:18) εδ (cid:19) := (cid:18) D − C − B A (cid:19) (cid:18) εδ (cid:19) + (cid:18) diag( C t D )diag( A t B ) (cid:19) where the addition in the right-hand-side is taken in Z . Notice that by this formula the action of the subgroup Γ g (2) on theset of characteristics is trivial, and thus the action of the entire groupSp(2 g, Z ) on the set of characteristics facters through the action ofSp(2 g, Z ) / Γ g (2) = Sp(2 g, Z ).One can thus study the orbits of characteristics or sets of charac-teristics under the symplectic group action. This was done by SalvatiManni in [24], where all of the following results can be found. Onefirst observes that the action of Γ g (4 , \ Γ g (2) on the set of theta con-stants differs from the modular one by extra signs, while Γ g (2) in ad-dition permutes the characteristics. It is also clear that the action ofSp(2 g, Z ) on the set of characteristics (which factors through the actionof Sp(2 g, Z )) is transitive. To study the action on tuples of character-istics (i.e. the orbits of the Sp(2 g, Z ) acting on (cid:0) Z g (cid:1) n diagonally), weneed more definitions. Definition 7.
The Weil symplectic form on the space Z g of charac-teristics is defined to be (cid:28)(cid:20) αβ (cid:21) , (cid:20) εδ (cid:21)(cid:29) := α · δ + β · ε. Notice that this symplectic form is not preserved by the action ofSp(2 g, Z ) on the set of characteristics: the pairing of the zero char-acteristic with any characteristic is zero, and it is the only such char-acteristic, while the action Sp(2 g, Z ) given by (1) is affine and does notpreserve zero. Definition 8.
A triple of characteristics (cid:20) ε δ (cid:21) , (cid:20) ε δ (cid:21) , (cid:20) ε δ (cid:21) is called syzygeticor azygetic depending on whether the sum ε · δ + ε · δ + ε · δ + ( ε + ε + ε ) · ( δ + δ + δ )= (cid:28)(cid:20) ε δ (cid:21) , (cid:20) ε δ (cid:21)(cid:29) + (cid:28)(cid:20) ε δ (cid:21) , (cid:20) ε δ (cid:21)(cid:29) + (cid:28)(cid:20) ε δ (cid:21) , (cid:20) ε δ (cid:21)(cid:29) ∈ Z is 0 or 1, respectively. Notice in particular that a triple of even charac-teristics is syzygetic or azygetic if their sum is even or odd, respectively.This notion is in fact invariant under the symplectic group action. SAMUEL GRUSHEVSKY
The orbits of the action (1) of the symplectic group on sets of char-acteristics are completely described by the following
Theorem 9 ([16] p. 212, [24]) . There exists an element of the sym-plectic group mapping a set of n characteristics to another set of n characteristics if and only if there exists a way to number the charac-teristics in the first set a . . . a n , and the characteristics in the secondset b . . . b n in such a way that • for any i the parity of a i and b i is the same • for any linear relation among a i with an even number of terms,i.e. if a i + . . . + a i k = 0 , there is a corresponding linear relation b i + . . . + b i k = 0 and vice versa. • any triple a i , a j , a k is a/syzygetic if and only if the correspondingtriple b i , b j , b k is a/syzygetic. Results of Cacciatori, Dalla Piazza, van Geemen interms of syzygy conditions
The main new ingredient of the superstring measure in genus 3 pro-posed by Cacciatori, Dalla Piazza, and van Geemen in [3] is the modu-lar form G , of weight 8 with respect to the group Γ(1 , ⊂ Sp(6 , Z ) —it is described there in terms of certain quadrics on Z . However, since G is a polynomial in theta constants with characteristics, from theorem9 it follows that the monomials appearing in it should be characterizedby the syzygy properties and linear dependencies of the characteristicsinvolved. We now obtain such a description of the modular form G ,by unraveling the definition of G given in [3] in terms of syzygies. Wewould like to thank Eric D’Hoker and Duong Phong for encouragingus to do this translation — which then gave a formula amenable togeneralizing to higher genus.Given an even characteristic ∆ = (cid:20) abcdef (cid:21) — or (cid:20) αβ (cid:21) in our notations— one can define a corresponding quadratic form on the set of charac-teristics, i.e. ([3], p. 12) for v ∈ Z one defines q ∆ ( v ) := v v + v v + v v + av + bv + cv + dv + ev + f v . If we write v = (cid:20) εδ (cid:21) , this is simply q ∆ ( v ) = ε · δ + α · ε + β · δ, UPERSTRING SCATTERING AMPLITUDES IN HIGHER GENUS 7
Since (cid:20) αβ (cid:21) is an even characteristic, we have α · β = 0, and thus q αβ (cid:18)(cid:20) εδ (cid:21)(cid:19) = ( ε + β ) · ( δ + α ) . This looks strange as the α and β are strangely swapped, and I believethat there is a small typo in [3] of interchanging the top and the bottomvector of the characteristic. For the characteristic (cid:20) (cid:21) that is used forexplicit calculations in [3] there is of course no difference, but otherwisemodularity would not hold. The correct definition should thus be(2) q αβ (cid:18)(cid:20) εδ (cid:21)(cid:19) = ( ε + α ) · ( δ + β ) . In [3] a quadric is now introduced Q ∆ := { v | q ∆ ( v ) = 0 } . From definition (2) it follows that this is the set of characteristics (cid:20) εδ (cid:21) such that (cid:20) ε + αδ + β (cid:21) is even, i.e. this is just the set of even character-istics, to which (cid:20) αβ (cid:21) is added. The symplectic form h v, w i on Z isdenoted E ( v, w ) in [3]. Notice that if both characteristics are even,this is the same as the quadratic form q . Considering Lagrangian (alsocalled maximal isotropic classically, see [16] and [23]) subspaces of Z with respect to h· , ·i means choosing three linearly independent char-acteristics (cid:20) ε i δ i (cid:21) i =1 .. such that the Weil pairing is zero on any pair, i.e.such that the sum of any pair of characteristics is again even. This isequivalent to saying that the triple of characteristics consisting of thispair and zero is syzygetic. The set of all even quadrics containing aLangrangian subspace is now considered in [3]. This means consideringthe set of all characteristics ∆ = (cid:20) αβ (cid:21) such that q ∆ | L = 0. Thus Q ∆ ⊃ L ⇐⇒ (cid:20) ε + αδ + β (cid:21) is even ∀ (cid:20) εδ (cid:21) ∈ L. We now notice (following 8.4 in [3], essentially) that if ∆ and ∆ ′ are twocharacteristics such that Q ∆ ∩ Q ∆ ′ ⊃ L , then we must have ∆ − ∆ ′ ∈ L . SAMUEL GRUSHEVSKY
Thus the definition of G at the top of page 13 in [3] becomes G [∆] = X L ⊂ Q ∆ Y v ∈ L θ [ v + ∆] . The condition L ⊂ Q ∆ means that the sum in the definition of G istaken over linear spaces generated by triples of characteristics (cid:20) ε i δ i (cid:21) i =1 .. such that all (cid:20) α + ε i β + δ i (cid:21) are even and all (cid:28)(cid:20) ε i δ i (cid:21) , (cid:20) ε j δ j (cid:21)(cid:29) = ε j · δ i + ε i · δ j = 0 . Adding these conditions together shows that all characteristics (cid:20) α + ε i + ε j β + δ i + δ j (cid:21) = (cid:20) αβ (cid:21) + (cid:20) α + ε i β + δ i (cid:21) + (cid:20) α + ε j β + δ j (cid:21) are even. Thus we get thefollowing alternative formula Proposition 10.
The modular form G [∆] defined in [3] (of weight 8with respect to Γ (1 , ) is equal to the sum over all sets of 8 evencharacteristics { u i } i =1 .. such that any pair of characteristics togetherwith ∆ form a syzygetic triple, of the products Q θ [ u i ] . Denote now v i := u i + ∆ and observe h v i , v j i = h u i , u j i + h u i , ∆ i + h ∆ , u j i = 0since the right-hand-side is exactly the condition that the triple ∆ , u i , u j is syzygetic. We thus get yet another formula Corollary 11.
The modular form G can be written as (3) G [∆] = X V ⊂ Z dim V =3 Y v ∈ V θ [ v + ∆] . We remark that a given summand on the right-hand-side is not iden-tically zero if and only if the set V +∆ contains only even characteristics(is a purely even coset in the language of [23]), in which case as de-scribed there it follows that V is totally isotropic. This expression for G yields itself to a straightforward generalization, and in this terms therestriction of G to the locus of decomposable abelian varieties is easyto understand. We undertake this study in the next two sections.5. Modular forms corresponding to subspaces of Z g Motivated by the study of quartic relations among theta constantsundertaken by Salvati Manni in [23] and by our reinterpretation of theform G above, in this section we investigate the properties of prod-ucts of theta constants with characteristics forming a translate of anisotropic subspace. We thank Riccardo Salvati Manni for telling us UPERSTRING SCATTERING AMPLITUDES IN HIGHER GENUS 9 about [23] and encouraging us to explore the behavior of these modu-lar forms.Following [23], we denote P M ( τ ) := Y v ∈ M θ [ v ]( τ ) for any M ⊂ Z g Notice that if M contains any odd characteristics, then P M is identicallyequal to zero, as all odd theta constants vanish identically. Let V ⊂ Z g be a vector subspace with basis v . . . v n . V is called isotropic ifthe symplectic form restricts to zero on it, i.e. if h v, w i = 0 for any v, w ∈ V . Since for even characteristics v and w the value of thesymplectic form h v, w i is equal to the parity of v + w , the space withbasis { v i } is isotropic if and only if it only contains even characteristics,or equivalently, if P V ( τ ) is not identically zero. Definition 12.
We define a function P ( g ) i,s on H g as the sum P ( g ) i,s ( τ ) := X V ⊂ Z g ; dim V = i P V ( τ ) s . Here the sum is taken over all i -dimensional (and thus of cardinality2 i ) vector subspaces, but note that if V is not isotropic, it containsodd characteristics, and the corresponding summand is zero. We willonly consider these functions subject to the condition i s = 2 k for someinteger k ≥ . For the case of the superstring measure k = 4 . Proposition 13.
The function P ( g ) i,s is a modular form (assuming i s =2 k for k ≥ ) of weight i − s for the subgroup Γ g (1 , .Proof. This can be seen from the discussion in [23], [16], [17]. To seethis, one notes that the action of Sp(2 g, Z ) on the set of characteristicsis affine; however, for an element γ = (cid:18) A BC D (cid:19) ∈ Γ g (1 ,
2) by definitiondiag( C t D ) = diag( A t B ) = 0 ∈ Z g , and thus the action of Γ g (1 ,
2) onthe set of characteristics fixes zero and is linear. Thus the summandsin the definition of P i,s get permuted — vector subspaces are mappedto vector subspaces by a linear action. The multiplicative factors aredet( Cτ + D ) / for each theta constant, giving the overall factor ofdet( Cτ + D ) i − for each P V . The other factor in the transformationformula (1) is the 8th root of unity φ . Since 2 i s = 2 k is divisible by16, it can be shown that the product of the 8th roots will turn out tobe equal to 1. We refer to [25] for a complete discussion and rigorousproof. (cid:3) The action of the full group Sp(2 g, Z ) on the set of characteristics isaffine — it shifts the zero to some characteristic ∆. Acting by Sp(2 g, Z )on the modular form P ( g ) i,s we then get Corollary 14.
Assuming i s = 2 k for k ≥ , for any even character-istic ∆ the function P ( g ) i,s [∆]( τ ) := X V ⊂ Z g ; dim V = i P V +∆ ( τ ) s is a modular form of weight i − s with respect to the subgroup Γ[∆] ⊂ Sp(2 g, Z ) that stabilizes ∆ under the action (1). This subgroup is con-jugate to Γ g (1 , (which, recall, is not a normal subgroup of Sp(2 g, Z ) ).The conjugation is provided by any element of Sp(2 g, Z ) that maps thezero characteristic to ∆ under the action (1). Note that if ∆ is odd, thecorresponding expression would be zero, as all summands would contain θ [∆] .Proof. To prove modularity, one can observe that P ( g ) i,s = P ( g ) i,s [0], whichwe know to be a modular form, is conjugated to P ( g ) i,s [∆] by the Sp(2 g, Z )action. For a direct proof, note that v + ∆ is an even characteristic forall v ∈ V if and only if for any triple v , v , ∆ for v , v ∈ V is syzygetic— this is the argument used to obtain the expression (3) in genus 3at the end of the previous section — and the syzygy is preserved bythe Sp(2 g, Z ) action. See [25] for more discussion, especially on thepossible 8th roots of unity. (cid:3) Restrictions of modular forms corresponding
In this section we determine the restrictions of modular forms P ( g ) i,s tothe loci of decomposable abelian varieties (products of lower-dimensionalones). Note that it is enough to determine the restriction of the mod-ular form P ( g ) i,s with zero characteristic to H k × H g − k — its restrictionsto the conjugates of this locus under Γ g (1 ,
2) can be obtained by actingby Γ g (1 , P ( g ) i,s [∆] with non-zero characteristic can then be obtained by conjugat-ing by an element of Sp(2 g, Z ) that maps P ( g ) i,s to P ( g ) i,s [∆]. Theorem 15.
The modular form P ( g ) i,s restricts to the locus of decom-posable abelian varieties (reducible period matrices in [3] ) as follows: (4) P ( g ) i,s | H k ×H g − k = X ≤ n,m ≤ i ≤ n + m N n,m ; i P ( k ) n, i − n s · P ( g − k ) m, i − m s , UPERSTRING SCATTERING AMPLITUDES IN HIGHER GENUS 11 where (5) N n,m ; i = n + m − i − Y j =0 (2 n − j )(2 m − j )(2 n + m − i − j ) . Remark 16.
Notice that many of the summands can be zero, as P ( c ) a,b ≡ a > c . In particular for P ( g ) g,s (the case of maximal isotropicsubspaces; the form G constructed in [3] is P (3)3 , in our notations) theonly non-zero term on the right is P ( k ) k, g − k s P ( g − k ) g − k, k s . Proof.
Indeed, let V ∼ = Z i be a vector subspace of Z g (notice thatwe never need to worry about parity or isotropy — the summands fornon-isotropic subspaces will vanish automatically). If a period matrixis a product of two lower-dimensional ones, τ g = τ k × τ g − k , the groupof points of order two on A τ is the direct sum of the groups of points oforder two on the factors. Let Z g = Z k ⊕ Z g − k )2 be this decomposition,and let π and π denote the projections onto the two summands. Since V ⊆ π ( V ) ⊕ π ( V ), we must have V ≤ π ( V ) · π ( V ). Since theseare all vector spaces over Z , denoting by n, m the dimensions of π ( V )and π ( V ) respectively, this implies the inequality 2 i ≤ n · m or,equivalently, i ≤ n + m . The projections maps π : V → π ( V ) and π : V → π ( V ), being group homomorphisms, are then 2 i − n -to-1 and2 i − m -to-1, respectively. Since any theta constant with characteristicrestricts to the product θ [ v ]( τ g ) = θ [ π ( v )]( τ k ) · θ [ π ( v )]( τ g − k ) , it follows that for n and m fixed we have P V ( τ g ) s = Y v ∈ V θ [ v ]( τ g ) s = Y v ∈ V θ [ π ( v )]( τ k ) s · θ [ π ( v )]( τ g − k ) s = Y v ∈ π ( V ) θ [ v ]( τ k ) i − n s Y v ∈ π ( V ) θ [ v ]( τ g − k ) i − m s = P i − n sπ ( V ) · P i − m sπ ( V ) To compute P ( g ) i,s , we need to sum over all V . Let us first sum over allthe spaces V for which the spaces π ( V ) and π ( V ) are fixed. Noticethat the product on the right is the same for all V with fixed projec-tions. The number of V with fixed projections only depends on thedimensions, and we compute it in the following combinatorial lemma. Lemma 17.
The number of i -dimensional vector subspaces V of Z n ⊕ Z m surjecting onto both summands n + m − i − Y j =0 (2 n − j )(2 m − j )(2 n + m − i − j ) , (where this number is understood to be zero if n + m < i , and to be oneif n + m = i . Note also that the product has a zero factor if n > i or m > i ).Proof. Fix a scalar product on Z n + m such that the chosen decomposi-tion Z n + m = Z n ⊕ Z m is orthogonal. The projections from V to Z n and Z m are then the orthogonal projections; the image of such a projectionmisses a vector v if and only if it is orthogonal to V . Thus what weneed to count is (for a fixed scalar product) the number of V ∼ = Z i suchthat V ⊥ ∼ = Z n + m − i does not intersect the coordinate subspaces Z n and Z m away from zero. Let us construct such a V ⊥ by choosing a basis v , . . . , v n + m − i for it. To choose such a basis, we will choose indepen-dently the projections π ( v ) , . . . , π ( v n + m − i ) — note that in order tohave V ⊥ ∩ Z n = { } , these vectors must be linearly independent — andsimilarly choosing linearly independent π ( v ) , . . . , π ( v n + m − i ) ∈ Z m .Thus π ( v ) can be chosen in 2 n − π ( v ) can bechosen in 2 n − π ( v ) in2 m − V ⊥ ⊂ Z n + m not intersecting Z n and Z m , together with a choice of an ordered basisof it, is equal to n + m − i − Y j =0 (2 n − j )(2 m − j )while the number of ordered basis in a fixed space V ⊥ ∼ = Z n + m − i is n + m − i − Y j =0 (2 n + m − i − j ) , and dividing one by the other gives the lemma. (cid:3) Now observe that if we take the sum over all V for fixed dimensions n and m , the projections π ( V ) and π ( V ) range over all n -dimensionalsubspaces of Z k , and m -dimensional subspaces of Z g − k )2 , respectively.We thus get X V ⊂ Z g ; dim V = i ; dim π ( V )= n ; dim π ( V )= m P V ( τ g ) s UPERSTRING SCATTERING AMPLITUDES IN HIGHER GENUS 13 = N n,m ; i X V ⊂ Z k dim V = n P i − n sV · X V ⊂ Z g − k )2 dim V = m P i − m sV = N n,m ; i P ( k ) n, i − n s · P ( g − k ) m, i − m s This is of course zero if n > k or m > g − k . Taking the sum over allpossible n and m (recall that we have n + m ≥ i and n, m ≤ i ) givesthe theorem. (cid:3) Ans¨atze for genera ≤ in terms of vector subspaces We now rewrite the low genus superstring measure proposed byD’Hoker and Phong [8] for genus 2 and by Cacciatori, Dalla Piazza,and van Geemen [3] for genus 3 in terms of the modular forms P ( g ) i,s constructed and studied above. Following the earlier works, we try tofind the superstring measure as the product of the bosonic measure(some formulas for which are known, but explicit point-independentexpressions for which are apparently not known for high genus — seethe discussion in [8]) and a function Ξ ( g ) [∆] depending on the charac-teristic ∆. As argued in [3], p.5 the factorization property can then berewritten as a condition on Ξ ( g ) [∆]. The arguments in [3], 2.7 show (es-sentially arguing that if Ξ ( g ) [∆] is equal to γ Ξ ( g ) for some γ ∈ Sp(2 g, Z ),then its degenerations can be computed by acting by γ on the degen-erations of Ξ ( g ) ) that for g ≤ ( g ) of weight 8 with respectto Γ g (1 ,
2) satisfying the factorization constraintΞ ( g ) | H k ×H g − k = Ξ ( k ) · Ξ ( g − k ) for any k < g . The reason this statement is only proven for g ≤ M g and not on the entire space A g . For genus highenough the locus of decomposable abelian varieties A k × A g − k does notlie in (the closure of) the locus of Jacobians, and the superstring mea-sure on M g may not give rise to a modular form on A g . However, forgenus 4 any decomposable abelian variety is still a product of Jacobians(possibly of nodal curves), and the statement still holds.We will now rewrite the known low genus superstring measures interms of our forms P ( g ) i,s . Notice that we are looking for a form Ξ ( g ) ofweight 8, and thus we will look for it as a linear combination of(6) G ( g ) i := P ( g ) i, − i , which are the only G ′ s of appropriate weight. Since the only 0-dimensionalvector space is zero, and all the 1-dimensional spaces over Z consistof zero and another vector, we have G ( g )0 = θ [0] ; G ( g )1 = θ [0] X v ∈ Z g \{ } θ [ v ] . In genus 1 the superstring measure is known to be given byΞ (1) = θ (cid:20) (cid:21) η = θ (cid:20) (cid:21) θ (cid:20) (cid:21) θ (cid:20) (cid:21) . The modular forms we have are(7) G (1)0 = θ (cid:20) (cid:21) and G (1)1 = θ (cid:20) (cid:21) θ (cid:20) (cid:21) + θ (cid:20) (cid:21) ! Using the Jacobi relation — the only algebraic identity among the threeeven theta constants with characteristics in genus 1 — we can expressΞ (1) in terms of these, obtainingΞ (1) = 12 (cid:16) G (1)0 − G (1)1 (cid:17) . In fact looking in [3], p.9 we see that G (1)0 = θ (cid:20) (cid:21) (cid:18) f + η (cid:19) and G (1)1 = θ (cid:20) (cid:21) (cid:18) f + η (cid:19) . The genus 2 superstring measure was computed in [8], but us let ustry to look for it in the formΞ (2) = a G (2)0 + a G (2)1 + a G (2)2 . The decomposable locus in this case is H × H (together with itsSp(4 , Z ) conjugates), the restriction of Ξ (2) by theorem 15 is a G (1)0 · G (1)0 + a (cid:16) G (1)0 · G (1)1 + G (1)1 · G (1)0 + G (1)1 · G (1)1 (cid:17) + a G (1)1 · G (1)1 . Notice that here all the combinatorial coefficients N n,m ; i from theorem15 are equal to one, which is easy to see geometrically by remember-ing that this is the count of the number of i -dimensional subspaces of Z n ⊕ Z m projecting onto both factors. Some of the coefficients for therestrictions we compute for higher genus are not as obvious, and wemake substantial use of the theorem. For this restriction to be equalto Ξ (1) · Ξ (1) = 14 G (1)0 · G (1)0 − G (1)0 G (1)1 − G (1)1 · G (1)0 + 14 G (1)1 · G (1)1 we must choose a = , a = − , a = , thus verifying UPERSTRING SCATTERING AMPLITUDES IN HIGHER GENUS 15
Proposition 18.
The following ansatz: Ξ (2) = 14 (cid:16) G (2)0 − G (2)1 + 2 G (2)2 (cid:17) , while being a holomorphic modular form of weight 8 with respect to Γ (1 , , satisfies the factorization constraint. Remark 19.
Note that unlike the formula for the genus 2 amplitudeobtained in [8], and the various expressions for it studied in [12], ourformula for Ξ (2) involves syzygetic rather than azygetic sets — andthey all include the zero characteristic, so that the θ [0] factors appearsnaturally. To show that our ansatz is equal to the formulas given in [8]and [3] one expresses all theta functions with characteristics in terms oftheta functions of the second order using the bilinear addition theoremand verifies that an identity is obtained (using Maple, not by hand).We now search for a genus 3 ansatz in the formΞ (3) = a G (3)0 + a G (3)1 + a G (3)2 + a G (3)3 . Notice that this form is equivalent to the one used in [3]: our G (3)3 istheir G , and their F i ’s can be expressed as linear combinations of our G (3)0 , G (3)1 , G (3)2 , as can be verified by implementing the bilinear additiontheorem in Maple. In our form we can use theorem 15 to easily computethe restriction of Ξ (3) to the locus of decomposable abelian varieties,which again has only one component, H × H , to beΞ (3) | H ×H = a G (1)0 · G (2)0 + a (cid:16) G (1)0 · G (2)1 + G (1)1 · G (2)0 + G (1)1 · G (2)1 (cid:17) + a (cid:16) G (1)0 · G (2)2 + G (1)1 · G (2)1 + 3 G (1)1 · G (2)2 (cid:17) + a G (1)1 · G (2)2 . Requiring this to be equal toΞ (1) · Ξ (2) = 18 (cid:16) G (1)0 − G (1)1 (cid:17) (cid:16) G (2)0 − G (2)1 + 2 G (2)2 (cid:17) = 18 (cid:16) G (1)0 · G (2)0 − G (1)0 · G (2)1 − G (1)1 · G (2)0 +2 G (1)0 · G (2)2 + G (1)1 · G (2)1 − G (1)1 · G (2)2 (cid:17) (we arranged the terms to be in the same order as in the formula for therestriction, where we note that G (1)1 · G (2)1 and G (1)1 · G (2)2 appear twice)allows us to compute the coefficients a i starting from i = 0 uniquely toget Proposition 20.
The following ansatz: Ξ (3) = 18 (cid:16) G (3)0 − G (3)1 + 2 G (3)2 − G (3)3 (cid:17) , while being a holomorphic modular form of weight 8 with respect to Γ (1 , , satisfies the factorization constraints. We now look for a genus 4 ansatz in the formΞ (4) = a G (4)0 + a G (4)1 + a G (4)2 + a G (4)3 + a G (4)4 . This is the first case when we have two different reducible loci: H ×H and H × H . We again use theorem 15 to compute the restrictions.To unclutter the formulas we drop the upper indices on the P ’s andcomputeΞ (4) | H ×H = a G · G + a ( G · G + G · G + G · G )+ a ( G · G + G · G +3 G · G )+ a ( G · G + G · G +7 G · G )+ a G · G Requiring this to be equalΞ (1) · Ξ (3) = 116 ( G − G ) · ( G − G + 2 G − G )we can solve for a i term by term (and the solution is unique!). On theother hand, we must also have a correct factorization on H × H : wemust haveΞ (4) | H ×H = a G · G + a ( G · G + G · G + G · G )+ a ( G · G + G · G + 3 G · G + G · G + 3 G · G + 6 G · G )+ a ( G · G + G · G + 9 G · G ) + a G · G equal to Ξ (2) · Ξ (2) = 116 ( G − G + 2 G ) · ( G − G + 2 G ) . This can again be solved term by term to give a unique solution for all a i . Miraculously these solutions are the same (this is best checked byMaple, or, very tediously, by hand)) and we thus get Theorem 21.
The expression Ξ (4) := 116 (cid:16) G (4)0 − G (4)1 + 2 G (4)2 − G (4)3 + 64 G (4)4 (cid:17) is a modular form of weight 8 with respect to Γ (1 , , and satisfies thefactorization constraints. This is the unique such linear combinationof G (4) i , and is thus a natural candidate for the genus 4 superstringmeasure. UPERSTRING SCATTERING AMPLITUDES IN HIGHER GENUS 17 Further directions
There seem to be three natural further questions to ask, which wenow discuss one by one.
Question 1: Propose an ansatz for the superstring measure in anygenus.
The computations above seem to work miraculously, but thisis of course not a coincidence. By working carefully with the com-binatorics of the coefficients in the restriction formula in theorem 15one can always get a unique linear combination of G ( g ) i that restrictscorrectly. Theorem 22.
For any genus g the (possibly multivalued) function Ξ ( g ) := 12 g g X i =0 ( − i i ( i − G ( g ) i is a modular form of weight 8 (up to a possible inconsistency in thechoice of roots of unity for the different summands) with respect to Γ g (1 , , such that its restriction to H k × H g − k is equal to Ξ ( k ) · Ξ ( g − k ) .Moreover, Ξ ( g ) is the unique linear combination of G ( g ) i that restrictsto the decomposable locus in this way.Proof. Uniqueness is easy to see: indeed, the coefficients of G ( g ) i canbe computed inductively; notice that the coefficient of G i has to bethe same for all genera, as there is no dependence on g in the formulafor restrictions — see (8) below. Thus in genus g there is in fact onlyone new coefficient to compute, that in front of G ( g ) g , and this canbe computed from the coefficient of G (1)1 · G ( g − g − when restricting to H × H g − .The hard part is verifying that this ansatz works — a priori it couldhappen that imposing the restriction constraint for some H k × H g − k would be incompatible with the constraint for some other k ′ . Thus weneed to verify that for the ansatz above we have for all k Ξ ( g ) | H k ×H g − k = Ξ ( k ) · Ξ ( g − k ) . Notice that the product on the right-hand-side is a sum of products ofthe type G ( k ) n · G ( g − k ) m with coefficients given by the lower-genus ans¨atze.Theorem 15 shows that the left-hand-side is also a sum of terms of thesame kind, and thus what we need to prove is that the coefficients of G ( k ) n · G ( g − k ) m on both sides agree, i.e. that (notice that the powers of all cancel)(8) n + m X i =0 ( − i i ( i − N n,m ; i = ( − n n ( n − · ( − m m ( m − , where N n,m ; i is the number of i -dimensional subspaces of Z n ⊕ Z m surjecting onto both summands, given explicitly by (5). Note thatthe summands for i < max( n, m ) are automatically zero, but it isconvenient to include them formally. Notice that this identity does notdepend on g and k , which is why the coefficient of G ( g ) i in Ξ ( g ) does notdepend on i .While the quantity N has a geometric interpretation and thus (8)seems amenable to a geometric inclusion-exclusion proof, we give aneasy proof by induction, still using some geometry for the inductivestep. Note that n and m enter the formula symmetrically, so we caninduct in either, and note that identity (8) is obviously true for n = 0or m = 0, when there is only one summand on the left, and thereis no product to take. To perform induction, we use the followingcombinatorial Lemma 23.
The counting functions N n,m : i given explicitly by (5) sat-isfy the following recursion: (9) N n,m +1; i +1 = N n,m ; i + (2 i +1 − m ) N n,m ; i +1 . Proof.
Recall that N n,m +1; i +1 is the number of ( i + 1)-dimensional sub-spaces V of Z n ⊕ Z m +12 surjecting onto both summands under the pro-jection maps π and π . Let p : Z n ⊕ Z m +12 → Z n ⊕ Z m be the pro-jection forgetting the last basis vector (denote this vector by e m +1 ).Since p is a homomorphism, the map p : V → p ( V ) is either 2-to-1, in which case dim p ( V ) = i and V = p − ( V ), or it is 1-to-1 anddim p ( V ) = i + 1. We now count how many different V can give riseto a given p ( V ) ⊂ Z n ⊕ Z m (which still surjects onto both summands).If dim p ( V ) = i , then V = p − ( V ) is unique — thus we get the firstsummand in the lemma, with no coefficient.For the second case, choose ( p , . . . , p m ) ∈ p ( V ) such that p ◦ π ( p k ) = e k is the k ’th basis vector of Z m (this is possible since p ◦ π : V ։ Z m ).The vectors { p , . . . , p m } ∈ p ( V ) are linearly independent since theirprojections are. We can complete them to a basis { p , . . . , p i +1 } of p ( V )such that p ◦ π ( p k ) = 0 for m < k ≤ i + 1: to accomplish this, takeany basis of p ( V ) and subtract the appropriate sums of p . . . p m fromthe rest to make p ◦ π zero.To determine V given this p ( V ) it suffices to choose v , . . . , v i +1 suchthat p ( v k ) = p k , which amounts to choosing the e m +1 -coordinate of UPERSTRING SCATTERING AMPLITUDES IN HIGHER GENUS 19 each v k — thus there are 2 i +1 choices. For any such choice V willsurject onto Z n and onto Z m , but for V to surject onto Z m +12 it isnecessary and sufficient for there to exist m + 1 vectors in V withlinearly independent π projections. We can choose v , . . . , v m as m ofthis vectors, and thus the condition for π : V → Z m +12 to be surjectiveis for there to exist some vector in the span of v m +1 , . . . , v i +1 with non-zero e m +1 -coordinate. Thus unless the e m +1 coordinate of all v k is zero,i.e. unless v k = p k for all k = m + 1 , . . . , i (and then we have twochoices for each of v , . . . , v m — so there are 2 m such cases), V surjectsonto both summands of Z n ⊕ Z m +12 and is counted in N n,m +1; i +1 . Thusfor each p ( V ) of dimension i + 1 there are exactly 2 i +1 − m differentsubspaces V projecting to it that are counted in N n,m +1; i +1 . (cid:3) Remark 24.
Note that this proof works also in the case when someof the N ’s appearing in the formula are zero. Another proof of thelemma can be obtained (but not so easily guessed!) by writing out theformulas (5) for N ’s in terms of products and manipulating them using2 k +1 − j +1 = 2(2 k − j ), etc. Then one also has to check the caseswhen some N is zero separately, while the geometric argument worksin all cases.We now complete the proof of the theorem by inducting from m to m + 1. We substitute the recursive expression (9) from the lemmainto the left-hand-side of (8) to get (we use I = i + 1 for the index ofsummation) n + m X I =0 ( − I I ( I − N n,m +1; I = n + m +1 X I =0 ( − I I ( I − (cid:0) N n,m ; I − + (2 I − m ) N n,m ; I (cid:1) = − m n + m X I =0 ( − I I ( I − N n,m ; I + n + m X i =0 ( − i +1 ( i +1) i N n,m ; i + n + m X I =0 ( − I I ( I − I N n,m ; I where we used the fact that the i = − I = n + m + 1 summandsare zero in the last two sums, and the fact that formula (9) worksfor N ’s some of which are zero as well. Now we note that the twoexpressions in the last line are the same up to sign and renaming the variable from i to I , and thus they cancel, so that we finally use theinductive assumption that (8) holds for m to obtain − m n + m X I =0 ( − I I ( I − N n,m ; I = − m ( − n + m n ( n − + m ( m − which is equal to the expression( − n + m +1 n ( n − + m ( m +1)2 , the right-hand-side of (8) for n and m + 1. The step of the inductionis thus proven. (cid:3) Remark 25.
This ansatz is a direct generalization of the formulaswe obtained above for g ≤ G = P , is the sum of square roots of products ofthetas. It could well happen, and seems perhaps not quite unlikelyin view of Riemann’s quartic relations and Schottky-Jung identities(for an example of the Riemann quartic relation and the identitiesfor theta constants on the Schottky locus, see the discussion of thegenus 3 situation in [4]), that the product of 32 theta constants withcharacteristics in a vector subspace may indeed admit a holomorphicroot over M g . In this case the expression above would be a naturalcandidate for the superstring measure. Note that the product of alltheta constants has a holomorphic square root in genus 3 by resultsof Igusa; this kind of condition for the square root to be holomorphicwas also encountered in the first attempts to compute the genus 3superstring measure in [12, 13]. Question 2: Verify that the proposed ansatz satisfies further physicalconstraints, for example that it yields a vanishing cosmological constantand vanishing 2- and 3-point functions.
Showing that the cosmologicalconstant vanishes is equivalent to showing that the sum P ∆ Ξ ( g ) [∆] isidentically zero. This has been verified for genus 2 in [8] and for genus3 in [3]. In general notice that this sum, if non-zero, is a modular formwith respect to the entire group Sp(2 g, Z ) of weight 8. From the factor-ization constraint being satisfied we know that it restricts to the locusof decomposable abelian varieties as the product of the correspondinglower-dimensional sums, which we can inductively assume to vanish.In particular in genus 4 this sum vanishes on the locus A × A andthus on the boundary of A g , which implies that this sum, if non-zero,is a modular form of slope at most 8.However, it is known that the slope of the effective cone of M isequal to 6 + >
8, and it is in fact known that the Schottky locus
UPERSTRING SCATTERING AMPLITUDES IN HIGHER GENUS 21 M ⊂ A is the zero locus of the unique modular form of slope 8 on A ,the Schottky equation. Thus the form P ∆ Ξ (4) [∆] must be a (possiblyzero) multiple of this Schottky equation, and thus vanishes identicallyon M , so our ansatz does produce a vanishing cosmological constantin genus 4.It seems very hard to extend a similar kind of argument to highergenus, where the slopes of effective divisors on M g and A g are notknown.Another constraint on the measure is to verify that all the 2- and3-point functions vanish. This was verified for the genus 2 measure in[14, 15], and checking this for the proposed ans¨atze in genera 3 and 4would be a good indication of their potential validity. Question 3: Investigate whether the above restrictions are sufficient toguarantee the uniqueness of the solution for the superstring measure.
Ifwe restrict ourselves to looking for the superstring measure as a prod-uct of the bosonic measure and a modular form of weight 8, suppose g is the lowest genus for which the ansatz is not unique, i.e. whenthere exist two distinct modular forms of weight 8 for Γ g (1 ,
2) with theidentical restriction to the decomposable locus. Then their difference F would be a modular form F of weight 8 with respect to Γ g (1 ,
2) van-ishing on all the components A k × A g − k of the locus of decomposableabelian varieties in A g . If it could be shown from the theory of modularforms that such an F is then identically zero, then uniqueness of theansatz in genus g would follow. This seems to be a really hard ques-tion, as the ring of modular forms for Γ g (1 ,
2) for genus g ≥ g (1 ,
2) and obtainingconditions guaranteeing the vanishing of such a form seems very hard— see [22]. The authors of [3] indicate that they will give a proof ofuniqueness in genus 3 in a forthcoming paper.Also note that while in [8, 9, 10, 11] the formula for the genus 2superstring measure was derived from the first principles and as suchhas to be unique, the derivations in higher genus are based on theassumption of the superstring measure being a product of a modularform and the bosonic measure, which then needs to be justified in somephysical way.
Acknowledgements
I learned about the problem from Duong Phong, to whom I amvery grateful for his constant encouragement, for explaining the basicquestions and computations for the string measure, and for detailed comments on the draft of this text. I am very thankful to Eric D’Hokerand Duong Phong for many conversations on the subject, for sharingtheir Maple code, for ideas on how the higher genus superstring measurecould be constructed, and especially for suggesting that I express thegenus 3 ansatz of Cacciatori, Dalla Piazza, and van Geemen in termsof syzygy conditions, which allowed me to then further generalize it.I am also very grateful to Riccardo Salvati Manni for bringing tomy attention his work [23] and the literature on polynomials P ( N m )corresponding to even cosets, and for the encouragement in exploringwhether these can be used to obtained an ansatz. I am very thankfulto Riccardo Salvati Manni for pointing out and giving a rigorous proofin [25] that P ( g ) i,s are only known to be modular forms for 2 i s = 2 k for k ≥ P i,s [∆] are modular with respect toconjugates of Γ g (1 ,
2) rather than Γ g (1 ,
2) itself, and for comments onthe manuscript.
References [1] Beilinson, A., Manin, F.:
The Mumford form and the Polyakov measure instring theory.
Commun. Math. Phys. (1986) 359–376.[2] Cacciatori, S.L., Dalla Piazza, F.:
Two loop superstring amplitudes and S representations. Lett. Math. Phys. (2008); arXiv:0707.0646.[3] Cacciatori, S.L., Dalla Piazza, F., van Geemen, B.:
Modular Forms and ThreeLoop Superstring Amplitudes. arXiv:0801.2543.[4] van Geemen, B., van der Geer, G.:
Kummer varieties and the moduli spaces ofabelian varieties , Amer. J. of Math. (1986) 615–642.[5] Green, M.B., Schwarz, J.H.:
Supersymmetrical string theories.
Phys. Lett. B (1982) 444–448.[6] Gross, D.J., Harvey, J.A., Martinec, E.J., Rohm, R.
Heterotic String Theory(II). The interacting heterotic string.
Nucl. Phys. B (1986) 75.[7] D’Hoker, E., Phong, D.H.:
Multiloop amplitudes for the bosonic Polyakov string ,Nucl. Phys. B (1986) 205–234.[8] D’Hoker, E., Phong, D.H.:
Two-Loop Superstrings I, Main Formulas.
Phys.Lett. B (2002) 241–255; hep-th/0110247.[9] D’Hoker, E., Phong, D.H.:
Two-Loop Superstrings II, The chiral Measure onModuli Space.
Nucl. Phys. B (2002) 3–60; hep-th/0110283.[10] D’Hoker, E., Phong, D.H.:
Two-Loop Superstrings III, Slice Independence andAbsence of Ambiguities.
Nucl. Phys. B (2002) 61–79; hep-th/0111016.[11] D’Hoker, E., Phong, D.H.:
Two-Loop Superstrings IV, The Cosmological Con-stant and Modular Forms.
Nucl. Phys. B (2002) 129–181; hep-th/0111040.[12] D’Hoker, E., Phong, D.H.:
Asyzygies, modular forms, and the superstringmeasure I.
Nucl. Phys. B , 58 (2005); hep-th/0411159.[13] D’Hoker, E., Phong, D.H.:
Asyzygies, modular forms, and the superstringmeasure. II.
Nucl. Phys. B , 83 (2005); hep-th/0411182.[14] D’Hoker, E., Phong, D.H.:
Two-Loop Superstrings V, Gauge Slice Indepen-dence of the N-Point Function.
Nucl. Phys. B (2005); hep-th/0501196.
UPERSTRING SCATTERING AMPLITUDES IN HIGHER GENUS 23 [15] D’Hoker, E., Phong, D.H.:
Two-Loop Superstrings VI, Non-RenormalizationTheorems and the 4-Point Function.
Nucl. Phys. B (2005); hep-th/0501197.[16] Igusa, J.-I.: Theta functions. Die Grundlehren der mathematischen Wis-senschaften, Band 194. Springer-Verlag, New York-Heidelberg, 1972.[17] Krazer, A.: Lehrbuch der Thetafunktionen, B. G. Teubner, Leipzig, 1903.[18] Manin, Y.,
The partition function of the Polyakov string can be expressed interms of theta functions , Phys. Lett. B (1986) 184–185.[19] Matone, M., and Volpato, R.
Higher genus superstring amplitudes from thegeometry of moduli space.
Nucl. Phys. B (2006) 321–340; hep-th/0506231.[20] Oura, M., Poor, C., Yuen, D.S.,
Toward the Siegel ring in genus four,
Int. J.Number Th., to appear.[21] Oura, M., Salvati Manni, R.:
On the image of code polynomials under thetamap , preprint.[22] Poor, C., Yuen, D.S.,
Linear dependence among Siegel modular forms.
Math.Ann. (2000) 205–234.[23] Salvati Manni, R.:
On the dimension of the vector space C [ θ m ] , Nagoya Math.J. (1985) 99–107.[24] Salvati Manni, R.: Modular varieties with level 2 theta structure , Amer. J.Math. (1994) 1489–1511.[25] Salvati Manni, R.:
Remarks on Superstring amplitudes in higher genus , Nucl.Phys. B, to appear, arXiv:0804.0512.[26] Verlinde, E., Verlinde, H.
Chiral Bosonization, determinants and the stringpartition function.
Nucl. Phys. B (1987) 357–396.(1987) 357–396.