On a formula of spin sums, Eisenstein-Kronecker series in higher genus Riemann surfaces

We discuss a decomposition formula of simple products of fermion correlation functions with cyclic constrains and its applications to spin sums of super string amplitudes. Based on some facts which are noted or derived in this paper, we propose a candidate of the form of this decomposition formula for some of higher genus cases which includes genus two case. Although we had to use several conjectures and assumptions due to unsolved mathematical difficulties, the method described in the text may be an efficient way to obtain the decomposition formula in higher genus cases. In particular, for those cases, we propose a concrete method to sum over non singular even spin structures for the product of arbitrary number of the fermion correlation functions with cyclic constraints in super string amplitudes. We also propose an explicit generalization of Eisenstein-Kronecker series to the higher genus cases in the process of considerations above.


Genus one
What we consider in this paper is a simple product of fermion correlation functions In this chapter we review the case of genus 1 as simple as possible, in a form which may be useful to generalize to higher genus cases. Many of the details will be analogously mentioned in the following chapters.
At genus one, we discuss Π =1 ( − +1 ) under the constraint +1 = 1 where (1.12) The main point of this formula is that the boundary dependence ( dependence) of the product of fermion correlation functions is factored out as (2 −2) ( ) that does not depend on the vertex inserting points 1 , 2 , … but on moduli parameter only [2]. Therefore, the whole of spin sum ∑ =1,2,3 ∏ ( ) =1 can be simplified as ] 4 can be written as , , ( 1 − 2 )( 2 − 3 )( 3 − 1 ) for = 1,2,3 respectively. (1.14) For arbitrary value of , if one multiplies on the simple product of fermion correlation functions by partition functions and perform a spin sum, it becomes a simple algebraic sum on the branch points that depends only on the moduli. For example, . (1.15) The numerator contains the denominator for arbitrary values of , and the whole of (1.15) can be always written as symmetric functions of 1 , 2 , 3 and hence by Eisenstein series [2] [4].

( , )
, (1.17) or use The product Π =1 ( , , ) can be regarded as an elliptic function of parameter of because of the condition ∑ =1 = 0, and so can be expanded by the derivatives of Weierstrass Pe function as where , ( ) are expansion coefficients defined here which does not depend on .
There is no , −1 ( ) term because no single order pole of should exist in an elliptic function of . The order of the pole in the last term ( −2) ( ) is which is the same order of poles on the right-hand side of (1.22) as a function of . There is no need to consider terms with higher order poles of .
Here we modify log of Π =1 ( , , ) . Using the right-hand side of (1.20) can be written as where the factor in Π =1 ( , , ) was extracted out from the exp function.
The whole is already the same as the contents of function except the pole factor 1 .
We multiply on the both side of (1.22) and consider differentiating times and set = 0 . Since Pe function has the following single term singularities: The term , ( ) ( −2− ) ( ) in eq. (1.22) multiplied has the form On both sides of eq. (1.22) multiplied , there is no singular term, and for a fixed value of ( first assume < ) there is only one term of order in the whole of the right-hand side of the equation, which is , ( ) (−1) − ( − − 1)! .
Therefore, for the case < , differentiating the both sides of times and set = 0 we have , ( ) : To obtain , ( ) ( = ), we note that every term on the right-hand side of (1.22) after multiplied has a constant term except for the case of Differentiating the both sides of (1.22) times after multiplying , we have (1.28) under the condition ∑ =1 = 0 can be obtained by setting equals to half period : Since ( ) ( ) = 0 , by defining a new variable as − = 2 , this can be simplified as: This has the same form of (1.9), (1.10), (1.11). We describe some simple calculations and observations for later convenience in chapter 2 and 3 in the below.
For a single factor of ( , , ) , we modify as We used the k=1 term in the exp function 2 2 2 ( ) to express the sigma function by 1 ( , ). 2 ( ) relates to the zeta function value at one of the half periods as We note that ( − 1)! exactly cancels the constant term ( ( ) 0 term) of the function 1 ( , ) when it is Laurant expanded. We temporally call this ( ) 0 term −( − 1)! "zero mode" of the 1 ( , ) , just for short calling, for later chapter convenience.
We define the "zero mode subtracted" functions 1 ( ) as In the calculation of , only 1 ( ) function appears. This can be simply understood by looking at the form of as follows.
The three sets of terms of sum in the exp function of come from all the same function, , is the same form of the function and gives 1 ( , ) .
One more set of terms − ∑ 1 ( ) =1 is only to make the inside of exp function zero after the calculation. Then it is obvious that the ( ) 0 term in the expansion form of 1 ( , ) should always be the same as the left function, ( ) 0 term with opposite sign.
The reason why we describe this simple observation in detail is as follows.
As was pointed out in [5], the factor 1 ( , ) in the denominator in the right-hand side of the product of Eisenstein-Kronecker series is a key to the residue condition As we saw, because the product of fermion correlation is related to Π =1 ( , , ) as , in the decomposition formula of Π =1 ( ), the effect of such 1 ( , ) are all included in a form as 1 ( ) . As we will see later, in some cases of higher genus, similar formula holds, and hence the zero mode subtractions will also hold in those cases. The description here is for the preparation of such higher genus cases.
The set of these "zero modes" is apparently a set of all coefficients of non-singular terms in the expansion of 1 ( , ) . In genus 1, those are −( − 1)! , essentially holomorphic Eisenstein series. The first one is − 2 , which is equal to −2 1 , the constant used in the definition of Pe function.
When we expand the right-hand side of (1.32), the coefficient of for arbitrary can be explicitly written down by considering the form of weight Schur polynomial.
For any and Λ the following equation holds: where ( ) is a polynomial of s given as Below are some examples to be used later: We calculate the right-hand side of (1.9): This says that, since = −2 1 − +1 (2) (0) +1 (0) , the spin structure dependence of the product Π =1 ( ) under the condition (1.2) is entirely through only one kind of theta constants, +1 (2) (0) +1 (0) . There exists a natural generalization of this formula in hyperelliptic arbitrary genus cases when represents non-singular even characteristics.
(1.49) Then eq. (1.47) can be also written as  ] . (1.57) As long as fermion field correlation functions are concerned, we have to set the common parameter to in the end in the whole of the product , to derive a decomposition formula of the product of fermion correlation functions.
Setting = 0 is only to obtain the coefficients , .
In the decomposition formula 2. higher genus
These branch points 1 , 2 , . . . . . 2 +2 are from left to right, and there are cuts between 1 and 2 ; 3 and 4 ; … 2 +1 and 2 +2 = ∞ . (Please see Figure 1 of [11].) What we consider here is a simple product of fermion correlation functions The Jacobi theta function in higher genus is defined in a standard notation: The theta function is called odd or even depending on whether the inner product 4 • is even or odd.
Let 1 , 2 , . . . and 1 , 2 , . . . . be a canonical homology basis. We choose canonical holomorphic differentials of the first kind 1 , 2 , . . . and associated meromorphic differentials of the second kind 1 , 2 , . . . . The periods are given as Please note that in this paper ∮ is normalized, ∮ is denoted as . The factor − 1 2 in front of the definition is introduced to get the consistency to the genus 1 case of 1 relations to Pe function and Weierstrass relations (1.43).
In the following, , which corresponds to 1 in the genus 1, will play an important role.
As long as fermion correlation functions are concerned, in the case of = 2 for example, the theta function which has a set of following characteristics may play the same role of 1 ( ) of the genus 1: ) . (2.5) There is natural generalization for any higher genus in the hyperelliptic case, but as long as I know, the Laurent expansion form of [ ] ( ) is not known in any genus except = 1.
Consider the abelian image of each of branch points as Ω ≡ ∫ This is the fundamental tool on considering half periods in higher genus Riemann surface.
Each of Ω is the component vector for all of holomorphic one forms , The index is attached to each of branch points , = 1,2, … 2 + 1.
When = 2 + 2 , the integral is zero. The results of the integrals have a form Both of and for each are dimension vectors whose all elements are either of Ω is also dimension vectors whose components depends on holomorphic 1 forms for = 1,2, … . Strictly speaking Ω should be written as Ω , but we will sometimes abbreviate the index .
Please also note that the result of the integral is not denoted as + , but as + .
Since and are often used as characteristics of theta functions, we call the pair even or odd when the inner product 4 • is even or odd.
For example, in the case of = 2, since there are 2 + 2 = 6 branch points, We are also interested in the following combinations of Ω : Select any of combination of 2 indices , out of non-zero 2 + 1 number of Ω in the case of genus , we define Similarly, we define Ω , which is any three combinations of Ω , that is, ( + + ) + ( + + ) .
We define the following sum, selecting only even number of in Ω and summing up all of them in genus : ) . (2.9) is called the vector of Riemann constants with base point ∞ .
In this sense, we call that Ω + are non-singular and odd half-periods, and Ω + are non-singular even half periods at genus 2, even though [Ω ] are not always odd, and [ Ω ] are not always even. For even characteristics , the word non-singular means that the value of the theta function with zero characteristic at zero is not vanishing, (0) ≠ 0 .
We are only interested in non-singular even half periods here, and so 10 Ω + are the = 2 extension of half periods , = 1,2,3 in genus 1.
The argument above can be extended to arbitrary genus g in hyperelliptic case as follows. For the details, please see [11] for example.
It is known that in genus g, there are Then [ + Ω ] will be the characteristic of non-singular even spin structures.
In the following, the theta function in genus whose characteristics are Riemann constants will be denoted as: is odd, and even if ( +1) 2 is even, for the genus .
The following fact will be the starting point on the arguments on decomposition formula in higher genus.
Consider a function at any genus in hyperelliptic case defined as where is the -dimension vector = ( 1 , 2 , … ) . The is a -dimension parameter, ( , ) is the Prime form, and set = ( − ) .
As long as fermion correlation functions are concerned, the function ( , ) defined in (2.13) may be regarded as a generalization of Eisenstein-Kronecker series in genus 1.
We list up some facts and assumptions as (A) ~ (G) which will be helpful to pursue further arguments below. (A) is about the main direction of the methods in this paper, and also theorems (D) and (E) are important for our purpose.
Higher genus Pe functions are defined using theta functions with Riemann constant characteristics as:  (A) What we mainly consider is the product of ( , ) which satisfies cyclic condition +1 = 1 : . (2.20) We assume that the right-hand side of (2. The order of poles 2 of these functions of are 2, 3, 4, . , and the set of those will have enough dimension of function space to obtain the coefficients of the expansion. On the base functions of expansion, we will discuss some more in Appendix B. From the form of the Prime forms in the product of fermion correlation function of the left-hand side of (2.20) under the condition +1 = 1 , the whole of the product of eq.
(2.20) will be holomorphic 1 forms, and the dependence of 1 , 2 , … will be all different. This is also supported by the results of ref.
[7] [8]. That is, one may have to seek possible form as does not depend on Pe function, depends on 1 2 3 , …., . Implicit sums over 1 2 … = 1,2, … are assumed. In higher genus, the notation of , 1 2 … is used instead of , due to the intricate contractions of indices between the modular covariant functions parts and Pe functions parts.
Basically, we are interested in the pole structures on the difference of the vertex inserting points − +1 . The product of fermion correlation functions should be the one which can reproduce "simultaneously single poles of the products of − +1 for different " , as in the case of genus one. However, in the following, only the differentiations with respect to defined as = ( 1 , 2 , … ) with will appear. These are the variables in higher genus theta functions. We denote those differentiations as or . We will seek possible forms of (E) It holds that, as the generalization of Weierstrass formula of genus 1 to the arbitrary genus (See eq. (2.100) in [10]. Original paper is [12] ) , in hyperelliptic case, for , = 1,2, … . The represents any non-singular even spin structure. See eq.
The following (F) and (G) will also be validated in general. In particular, there is a factor ( ) in the denominator of the product. Therefore, the same argument in the case of genus one, on page 8 and 9 will hold. That is, the "zero mode subtraction" procedure which we saw in the case of genus 1 will hold too by the same reason.
We define the zero mode as We also define In the following, we adopt one more procedure. We restrict our argument to the case that ( ) is odd, that is the genus g satisfies

(2.35)
One natural candidate of the form of 3, may be constructed as follows.
Define as The strict meaning of differentiations is as follows. Define ≝ ( − +1 )

Each is a dimension vector, = (∫ +1 )
The meaning of is that for each term of ( ) inside the sum of , differentiate with respect to , and the index is contracted with .
This is well-managed by using a g-component parameter = ( 1 , 2 , … ) and for ∑ ln ( ( − +1 ) + ) =1 , do the differentiation with respect to and set = 0 , to give the result of differentiating with respect to different variables for each theta functions automatically, as we saw in the genus one case.
Differentiation with respect to may be contracted with ( ) even when and are different from each other. Therefore, even after using chain rule of differentiation, 3 is not equal to − 1 2 3 3 3 (which is equal to zero.) The middle terms in the right-hand side of ( We write the form There will be another kind of numerical factors which must be considered. In genus one, the origin of the factor 1 (2 −1)! in front of eq. (1.10) was naturally explained in the proof of the decomposition formula. This factor was first correctly guessed in [3].
The similar over-all factor will appear in the higher genus case too and we may have to guess it. This is not a trivial issue and I have no idea on it at present. The difficulty comes from the fact that it is not so straightforward to express 1  in the proof given in chapter 1, the product of fermion correlation functions will have the following closed form: In the last line, the sum is beginning from = 2 because the first zero mode lambda Λ , which will be −2η for any genus, is already included in the Pe function in the second line. This can also be written as In the genus 1 case we can apply (2.49) rigorously. we can write the formula (1.9) as .
On the other hand, zero mode subtraction in 2 1 ( ) originates from the zero modes ( 0 terms) in the set of expansion base functions , (1) , (2)  The conventions used in the equations above are as follows.
In genus 2 , 4-point amplitudes, the following spin sum was also described in [7]: The even order of poles, we prepare the product of ( ) s. For odd order of poles, we multiply odd order pole functions, ( ), ( ) , … to the product of ( ) s.
We assume that these base functions gives enough function space to expand the function Π =1 ( ( − +1 ), ) under the cyclic condition +1 = 1 . In the genus 1, there is different way of expansion from that of (1.22), and it has similar features of this expansion base functions. We briefly described those in appendix B.
Up to = 3, only ( ) , ( ) are necessary as such base functions, so the form of the expansion formula will be the same as in the last chapter: where numerical factors in each term are all abbreviated because it is difficult to determine those without clarifying the pole structures of ( ) .
As for = 4,5, we need ( ) , ( ), ( ) ( ) as the base functions of expansion, and the result may be written after setting = Ω as follows: At present, details of spin structure independent term ⏞ are not clear.
We can't guess the form of .
In higher point amplitudes, because the pole structures of ( ) ( ) ( ) … becomes complicated, it becomes difficult to guess the form of the coefficients of the expansion at present. (See Appendix B.) The may include "the second zero mode" Λ , and may include combinations of terms which appeared in = 4 case.
In general, to clarify the coefficients of expansion of the product Π =1 ( , ) defined in (2.20), we first need to know the mathematical structure of the expansion form of ( ) .
All of the spin structure dependent terms in Π =1 ( , +1 ) have the form of the direct products (simple products) of (Ω ), that is, When the decomposition formula is applied to string amplitudes, partition functions , which will also be functions of branch points, are multiplied to the simple products of (Ω ) and summed over . This spin sum is simple algebra of 2 + 1 branch points only, and there is no principal difficulty for arbitrary by the procedure already described, although calculations will be quite heavy for large values of or .
The general results of the spin sum will be symmetric functions of branch points 1 , 2 , … 2 +1 , as in the case of genus 1.
These symmetric functions will be generalized holomorphic Eisenstein series. All zero modes in the expression of the coefficients written in terms of

Summary and Conclusions
In this paper, we tried to clarify some properties of simple product of fermion in genus ≥ 2 to apply to simplified RNS formalism calculations of superstring amplitudes.
As is well known, as clarified in [7] and related papers, for higher genus ≥ 2 , even in the case of calculating amplitudes of external massless bosons in Type I and Type IIB super strings, we need to incorporate other type of vertex operators in which cyclic conditions are not satisfied. The main purpose of this paper was to consider efficient way of summing over spin structures of the type with cyclic condition only.
In genus 1, under such cyclic condition, spin structure dependent part of the product can be extract out as We expect that this function will play the same role as in the genus 1 case.
We demonstrated some concrete calculations using a conventional function  with cyclic condition will have the form of direct products of (Ω ), that is, This fact itself may be validated for the non-singular and even cases for arbitrary genus, at least for the hyper elliptic curves.
Then the method described in chapter 2 will give a concrete way of calculating summing over non-singular even spin structure for arbitrary naturally. This calculation will also give various combinations of holomorphic 1 forms in the results.
The simplest examples are shown in subsection 2.4.
It is desirable to have a closed form of the decomposition formula in future.
When the decomposition formula is applied to calculate super string amplitudes in RNS formalism, the "generalized holomorphic Eisenstein series" may appear in the integrand of the amplitudes in 3 different ways.
The first one is through 1 2 … ln ( ) in the function 1 , 2 ,.. ( 1 , 2 , … . ). The second one will be through zero modes of Pe functions as the base functions of expansion themselves, as seen in the subsection 2.3. These will be included in the spin structure independent terms in the decomposition formula such as ⏞ and ⏞ in (2.73) and (2.74).
The third one will be through the result of spin sum. The result of the spin sum of the products of (Ω ) will be in general symmetric functions of branch points 1 , 2 , … 2 +1 . These symmetric functions will also will be expressed by generalized holomorphic Eisenstein series. All these as well as the boson field Wick contractions will consist integrands of string amplitudes.
In genus 1, Pe function is expressed by 1 ( ) and 1 , and the product of fermion correlation functions with cyclic condition is expressed by 1 ( ) and through where m, n are integers.
Starting from the infinite product representation of sigma function at genus 1, The coefficient of the highest degree term in (2 −2) is (2 − 1)! . This determinant will be denoted by If is odd, the last term of the right hand side of (B.1) should be replaced with If one adopts (1.22), it is possible to write down a compact closed form of the decomposition formula as (1.9). This is the reason we adopt (1.22) in chapter 1.
Instead, the spin structure dependent part becomes a little bit complicated form, Instead, the forms of coefficients , ( ) become rather complicated, because the expansion base functions in the right hand side of eq.(B.1) contain products of Pe functions such as ( ) and (1) ( ) ( ) . Inserting the expansion form of and (1) given in (A.8), the process of getting the coefficients , ( ) as in the proof of decomposition formula in chapter 1 becomes not so straightforward 6 .
In higher genus, as we saw in chapter 2, we have to consider the dimension of function space spanned by the expansion base functions. Very roughly speaking, as for "even order" of poles of expansion base functions in in higher genus, the set of the products of Pe function, ( ) , ( ) ( ), ( ) ( ) ( )…. will span the largest function space. Or these direct products of Pe functions should be included in the desired base functions, instead of ( ), ( ), …. . We will also need to prepare other functions which have the "odd order" poles. In genus 1, if base functions of expansion, ( ), 2 ( ), 3 ( ), 4 ( ) … are prepared as functions with even order poles, only one more, (1) ( ) , is multiplied to those, and odd order poles are prepared. In genus one this is enough as preparing all of base functions.
In higher genus, if the following formula is valid as a generalization of (A.17), the same procedure may be validated: For odd ( ≥ 5), 1 2 … ( ) can be expressed as follows: for example. Those functions with "odd order " poles may affect the forms of the expansion coefficients and so this should be clarified in future.
In a sense, for higher genus cases, we are forced to generalize the expansion of the form (B.1), instead of (1.22). By setting = Ω , the base functions which have factors of the Pe function 1 2 .... for odd will be zero, so only the direct products of Pe functions (Ω ) , (Ω ) (Ω ), (Ω ) (Ω ) (Ω )…. will be remained non-zero in the final form of the decomposition formula of products of fermion correlation functions with cyclic conditions, as the most general result of spin structure dependent terms. Therefore, spin sum will be performed by the method described in the text.