Multiloop Amplitudes of Light-cone Gauge Superstring Field Theory: Odd Spin Structure Contributions

We study the odd spin structure contributions to the multiloop amplitudes of light-cone gauge superstring field theory. We show that they coincide with the amplitudes in the conformal gauge with two of the vertex operators chosen to be in the pictures different from the standard choice, namely (-1,-1) picture in the type II case and -1 picture in the heterotic case. We also show that the contact term divergences can be regularized in the same way as in the amplitudes for the even structures and we get the amplitudes which coincide with those obtained from the first-quantized approach.


Introduction
String field theory is expected to provide a nonperturbative formulation of string theory. It is a second-quantized string theory from which one can calculate Feynman amplitudes which agree with those of the first-quantized theory. For bosonic strings, there are several proposals of such string field theories. For superstrings, because of the problems with the method to calculate multiloop amplitudes using the picture changing operators, the construction of a string field theory has been a difficult problem. Recently, Sen has constructed a gauge invariant formulation of the string field theory for closed superstrings [1][2][3][4][5], based on the formulation [6] of closed string field theory for bosonic strings with a nonpolynomial action.
Light-cone gauge closed superstring field theory is a string field theory for superstrings which involves only three-string interaction terms. It can be proved formally that the Feynman amplitudes of the string field theory coincide with those of the first-quantized theory [7].
The proof was formal because there appear unphysical divergences which are called the contact term divergences [8][9][10][11][12]. In a previous paper [13], we have shown that these divergences can be dealt with by dimensional regularization. In the case of type II superstrings, for example, one formulates a light-cone gauge superstring field theory in noncritical dimensions or the one whose worldsheet theory for transverse variables is a superconformal field theory with central charge c = 12 [14]. Although Lorentz invariance is broken by doing so, it does not cause so much trouble because the light-cone gauge theory is a completely gauge-fixed theory. In [13], we have shown that the multiloop amplitudes corresponding to the Riemann surfaces with even spin structure involving external lines in the (NS,NS) sector can be calculated using the dimensional regularization and the results coincide with those of the first-quantized approach.
What we would like to do in this paper is to generalize these results to the case of the surfaces with odd spin structure. On the Riemann surfaces with odd spin structure, there exist zero modes of the fermionic variables on the worldsheet which make the manipulations of the amplitudes complicated. We will show that it is possible to deal with these zero modes and prove that the amplitudes are equal to those of the first-quantized method, when all the external lines are in the (NS,NS) sector, in the case of type II superstrings. It is straightforward to obtain similar results for heterotic strings.
The organization of this paper is as follows. In section 2, we review the results in [13] and the problems with the odd spin structures. In section 3, we deal with the amplitudes for the odd spin structure and show that these also coincide with those from the first-quantized approach. Section 4 is devoted to discussions. In the appendices, we present details of the manipulations given in the main text.

Light-cone gauge superstring field theory
In this section, we review the known results for the multiloop amplitudes of light-cone gauge superstring field theory and the problems with the odd spin structures.

Light-cone gauge superstring field theory
In the light-cone gauge string field theory, the string field is taken to be an element of the Hilbert space H of the transverse variables on the worldsheet and a function of In this paper, we consider the string field theory for type II superstrings in 10 dimensional flat spacetime as an example. |Φ(t, α) should be GSO even and satisfy the level-matching condition where L 0 ,L 0 are the zero modes of the Virasoro generators of the worldsheet theory.
The action of the string field theory is given by  which consists of the kinetic terms and the three-string interaction terms. B and F denote the sums over bosonic and fermionic string fields respectively. The three-string vertices V 3 | are elements of H * ⊗ H * ⊗ H * whose definition can be found in [13,15,16].  N is given as a path integral over the transverse variables X i , ψ i ,ψ i on the light-cone diagram. A light-cone diagram consists of cylinders which correspond to propagators of the closed string. On each cylinder one can introduce a complex coordinate ρ = τ + iσ , (2.6) whose real part τ coincides with the Wick rotated light-cone time it and imaginary part σ ∼ σ + 2πα r parametrizes the closed string at each time. The ρ's on the cylinders are smoothly connected except at the interaction points and we get a complex coordinate ρ on Σ. The path integral on the light-cone diagram is defined by using the metric ρ is not a good coordinate around the interaction points and the punctures, and the metric (2.7) is not well-defined at these points. F (g) N can be expressed in terms of correlation functions defined with a metric dŝ 2 = 2ĝ zz dzdz which is regular everywhere on the worldsheet, Here z is a complex coordinate of the Riemann surface and the coordinate ρ becomes a function ρ(z) of z (see e.g. [18][19][20]). S LC X i , ψ i ,ψ i denotes the worldsheet action of the transverse variables and the path integral measure dX i dψ i dψ i ĝzz is defined with the metric dŝ 2 = 2ĝ zz dzdz. Since the integrand was defined by using the metric (2.7), we need the anomaly factor e − 1 2 Γ[σ;ĝzz] , where As was demonstrated in [13], if all the external lines are in the (NS,NS) sector and the spin structure for the left and right fermions are both even, the term in the sum in (2.8) can be recast into a conformal gauge expression: Here S tot denotes the worldsheet action for the variables X µ , ψ µ ,ψ µ (µ = +, −, 1, . . . , 8), ghosts and superghosts, is the picture changing operator (PCO),X (z) is its antiholomorphic counterpart and T F denotes the supercurrent for ∂X µ , ψ µ . The contours C K and ε K = ±1 are chosen so that the antighost insertions correspond to the moduli parameters for the light-cone amplitudes.
The vertex operator V is the supersymmetric DDF vertex operator given by −s are similarly given for the antiholomorphic sector. Here we use the notation (2.15) z = (z, θ) denotes the superspace coordinate on the worldsheet and X + L denotes the leftmoving part of the superfield X + . 3 We take the vertex operators to satisfy the on-shell V DDF r (Z r ,Z r ) turns out to be a weight 1 2 , 1 2 primary field made from X µ , ψ µ ,ψ µ . Therefore V (−1,−1) r (Z r ,Z r ) is an on-shell vertex operator in (−1, −1) picture. It is easy to see that the expression of the amplitude (2.5) given as an integral of (2.10) is BRST invariant.
One way to derive the expression (2.10) is as follows [13]. Using a nilpotent fermionic charge, it is possible to show that the right hand side of (2.10) is equal to In this form, the path integral factorizes into the contributions from X µ , ψ µ ,ψ µ , ghosts and superghosts. Each of these contributions is calculated by takingĝ zz to be the Arakelov metric g A zz [21]. In the matter sector, integration over the longitudinal variables yields where 3 Although X + L is not a well-defined quantity, it is used as a short-hand notation to express the vertex operator (2.13), which is well-defined.
Here ϑ[α] denotes the theta function with characteristic α and Ω is the period matrix.
α L and α R denote the characteristics corresponding to the spin structures of the left-and right-moving fermions respectively. Z X ± [g zz ] and Z ψ ± [g zz ] are respectively the partition functions of the free variables X ± and ψ ± ,ψ ± on the worldsheet endowed with the metric ds 2 = 2g zz dzdz. z I (r) denotes the interaction point at which the r-th string interacts. The contributions from the ghosts and superghosts are given as

Dimensional regularization
The amplitudes of superstring theory were calculated using the first-quantized formalism in [22] in which an expression using the PCO's was given. The expression (2.10) is a special case of the one in [22], where the PCO's are placed at the interaction points of the light-cone Unfortunately, the amplitude (2.5) given as an integral of (2.10) or (2.8) is not welldefined. (2.8) diverges when some of the interaction points collide, because T LC F (z) has the OPE which makes the integral (2.5) ill-defined. This kind of divergence is called the contact term divergence.
Accordingly, the conformal gauge expression (2.10) suffers from the so-called spurious singularity. The holomorphic part of the correlation function of the superghost system has the form . (2.23) Here ω is the canonical basis of the holomorphic 1-forms, △ is the Riemann class, E(z, w) is the prime form of the surface and σ (z) is a holomorphic g 2 form with no zeros or poles. The base point P 0 is an arbitrary point on the surface. 5 This correlation function diverges when It also diverges when some of Z r collide, but such singularities are at the boundary of moduli space of the punctured Riemann surface. The singularities given above are called the spurious singularities. The first type of singularity corresponds to the contact term divergence mentioned above. The second type of singularity is due to existence of zero modes of γ. Singularities of this kind do not arise in our case. Since Z r (r = 1, . . . N) and z I (I = 1, . . . , 2g − 2 + N) are the poles and the zeros of the meromorphic one-form ∂ρ (z) dz respectively, I z I − r Z r is a canonical divisor on the surface. Therefore we obtain where Ω is the period matrix. This yields Therefore, in order to make the amplitudes given in the previous subsection well-defined, we should deal with the contact term divergences. In our previous works, we employ the dimensional regularization to do so. Let us summarize the results: 5 For the mathematical background relevant for string perturbation theory, we refer the reader to [19].
• One can formulate the light-cone gauge superstring field theory in d = 10 dimensional space time. The amplitudes are given in the form (2.5) with Taking d to be large and negative, the factor e − d−2 16 Γ[σ;ĝzz] tame the contact term divergences.
• More generally we can regularize the divergences by taking the worldsheet superconformal field theory to be the one with central charge c = 12. One convenient choice of the worldsheet theory is the one in a linear dilaton background Φ = −iQX 1 , with a real constant Q. The worldsheet action of X 1 and its fermionic partners ψ 1 ,ψ 1 on a worldsheet with metric dŝ 2 = 2ĝ zz dzdz becomes The amplitude is expressed in the form (2.26) with It was shown in [14] that (with the Feynman iε) by taking Q 2 > 10, the amplitudes become finite.
• We can define the amplitudes as analytic functions of Q 2 and take the limit Q → 0 to obtain those in d = 10. In order to study the limit, it is useful to recast the expression (2.26) into the conformal gauge one [13] which looks quite similar to the critical case (2.10). 6 The crucial difference is that the worldsheet theory for the longitudinal variables X ± , ψ ± ,ψ ± is a superconformal field theory called the supersymmetric X ± CFT, which has the central charge With this CFT, we can construct a nilpotent BRST charge. Using this expression, one can show that the amplitudes in the limit Q → 0 coincide with those given by the Sen-Witten prescription [23], if the latter exists.

The problems with odd spin structure
The light-cone gauge amplitudes can be defined and calculated for odd spin structure, and we get the expression (2.5) with the integrand given by (2.8 which cancel the divergent contribution from the β, γ system. Therefore we need to make sense out of the combination 0 × ∞ to obtain the BRST invariant expression corresponding to the light-cone gauge amplitudes. 6 Notice that the expression here is different from the one in [13] where the operators are inserted in place of e − iQ 2 αr X + ẑ I (r) ,ẑ I (r) . The properties of operators of this kind with operator valued argumentsẑ I ,ẑ I are explained in appendix A.

Odd spin structure
The problem mentioned at the end of the previous section can be avoided by considering the amplitudes with insertions of ψ + , ψ − and δ(β), δ(γ). We would like to show that such insertions can be realized in a BRST invariant way, if we consider the conformal gauge amplitudes taking some of the vertex operators to have 0 or −2 picture, when all the external lines are in the (NS,NS) sector.

Multiloop amplitudes
Let us consider the case where the spin structure α L for the left-moving fermions is odd and α R for the right-moving fermions is even. The case where α L is even and α R is odd or both of α L and α R are odd can be dealt with in the same way. We would like to show that the in the sum in (2.8) corresponding to such a spin structure can be recast into a conformal gauge expression up to a numerical factor. As we will see, the expression (3.2) is well-defined and free from the combination 0 ×∞. Here V which satisfy where X is the picture changing operator (2.11) and Q B denotes the BRST charge (C.12).
Since V DDF Z,Z is expressed as (2.13), V (0,−1) 2 Z 2 ,Z 2 can be rewritten as where the ellipses in the last line denote the terms which do not involve ψ − . It is possible to show that (3.2) is equal to A poof of this fact can be found in appendix C.2. Therefore, in order to show that (3.1) is proportional to (3.2), we evaluate (3.7) and prove that it is proportional to (3.1). In (3.7), and that of the superghosts is evaluated to be (3.10) Here h α L (z) defined in (B.10) is equal to the zero mode of spin 1 2 left-moving fermion with spin structure α L . The explicit form of S and its relation to e −Γ[σ;g A zz ] can be found in [13]. In the manipulations in (3.10), we have used (2.24) and the following identities: where C is a quantity independent of z. From this expression we can derive .

Dimensional regularization
The amplitudes given by the integral (2.5) with the integrand of the form (3.2) is not welldefined, because of the contact term divergences. In order to make them well-defined, we employ the dimensional regularization illustrated in subsection 2.2. The amplitudes are given in the form (2.5) with the integrand (2.26) with d = 10 − 8Q 2 . The light-cone gauge amplitudes are finite for Q 2 > 10. As in the case of even spin structure, we can define the amplitudes as analytic functions of Q 2 and take the limit Q → 0 to obtain those in d = 10.
In order to study the limit, we recast the light-cone gauge expression into a conformal gauge one. The noncritical version of (3.2) is given as (3.14) Here the worldsheet theory of the longitudinal variables are taken to be the supersymmetric X ± CFT. We discuss the correlation functions of the supersymmetric X ± CFT for odd spin structures in appendix C.1. As is shown in appendix C.2, this expression is equal to  We use the light-cone gauge expression of the amplitude for Q 2 > 10 to define it as an analytic function of Q 2 , which is denoted by A LC (Q 2 ). We would like to see what happens in the limit Q → 0. The conformal gauge expression (3.14) can be deformed to define the amplitudes following the Sen-Witten prescription [23,24]. We can divide the moduli space into patches and put the PCO's avoiding the spurious singularities as was explained in [23] and define the amplitude A SW (Q 2 ). Moving the locations of the PCO's, the amplitudes change by total derivative terms in moduli space. Taking Q 2 big enough, these total derivative terms do not contribute to the amplitudes, because the infrared divergences are regularized. Therefore A SW (Q 2 ) coincides with A LC (Q 2 ) as an analytic function of Q 2 .
Since A SW (Q 2 ) is free from the spurious singularities, it can be well-defined for Q 2 < 10 and if the right hand side is well-defined.

Conclusions and discussions
In this paper, we have shown that the Feynman amplitudes of the light-cone gauge closed superstring field theory can be calculated using the dimensional regularization technique, for higher genus Riemann surfaces with odd spin structure, if the external lines are in the (NS,NS) sector. In order to deal with the fermion zero modes peculiar to odd spin structures, we need to change the pictures of the vertex operators in the conformal gauge expression.
We obtain the amplitudes in noncritical dimensions which coincide with the ones defined by using the Sen-Witten prescription. The amplitudes in the critical dimensions correspond to the limit d → 10 or Q → 0, and the results coincide with those given by the Sen-Witten prescription.
There are several things remain to be done. One is to check how the amplitudes obtained by our procedure are related to the standard results in more detail. In particular, we should study the conditionally convergent integrals which appear in the Feynman amplitudes of superstrings. We expect that our regularization makes the integrals well-defined but in a way different from those in [25,26]. Another thing to be done is to generalize our results to the amplitudes with external lines in the Ramond sector. With the correlation functions involving spin fields given for example in [27], it will be straightforward to rewrite the lightcone gauge expression into the conformal gauge one. These problems are left to future work.

A Operator valued coordinate
It is convenient to introduce the operator valued coordinateẑ I and its supersymmetric versioñ z I , in order to express the conformal gauge form of the amplitudes for noncritical dimensions. Let us first considerẑ I which is defined in the bosonic case. In the light-cone gauge setup, we consider the situation where the variable X + (z,z) possesses an expectation value − i 2 (ρ (z) +ρ (z)). Therefore we decompose it into the expectation value and the fluctuation as Roughly speaking, we defineẑ I to be an operator valued coordinate which satisfies We takeẑ I so as to coincide with z I when δX + = 0. Assuming thatẑ I is expanded in terms of the fluctuation δX + asẑ where δ (n) z I is at the n-th order in the derivatives of δX + , in principle we can obtain δ (n) z I if ∂ 2 ρ (z I ) = 0. Lower order examples are given by (A.5) In general δ (n) z I becomes a polynomial of the derivatives of δX + at z = z I . Quantities of order n with n > N for some N > 0 do not contribute to the correlation functions we consider in this paper.ẑ I , which is the antiholomorphic counterpart ofẑ I , can be obtained in the same way.
The OPE ofẑ I with the energy-momentum tensor T (z) comes from the contractions of Taking the OPE of (A.2) with T (z), we get Using (A.2) and the fact that ∂ 2 X + (ẑ I ) = − i 2 ∂ 2 ρ (z I ) + · · · is invertible perturbatively, we obtain Expandingẑ I as in (A.4), we can see that the right hand side of (A.7) involves poles of arbitrarily high order at z = z I . Only finite number of them are relevant in the correlation functions (2.29), (3.14) and (3.15).
With the OPE (A.7) and its antiholomorphic version, we can show the OPE's T (z) e iαX + ẑ I ,ẑ I ∼ regular , for any constant α. Therefore e iαX + ẑ I ,ẑ I is a BRST invariant operator in the conformal gauge bosonic string theory in noncritical dimensions.
It is straightforward to define the operator valued supercoordinateẑ I . We defineẑ I = ẑ I ,θ I to be the operator valued supercoordinate which satisfies Notice thatẑ I is the operator version of the supercoordinatez I defined in [7,28,29] which is not superconformal, but it is sufficient for our purpose. Similarly to the bosonic case, we decompose X + (z,z) as where ρ s ,ρ s are the supersymmetric version of ρ,ρ whose explicit form is given in (C.5). S α in that equation is the one in (B.5) or in (B.12) according to whether the spin structure of the fermions is even or odd. Using this decomposition, we getẑ I ,θ I as expansions around z I ,θ I in terms of the fluctuation δX + aŝ assuming ∂ 2 ρ (z I ) = 0. For example, We obtain the OPE's From these OPE's, we get T (z) e iαX + ẑ I ,ẑ I ∼ regular , T (z) e iαX + ẑ I ,ẑ I ∼ regular , (A.14) for any constant α. Therefore e iαX + ẑ I ,ẑ I is BRST invariant in the conformal gauge superstring theory in noncritical dimensions.
Using the operator valued coordinate thus defined, we can define a BRST invariant operator e − iQ 2 αr X + ẑ I (r) ,ẑ I (r) , (A. 15) which can be replaced by e − iQ 2 αr X + (z I (r) ,z I (r) ) (A. 16) in evaluating (2.29), (3.14) and (3.15). This expression can be used instead of the complicated combination z (r) [13,30], which has a similar effect .

B Correlation functions of free fermions
In this appendix, we review a few basic facts about the correlation functions of free fermions on higher genus Riemann surfaces.
The correlation functions of a free Dirac fermion with spin structure α L for left and α R for right can be given by [19,31] When both α L and α R correspond to even spin structures, using the formula given in [32] for the case it is straightforward to show that (B.1) can be transformed into is the Szego kernel. The expression (B.4) implies that the partition function is given by and the propagators of the fermions are When the spin structures are not even, we need to take care of the fermion zero modes. For example, let us consider the case where α L corresponds to an odd spin structure and α R corresponds to an even one. In this case, using the formula (B.2) for in the limit q → p, we get [32] ϑ gives the zero mode of the fermion. Substituting (B.9) into the right hand side of (B.4), we . (B.12) S α L (x, y) can be identified with the propagator of the left-moving fermions and it involves the zero mode variables ψ † 0 , ψ 0 . ψ † 0 and ψ 0 should be integrated over after all the contractions are performed. The other cases where α R corresponds to an odd spin structure can be dealt with in the same way.

C Dimensional regularization for odd spin structure
In this appendix, we explain the details of how dimensional regularization works in the case of odd spin structure.

C.1 Supersymmetric X ± CFT
In order to get the expression of the amplitudes in the conformal gauge, we need to calculate the correlation functions of the supersymmetric X ± CFT on the surface with odd spin structure.
The action of the supersymmetric X ± CFT is given in the form where S free [ĝ zz , X ± ] denotes the free action of X ± . When the spin structures are both even, the correlation functions of the supersymmetric X ± CFT are evaluated as [13] dX Regarding the second and third lines as a correlation function of of the free theory with the source term we can calculate it by replacing the X + (z,z) by its expectation value − i 2 (ρ s (z) +ρ s (z)) and derive the fourth line. Here ρ s ,ρ s are the supersymmetric version of ρ,ρ and expressed where S α L and S α R are taken to be the Szego kernel (B.5). The partition function Z X super [ĝ zz ] 2 is described by using Z X ± [ĝ zz ] and Z ψ ± [ĝ zz ] in (2.19) as The explicit form of e − d−10 8 Γsuper[ĝzz, ρs+ρs] can be found in [13].
and we can deduce that the energy momentum tensor T X ± (z) satisfies the OPE it is possible to construct a nilpotent BRST charge These properties can be proved in the same way as in the even spin structure case, because we need only the behaviors of the fermion propagators around the singularities to do so.
C.2 A proof of equality of (3.14) and (3.15) In this appendix, we show that (3.14) is equal to (3.15). In the case Q = 0, this equality implies that (3.2) is equal to (3.7). Proving this can be done by using a fermionic charge 7 iX + L − 1 2 ρ (z) + β 2∂ρ ψ + (z) , (C.15) and its antiholomorphic counterpartQ ′ . Here iX + L − 1 2 ρ (z) is defined as with a generic point w 0 on the surface. iX + L − 1 2 ρ (z) thus defined is single valued on the surface in the correlation functions we consider here because − i 2 ρ coincides with the expectation value of X + L in the presence of the sources e −ip + r X − . In order to useQ ′ , we need to rewrite the ghost part of the correlation function. Inserting with z J = z I , because e − iQ 2 αr X + ẑ I (r) ,ẑ I (r) and S int are Grassmann even. Hence we conclude that z I dw 2πi ∂ρψ − (w) w−z I in (C.23) does not contribute to the path integral. We can do the same thing for the antiholomorphic part and prove that the X (z I ) ,X (z I ) which appear in (C. 18) can be replaced by −e φ T LC F (z I ) , −eφT LC F (z I ) for all I. By deforming the contours of the antighost insertions back, we can see that (3.14) is equal to (3.15).