Multiloop Amplitudes of Light-cone Gauge Superstring Field Theory: Odd Spin Structure Contributions
Nobuyuki Ishibashi Koichi Murakami
hep-th





footnote



UTHEP-710







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




 



Nobuyuki Ishibashi^a[e-mail: ishibash@het.ph.tsukuba.ac.jp]
and Koichi Murakami^b[e-mail: koichi@kushiro-ct.ac.jp] 






^a Tomonaga Center for the History of the Universe, 

University of Tsukuba

Tsukuba, Ibaraki 305-8571, JAPAN


 





^bNational Institute of Technology, Kushiro College,

 Otanoshike-Nishi 2-32-1, Kushiro, Hokkaido 084-0916, JAPAN 










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. 

footnote










§ 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 <cit.>,
based on the formulation <cit.> 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
<cit.>. The proof was formal because there appear unphysical
divergences which are called the contact term divergences 
<cit.>. In a previous
paper <cit.>, 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 c12 <cit.>. 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 <cit.>, 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 <ref>,
we review the results in <cit.> and the problems with
the odd spin structures. In section <ref>,
we deal with the amplitudes for the odd spin structure and show that
these also coincide with those from the first-quantized approach.
Section <ref> 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 

    |Φ(t,α)⟩

is taken to be an element of the Hilbert space ℋ of the
transverse variables on the worldsheet and a function of 

    t     =     x^+ ,
    α    =     2p^+ .

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 

    (L_0-L̅_0)|Φ(t,α)⟩ =0 ,

where L_0,L̅_0 are the zero modes of the Virasoro generators
of the worldsheet theory.

The action of the string field theory is given by 

    S     =    ∫ dt[1/2∑_B∫_-∞^∞α dα/4π⟨Φ_B(-α)|(i∂_t-L_0+L̅_0-1/α)|Φ_B(α)⟩.
           +1/2∑_F∫_-∞^∞dα/4π⟨Φ_F(-α)|(i∂_t-L_0+L̅_0-1/α)|Φ_F(α)⟩
           -g_s/6∑_B_1,B_2,B_3∫∏_r=1^3(α_rdα_r/4π)δ(∑_r=1^3α_r)⟨ V_3|Φ_B_1(α_1).⟩|Φ_B_2(α_2)⟩|Φ_B_3(α_3)⟩
           .-g_s/2∑_B_1,F_2,F_3∫∏_r=1^3(α_rdα_r/4π)δ(∑_r=1^3α_r)⟨ V_3|Φ_B_1(α_1).⟩α_2^-1/2|Φ_F_2(α_2)⟩α_3^-1/2|Φ_F_3(α_3)⟩],

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 <cit.>.
Here ℋ^∗denotes the dual space of ℋ. 



It is straightforward to calculate the amplitudes by the old-fashioned
perturbation theory starting from the action (<ref>)
and Wick rotate to Euclidean time. The propagator and the vertex are
given by the worldsheets depicted in figure <ref>,
where the left and right supercurrents T_F^LC,
T̅_F^LC
of the transverse variables 
X^i, ψ^i, ψ̅^i (i=1,…,8)
are inserted at the interaction points
of the three-string vertices. Each term in the expansion corresponds
to a light-cone gauge Feynman diagram for strings. 



A typical light-cone gauge Feynman diagram for strings is depicted
in figure <ref>. A Wick rotated g-loop N-string
diagram is conformally equivalent to an N punctured genus g
Riemann surface Σ. A g-loop N-string amplitude is given
as an integral over the moduli space of Σ as <cit.>

    𝒜_N^(g)=(ig_s)^2g-2+NC∫[dT][α dθ][dα] F_N^(g) ,

where ∫[dT][α dθ][dα] denotes the integration
over the moduli parameters and C is the combinatorial factor. The
integrand F_N^(g) is given as a path integral over the transverse
variables X^i, ψ^i, ψ̅^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 ,

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

    ds^2=dρ dρ̅ .

ρ is not a good coordinate around the interaction points and
the punctures, and the metric (<ref>) is not
well-defined at these points. F_N^(g) can be expressed in terms
of correlation functions defined with a metric dŝ^2=2ĝ_zz̅dzdz̅
which is regular everywhere on the worldsheet, as 

    F_N^(g)    =    (2π)^2δ(∑_r=1^Np_r^+)δ(∑_r=1^Np_r^-)e^-1/2Γ[σ;ĝ_zz̅]
          ×∑_spin structure∫[dX^idψ^idψ̅^i]_ĝ_zz̅e^-S^LC[X^i,ψ^i,ψ̅^i]
          ×∏_I=1^2g-2+N(|∂^2ρ(z_I)|^-3/2T_F^LC(z_I)T̅_F^LC(z̅_I))∏_r=1^NV_r^LC(Z_r,Z̅_r) .

Here  z is a complex coordinate of the Riemann surface
and the coordinate ρ becomes a function ρ (z) of z
(see e.g. <cit.>).
S^LC[X^i,ψ^i,ψ̅^i]
denotes the worldsheet action of the transverse variables and the
path integral measure [dX^idψ^idψ̅^i]_ĝ_zz̅
is defined with the metric dŝ^2=2ĝ_zz̅dzdz̅.
Since the integrand was defined by using the metric (<ref>),
we need the anomaly factor e^-1/2Γ[σ;ĝ_zz̅],
where 

    σ=ln∂ρ∂̅ρ̅-lnĝ_zz̅ ,
          Γ[σ;ĝ_zz̅]=-1/4π∫ dz∧ dz̅√(ĝ)(ĝ^ab∂_aσ∂_bσ+2R̂σ) .

z_I (I=1,⋯,2g-2+N) denote the z-coordinates
of the interaction points of the light-cone gauge Feynman diagram.
V_r^LC denotes the vertex operator for the r-th
external line inserted at z=Z_r (r=1,…,N). The right
hand side of (<ref>) does not depend on the choice of
ĝ_zz̅. 

As was demonstrated in <cit.>, 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 (<ref>)
can be recast into a conformal gauge expression:

    ∫[dX^μdψ^μdψ̅^μdbdb̅dcdc̅dβ dβ̅dγ dγ̅]_ĝ_zz̅e^-S^tot
              ×∏_K=1^6g-6+2N[∮_C_Kdz/∂ρb_zz+ε_K∮_C̅_Kdz̅/∂̅ρ̅b_z̅z̅]∏_I=1^2g-2+N[X(z_I)X̅(z̅_I)]
              ×∏_r=1^NV_r^(-1,-1)(Z_r,Z̅_r) .

Here
S^tot denotes the worldsheet action for the variables
X^μ,ψ^μ,ψ̅^μ (μ=+,-,1,…,8), ghosts
and superghosts, 

    X(z)=[c∂ξ-e^ϕT_F+1/4∂ bη e^2ϕ+1/4b(2∂η e^2ϕ+η∂ e^2ϕ)](z)

is the picture changing operator (PCO), X̅(z̅)
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_r^(-1,-1)(Z_r,Z̅_r)
is defined as

    V_r^(-1,-1)(Z_r,Z̅_r)≡ cc̅e^-ϕ-ϕ̅V_r^DDF(Z_r,Z̅_r) .

V_r^DDF(Z_r,Z̅_r) is the supersymmetric DDF
vertex operator given by 

    V_r^DDF(Z_r,Z̅_r)=A_-n_1^i_1(r)⋯A̅_-n̅_1^i̅_1(r)⋯ B_-s_1^j_1(r)⋯B̅_-s̅_1^j̅_1(r)⋯ e^-ip_r^+X^--i(p_r^--𝒩_r/p_r^+)X^++ip_r^iX^i(Z_r,Z̅_r) ,

with the DDF operators A_-n^i(r),B_-s^j(r) for the r-th
string defined as

    A_-n^i(r)    =    ∮_Z_rd𝐳/2π iiD𝒳^ie^-in/p_r^+𝒳_L^+(𝐳) ,
    
    
    B_-s^i(r)    =    ∮_Z_rd𝐳/2π iD𝒳^+/(ip_r^+∂𝒳^+)^1/2D𝒳^ie^-is/p_r^+𝒳_L^+(𝐳) ,

and A̅_-n^i(r),B̅_-s^i(r) are similarly given
for the antiholomorphic sector. Here we use the notation

    𝒩_r   ≡   ∑_kn_k+∑_ls_l=∑_k̅n̅_k̅+∑_l̅s̅_l̅ ,
    
    𝒳^μ(𝐳,𝐳̅)     =     X^μ(z,z̅)+iθψ^μ(z)+iθ̅ψ̅^μ(z̅)+θθ̅F^μ ,
    
    
    D    ≡   ∂/∂θ+θ∂/∂ z .

𝐳=(z,θ) denotes the superspace coordinate on the worldsheet
and 𝒳_L^+ denotes the left-moving part of the superfield 𝒳^+.[Although 𝒳_L^+ is not a well-defined quantity, it
is used as a short-hand notation to express the vertex operator (<ref>),
which is well-defined. ] We take the vertex operators to satisfy the on-shell condition

    1/2(-2p_r^+p_r^-+p_r^ip_r^i)+𝒩_r=1/2 .

V_r^DDF(Z_r,Z̅_r) turns out to be a weight
(1/2,1/2) primary field made from X^μ,ψ^μ,ψ̅^μ.
Therefore V_r^(-1,-1)(Z_r,Z̅_r) is an on-shell
vertex operator in (-1,-1) picture. It is easy to see
that the expression of the amplitude (<ref>) given as an integral
of (<ref>) is BRST invariant. 

One way to derive the expression (<ref>) is as follows
<cit.>. Using a nilpotent fermionic charge, it is
possible to show that the right hand side of (<ref>)
is equal to 

    ∫[dX^μdψ^μdψ̅^μdbdb̅dcdc̅dβ dβ̅dγ dγ̅]_ĝ_zz̅e^-S^tot
              ×∏_K=1^6g-6+2N[∮_C_Kdz/∂ρb_zz+ε_K∮_C̅_Kdz̅/∂̅ρ̅b_z̅z̅]∏_I=1^2g-2+N[e^ϕT_F^LC(z_I)e^ϕ̅T̅_F^LC(z̅_I)]
              ×∏_r=1^N[cc̅e^-ϕ-ϕ̅V_r^DDF(Z_r,Z̅_r)] .

In this form, the path integral factorizes into the contributions
from X^μ,ψ^μ,ψ̅^μ, ghosts and superghosts.
Each of these contributions is calculated by taking ĝ_zz̅
to be the Arakelov metric g_zz̅^A <cit.>.
In the matter sector, integration over the longitudinal variables
yields

    ∫[dX^±dψ^±dψ̅^±]_g^A_zz̅
       e^-S^±∏_r=1^NV_r^DDF(Z_r,Z̅_r)
    
        =    
    (2π)^2 δ( ∑_r=1^Np^-_r)
    δ( ∑_r=1^N p^+_r)
        Z_X^±[ g^A_zz̅]
        Z_ψ^±[ g^A_zz̅]
     ∏_r=1^N[
           1/α_r
           e^-ReN̅_00^rr
           V_r^LC(Z_r,Z̅_r)
        ] ,

where 

    Z_X^±[ g_zz̅]
        =    ( ^'( -g^zz̅∂_z∂_z̅)/∫ d^2z√(g))^-1 ,
    
    Z_ψ^±[ g_zz̅] 
        =    (^'(-g^zz̅∂_z∂_z̅)/ImΩ∫ d^2z√(g))^-1/2ϑ[α_L](0)ϑ[α_R](0)^∗ ,
    N̅_00^rr    =    lim_z→ Z_r[ρ(z_I^(r))-ρ(z)/α_r+ln(z-Z_r)] .

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_zz̅] and 
Z_ψ^±[ g_zz̅]
are respectively the partition functions of the free variables X^± 
and ψ^±,ψ̅^± on the worldsheet endowed with
the metric ds^2 = 2g_zz̅ dz dz̅. 
z_I^(r) denotes the interaction point
at which the r-th string interacts. The contributions from the
ghosts and superghosts are given as 

    ∫[dbdb̅dcdc̅]_g_zz̅^Ae^-S^bc∏_r=1^Ncc̅(Z_r,Z̅_r)∏_K=1^6g-6+2N[∮_C_Kdz/∂ρb_zz+ε_K∮_C̅_Kdz̅/∂̅ρ̅b_z̅z̅]
              
    =(Z_X^±[g^A_zz̅]
     )^-1
     e^-Γ[σ; g_zz̅^A]∏_r=1^N(α_re^2ReN̅_00^rr) ,
          ∫[dβ dγ dβ̅dγ̅]_g^A_zz̅
         e^-S_βγ∏_I=1^2g-2+N[ e^ϕ(z_I)
                    e^ϕ̅(z̅_I)
             ]
        ∏_r=1^N[ e^-ϕ(Z_r)
                     e^-ϕ̅(Z̅_r)
              ]
    
              
       =( Z_ψ^±[ g^A_zz̅]
        )^-1
        e^1/2Γ[σ; g_zz̅^A]∏_r=1^Ne^-ReN̅_00^rr∏_I=1^2g-2+N|∂^2ρ(z_I)
                            |^-3/2 .

Substituting eqs.(<ref>), (<ref>), (<ref>)
into (<ref>), we can easily see that (<ref>)
is equal to (<ref>). 



 §.§ Dimensional regularization


The amplitudes of superstring theory were calculated using the first-quantized
formalism in <cit.> in which an expression using the
PCO's was given. The expression (<ref>) is a special
case of the one in <cit.>, where the PCO's are placed
at the interaction points of the light-cone Feynman diagram.[Notice that in the light-cone setup, the positions of the PCO's have
the fixed coordinate in the coordinate patch on the surface and we
do not need ∂ξ terms.] 

Unfortunately, the amplitude (<ref>) given as an integral
of (<ref>) or (<ref>) is not well-defined.
(<ref>) diverges when some of the interaction points collide,
because T_F^LC(z) has the OPE

    T_F^LC(z_I)T_F^LC(z_J)∼2/(z_I-z_J)^3+⋯ ,

which makes the integral (<ref>) ill-defined. This kind of
divergence is called the contact term divergence. 

Accordingly, the conformal gauge expression (<ref>)
suffers from the so-called spurious singularity. The holomorphic part
of the correlation function of the superghost system has the form

    ⟨∏_I=1^2g-2+Ne^ϕ(z_I)∏_r=1^Ne^-ϕ(Z_r)⟩
            ∼[ϑ[α_L](-∑_r∫_P_0^Z_rω+∑_I∫_P_0^z_Iω-2∫_P_0^△ω)]^-1
            ×∏_I,rE(z_I,Z_r)/∏_I<JE(z_I,z_J)∏_r<sE(Z_r,Z_s)∏_rσ^2(Z_r)/∏_Iσ^2(z_I) .

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.[For the mathematical background relevant for string perturbation theory,
we refer the reader to <cit.>.] This correlation function diverges when


  * Some of z_I collide. 

  * ϑ[α_L](-∑_r∫_P_0^Z_rω+∑_I∫_P_0^z_Iω-2∫_P_0^△ω)=0.

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, ∑_Iz_I-∑_rZ_r is a canonical divisor
on the surface. Therefore we obtain

    -∑_r=1^N∫_P_0^Z_rω+∑_I=1^2g-2+N∫_P_0^z_Iω≡2∫_P_0^△ω    ℤ^g+ℤ^gΩ ,

where Ω is the period matrix. This yields 

    [ϑ[α_L](-∑_r∫_P_0^Z_rω+∑_I∫_P_0^z_Iω-2∫_P_0^△ω)]^-1=[ϑ[α_L](0)]^-1 ,

which is included in the factor (Z_ψ^±)^-1
in (<ref>). [ϑ[α_L](0)]^-1
may become singular at some points in the moduli space, but the [ϑ[α_L](0)]^-1
cancels the factor ϑ[α_L](0)
from the partition function Z_ψ^± of ψ^± and
the whole amplitude is free from this type of singularity. 

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:


  * One can formulate the light-cone gauge superstring field theory in
d10 dimensional space time. The amplitudes are given in the
form (<ref>) with 

    F_N^(g)    =    (2π)^2δ(∑_r=1^Np_r^+)δ(∑_r=1^Np_r^-)e^-d-2/16Γ[σ;ĝ_zz̅]
          ×∑_spin structure∫[dX^idψ^idψ̅^i]_ĝ_zz̅e^-S^LC[X^i,ψ^i,ψ̅^i]
          ×∏_I=1^2g-2+N(|∂^2ρ(z_I)|^-3/2T_F^LC(z_I)T̅_F^LC(z̅_I))∏_r=1^NV_r^LC .

Taking d to be large and negative, the factor e^-d-2/16Γ[σ;ĝ_zz̅]
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 c12.
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,ψ̅^1
on a worldsheet with metric dŝ^2=2ĝ_zz̅dzdz̅
becomes 

    S[X^1,ψ^1,ψ̅^1;ĝ_zz̅]     =    1/8π∫ dz∧ dz̅√(ĝ)(ĝ^ab∂_aX^1∂_bX^1-2iQR̂X^1)
            +1/4π∫ dz∧ dz̅i(ψ^1∂̅ψ^1+ψ̅^1∂ψ̅^1) ,

The amplitude is expressed in the form (<ref>) with 

    d=10-8Q^2 .

It was shown in <cit.> 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 (<ref>)
into the conformal gauge one <cit.>

    ∫[dX^μdψ^μdψ̅^μ
                   dbdb̅dcdc̅dβ dβ̅
                   dγ dγ̅]_g_zz̅^A
           e^-S^tot
              ×∏_K=1^6g-6+2N[
         ∮_C_Kdz/∂ρb_zz
         +ε_K∮_C̅_Kdz̅/∂̅ρ̅b_z̅z̅]   
    
    
    ∏_I=1^2g-2+N[X(z_I)X̅(z̅_I)
                        ]
    
              ×∏_r
       e^-iQ^2/α_r𝒳^+(𝐳̂̃̂_I^(r),
             𝐳̂̃̂̅̂̃̂_I^(r))
    
    
    ∏_r=1^N[V_r^(-1,-1)(Z_r,Z̅_r)
    ],

which looks quite similar to 
the critical case (<ref>).[Notice that 
the expression here is different from the one 
in <cit.> where the operators

    ∮_z_I^(r)d𝐳/2π i𝒮(𝐳,Z_r)
    ∮_z̅_I^(r)d𝐳̅/2π i𝒮̅(𝐳̅,Z̅_r)

are inserted in place of 
e^-iQ^2/α_r𝒳^+(𝐳̂̃̂_I^(r),
        𝐳̂̃̂̅̂̃̂_I^(r)).
The properties of operators of this kind with operator valued arguments
𝐳̂̃̂_I,𝐳̂̃̂̅̂̃̂_I
are explained in appendix <ref>. ] The crucial difference is that the worldsheet theory for the longitudinal
variables X^±,ψ^±,ψ̅^± is a superconformal
field theory called the supersymmetric X^± CFT, which has the
central charge

    c=3+12Q^2 .

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 <cit.>,
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 (<ref>) with
the integrand given by (<ref>). The correlation functions
of free fermions on higher genus Riemann surfaces are given in appendix
<ref>. Only the amplitudes with enough
fermions from the vertex operators and T_F insertions
to soak up the zero modes are nonvanishing. However, we have a problem in rewriting
the light-cone gauge expression (<ref>) into the BRST
invariant one (<ref>), if we proceed as in the previous
section. If α corresponds to an odd spin structure,

    ϑ[α](0)=0 .

 The correlation function (<ref>) of the β,γ
system diverges because it involves factors 

    (ϑ[α](0))^-1 ,

coming from (Z_ψ^±)^-1 on the right hand side of   (<ref>).
On the other hand, the partition function (<ref>) of the
ψ^± variables involves factors 

    ϑ[α](0) ,

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. 



§ 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 term 

    ∫[dX^idψ^idψ̅^i]_ĝ_zz̅e^-S^LC[X^i,ψ^i,ψ̅^i]
          ×∏_I=1^2g-2+N(|∂^2ρ(z_I)|^-3/2T_F^LC(z_I)T̅_F^LC(z̅_I))∏_r=1^NV_r^LC(Z_r,Z̅_r) ,

in the sum in (<ref>) corresponding to such a spin structure
can be recast into a conformal gauge expression 

    ∫[dX^μdψ^μdψ̅^μdbdb̅dcdc̅dβ dβ̅dγ dγ̅]_g_zz̅^Ae^-S^tot
              ×∏_K=1^6g-6+2N[∮_C_Kdz/∂ρb_zz+ε_K∮_C̅_Kdz̅/∂̅ρ̅b_z̅z̅]∏_I[X(z_I)X̅(z̅_I)]
              × V_1^(-2,-1)(Z_1,Z̅_1)V_2^(0,-1)(Z_2,Z̅_2)∏_r=3^N[V_r^(-1,-1)(Z_r,Z̅_r)] ,

up to a numerical factor. As we will see, the expression (<ref>)
is well-defined and free from the combination 0×∞. Here
V_r^(-1,-1)(Z_r,Z̅_r) is the (-1,-1)
picture vertex operator defined in (<ref>) and V_1^(-2,-1)(Z_1,Z̅_1), V_2^(0,-1)(Z_2,Z̅_2)
are given by 

    V_1^(-2,-1)(Z_1,Z̅_1)     =     -2/p_1^+cc̅e^-2ϕe^-ϕ̅ψ^+V_1^DDF(Z_1,Z̅_1) ,
    
    V_2^(0,-1)(Z_2,Z̅_2)     =    [-cc̅e^-ϕ̅∮_Z_2dz/2π iT_F(z)+1/4c̅γ e^-ϕ̅]V_2^DDF(Z_2,Z̅_2),

which satisfy

    XV_1^(-2,-1)(Z_1,Z̅_1)     =     V_1^(-1,-1)(Z_1,Z̅_1) ,
    
    
    XV_2^(-1,-1)(Z_2,Z̅_2)     =     V_2^(0,-1)(Z_2,Z̅_2) ,
    
    
    Q_BV_1^(-2,-1)(Z_1,Z̅_1)     =     0 ,

where X is the picture changing operator (<ref>) and Q_B
denotes the BRST charge (<ref>). Since V^DDF(Z,Z̅)
is expressed as (<ref>), V_2^(0,-1)(Z_2,Z̅_2)
can be rewritten as

    V_2^(0,-1)(Z_2,Z̅_2)
    
           =  -cc̅e^-ϕ̅A_-n_1^i_1(2)⋯A̅_-n̅_1^i̅_1(2)⋯ B_-s_1^j_1(2)⋯B̅_-s̅_1^j̅_1(2)⋯
            ×1/2[p_2^+ψ^-+(p_2^--𝒩_2/p_2^+)ψ^+-p_2^iψ^i]e^-ip_2^+X^--i(p_2^--𝒩_2/p_2^+)X^++ip_2^iX^i
              +1/4c̅γ e^-ϕ̅V_2^DDF(Z_2,Z̅_2)
    
           =  -1/2cc̅e^-ϕ̅p_2^+:ψ^-V_2^DDF(Z_2,Z̅_2):+⋯ ,

where the ellipses in the last line denote the terms which do not
involve ψ^-.

V_1^(-2,-1)(Z_1,Z̅_1), V_2^(0,-1)(Z_2,Z̅_2)
can be considered to be the BRST invariant vertex operators in (-2,-1), (0,-1)
pictures respectively. It is straightforward to define vertex operators
V^(-1,-2,),V^(-1,0), or V^(-2,-2),V^(0,0)
which can be used to express the amplitudes for the cases of the other
spin structures mentioned above. 

It is possible to show that (<ref>) is equal to 

    ∫[dX^μdψ^μdψ̅^μdbdb̅dcdc̅dβ dβ̅dγ dγ̅]_g_zz̅^Ae^-S^tot
              ×∏_K=1^6g-6+2N[∮_C_Kdz/∂ρb_zz+ε_K∮_C̅_Kdz̅/∂̅ρ̅b_z̅z̅]
    
    
    ∏_I[e^ϕT_F^LC(z_I)e^ϕ̅T̅_F^LC(z̅_I)]
              × V_1^(-2,-1)(Z_1,Z̅_1)V_2^(0,-1)(Z_2,Z̅_2)∏_r=3^N[V_r^(-1,-1)(Z_r,Z̅_r)] .

A poof of this fact can be found in
appendix <ref>. Therefore, in order to show that
(<ref>) is proportional to (<ref>), we evaluate
(<ref>) and prove that it is proportional to (<ref>).
In (<ref>), we can replace V_2^(0,-1)(Z_2,Z̅_2)
by

    -1/2cc̅e^-ϕ̅p_2^+:ψ^-V_2^DDF:(Z_2,Z̅_2) ,

because only this term can soak up the zero mode of ψ^-. After
such a replacement, (<ref>) factorizes into contributions
from the ghosts, superghosts, longitudinal modes and the transverse
modes. 






The parts of the transverse variables and ghosts are the same as
those in (<ref>)
and we can use (<ref>) to evaluate the latter.
In this case,
the correlation function of the longitudinal variables is
modified from (<ref>) into

    ∫[dX^±dψ^±dψ̅^±]_g^A_zz̅
       e^-S^± ψ^+(Z_1) ψ^-( Z_2)
       ∏_r=1^NV_r^DDF(Z_r,Z̅_r)
    
           =
    (2π)^2 δ( ∑_r=1^Np^-_r)
    δ( ∑_r=1^N p^+_r)
        Z_X^±[ g^A_zz̅]
       ( '( -g^A z z̅∂_z∂_z̅)/ImΩ∫ d^2z √(g^A))^-1/2ϑ[ α_R] (0)^∗
              × h_α_L( Z_1)
           h_α_L( Z_2)
     ∏_r=1^N[
           1/α_r
           e^-ReN̅_00^rr
           V_r^LC(Z_r,Z̅_r)
        ] ,

and that of the superghosts
is evaluated to be

    ∫[dβ dγ dβ̅dγ̅]_g^A_zz̅
          e^-S_βγ  e^-2ϕ(Z_1)
      ∏_I[e^ϕ(z_I)
                      e^ϕ̅(z̅_I)
                ]
      ∏_r=3^Ne^-ϕ(Z_r)
      ∏_r=1^Ne^-ϕ̅(Z̅_r)
    
       ∝   (^'(
                 -g^A zz̅∂_z∂_z̅)/ImΩ∫ d^2z√(g^A))^1/2
          
      ×[ϑ[α_L]
        (-∑_r=1^N∫_P_0^Z_rω
              +∑_I∫_P_0^z_Iω
              -2∫_P_0^+∫_Z_1^Z_2ω)  ]^-1
          
      ×[ ϑ[α_R]
      (-∑_r=1^N∫_P_0^Z_rω
            +∑_I∫_P_0^z_Iω
            -2∫_P_0^ω)^∗]^-1
          
      ×|
      ∏_I,rE(z_I,Z_r)/∏_I<JE(z_I,z_J)
             ∏_r<sE(Z_r,Z_s)∏_rσ^2(Z_r)/∏_Iσ^2(z_I)|^2e^-12S
          
      ×∏_r=3^NE(Z_2,Z_r)
                 ∏_IE(z_I,Z_1)/∏_IE(z_I,Z_2)
                 ∏_r=3^N E(Z_1,Z_r)·σ^2(Z_1)/σ^2(Z_2)· 
         E(Z_1,Z_2)
    
       ∝   α_1/α_2( ^'(
                      -g^A zz̅∂_z∂_z̅)/ImΩ∫ d^2z √(g^A))^1/2| 
        ∏_I,rE(z_I,Z_r)/∏_I<JE(z_I,z_J)
               ∏_r<sE(Z_r,Z_s)∏_rσ^2(Z_r)/∏_Iσ^2(z_I)|^2e^-12S
          
      ×
    
    
    1/h_α_L(Z_1)h_α_L(Z_2)
              ϑ[α_R](0)^∗
       =   α_1/α_2( ^'(
                      -g^A zz̅∂_z∂_z̅)/ImΩ∫ d^2z √(g^A))^1/2
     e^1/2Γ[ σ; g^A_zz̅]∏_r e^-ReN̅^rr_00∏_I| ∂^2 ρ (z_I) |^-3/2
            ×1/h_α_L(Z_1)h_α_L(Z_2)
              ϑ[α_R](0)^∗ .

Here h_α_L(z) defined in (<ref>)
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_zz̅]
can be found in <cit.>.
In the manipulations in
(<ref>), we have used (<ref>)
and the following identities:

    E(Z_2,Z_1) 
        =    ϑ[α_L]
                  (∫_Z_1^Z_2ω)/h_α_L(Z_1)h_α_L(Z_2) ,
    ∏_r=3^NE(Z_2,Z_r)∏_IE(z_I,Z_1)/∏_IE(z_I,Z_2)∏_r=3^NE(Z_1,Z_r)·σ^2(Z_1)/σ^2(Z_2)    =     -α_1/α_2 .

(<ref>) is proved by observing 

    |∂ρ(z)|^2
    =C |σ(z)|^4|∏_IE(z,z_I)/∏_rE(z,Z_r)|^2 ,

where C is a quantity independent of z. From this expression
we can derive 

    α_1/α_2    =    lim_z→ Z_1, w→ Z_2(z-Z_1)∂ρ(z)/(w-Z_2)∂ρ(w)
        =    lim_z→ Z_1, w→ Z_2z-Z_1/w-Z_2exp[∫_w^zdu ∂ln|∂ρ(u)
                                  |^2]
    
        =     -∏_r=3^N E(Z_2,Z_r)
                   ∏_IE(z_I,Z_1)/∏_IE(z_I,Z_2)∏_r=3^N E(Z_1,Z_r)·σ^2(Z_1)/σ^2(Z_2) .

Combining eqs.(<ref>), (<ref>) 
and (<ref>),
it is straightforward to show that (<ref>) is proportional
to (<ref>). 



 §.§ Dimensional regularization


The amplitudes given by the integral (<ref>) with the integrand
of the form (<ref>) is not well-defined, because of the
contact term divergences. In order to make them well-defined, we employ
the dimensional regularization illustrated 
in subsection <ref>.
The amplitudes are given in the form (<ref>) with the integrand
(<ref>) 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 (<ref>) is given as

    ∫[dX^μdψ^μdψ̅^μdbdb̅dcdc̅dβ dβ̅dγ dγ̅]_g_zz̅^Ae^-S^tot
              ×∏_K=1^6g-6+2N[∮_C_Kdz/∂ρb_zz+ε_K∮_C̅_Kdz̅/∂̅ρ̅b_z̅z̅]
    
    
    ∏_I[X(z_I)X̅(z̅_I)]
    
              ×∏_re^-iQ^2/α_r𝒳^+(𝐳̂̃̂_I^(r),𝐳̂̃̂̅̂̃̂_I^(r))
    
              × 
    V_1^(-2,-1)(Z_1,Z̅_1)V_2^(0,-1)(Z_2,Z̅_2)∏_r=3^N[V_r^(-1,-1)(Z_r,Z̅_r)].

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 <ref>. As is shown in appendix
<ref>, this expression is equal to 

    ∫[dX^μdψ^μdψ̅^μdbdb̅dcdc̅dβ dβ̅dγ dγ̅]_g_zz̅^Ae^-S^tot
              ×∏_K=1^6g-6+2N[∮_C_Kdz/∂ρb_zz+ε_K∮_C̅_Kdz̅/∂̅ρ̅b_z̅z̅]
    
    
    ∏_I[e^ϕT_F^LC(z_I)e^ϕ̅T̅_F^LC(z̅_I)]
    
              ×∏_re^-iQ^2/α_r𝒳^+(𝐳̂̃̂_I^(r),𝐳̂̃̂̅̂̃̂_I^(r))
              × V_1^(-2,-1)(Z_1,Z̅_1)V_2^(0,-1)(Z_2,Z̅_2)∏_r=3^N[V_r^(-1,-1)(Z_r,Z̅_r)] .

In (<ref>), we can replace V_2^(0,-1)(Z_2,Z̅_2)
by (<ref>) for the same reason as that in the critical
case. With the replacement, the path integral (<ref>)
can factorize into those of matter, ghosts and superghosts. For the
longitudinal variables, we get from (<ref>)

    ∫[d𝒳^+d𝒳^-]_g^A_zz̅
          e^-S_super^±[g^A_zz̅]∏_r=1^N[V_r^DDF(Z_r,Z̅_r)e^-iQ^2/α_rX^+(𝐳̂̃̂_I^(r),𝐳̂̃̂̅̂̃̂_I^(r))]ψ^+(Z_1)ψ^-(Z_2)
    
        =     (2π)^2δ(∑_sp_s^-)
                     δ(∑_rp_r^+)
    
    
     
     
     
     
     
    Z_X^±[ g^A_zz̅]
    (^'(-g^A zz̅∂_z∂_z̅)/ImΩ∫ d^2 z √(g^A))^-1/2ϑ[α_R](0)^∗
              × e^Q^2/2Γ[σ;g^A_zz̅]h_α_L(Z_1)h_α_L(Z_2)
    
    
    
    ∏_r[1/α_re^-ReN̅_00^rrV_r^LC] .

Combining eqs.(<ref>), (<ref>) and (<ref>),
it is straightforward to show that (<ref>) is proportional
to the light-cone gauge expression in (<ref>)
with d=10-8Q^2. 

The expression (<ref>),
summed over spin structures and 
integrated over the moduli parameters, 
gives a BRST invariant expression of the amplitude. The operators
X,X̅, 
e^-iQ^2/α_r𝒳^+(𝐳̂̃̂_I^(r),
         𝐳̂̃̂̅̂̃̂_I^(r)), 
V_1^(-2,-1), V_2^(0,-1), 
V_r^(-1,-1)
are all BRST invariant and the BRST variations of the antighost insertions
yield total derivatives on moduli space. Since
(<ref>) coincides with (<ref>), 
the amplitude is finite for Q^2>10. 

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 (<ref>)
can be deformed to define the amplitudes following the Sen-Witten
prescription <cit.>. We can divide the moduli space
into patches and put the PCO's avoiding the spurious singularities
as was explained in <cit.> 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 

    lim_Q→0A^LC(Q^2)=A^SW(0) ,

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 <cit.>. 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 <cit.>, it
will be straightforward to rewrite the light-cone gauge expression
into the conformal gauge one. These problems are left to future work. 



§ ACKNOWLEDGMENTS


N.I. would like to thank Ashoke Sen for useful comments. He would
also like to thank the organizers of “Recent Developments on Light
Front” at Arnold Sommerfeld Center for Theoretical Physics and “SFT@HIT”
at Holon, especially Ivo Sachs, Ted Erler, Sebastian Konopka and Michael
Kroyter, for hospitality. This work was supported in part by Grant-in-Aid
for Scientific Research (C) (25400242) and (15K05063) from MEXT.





§ OPERATOR VALUED COORDINATE


It is convenient to introduce the operator valued coordinate ẑ_I
and its supersymmetric version 𝐳̂̃̂_I, in
order to express the conformal gauge form of the amplitudes for noncritical
dimensions. 

Let us first consider ẑ_I which is defined in the bosonic
case. In the light-cone gauge setup, we consider the situation where
the variable X^+(z,z̅) possesses an expectation
value -i/2(ρ(z)+ρ̅(z̅)).
Therefore we decompose it into the expectation value and the fluctuation
as 

    X^+(z,z̅)=-i/2(ρ(z)+ρ̅(z̅))+δ X^+(z,z̅) .

Roughly speaking, we define ẑ_I to be an operator valued
coordinate which satisfies

    ∂ X^+(ẑ_I)=0 .

Substituting (<ref>) into (<ref>), we get 

    -i/2∂ρ(ẑ_I)+∂δ X^+(ẑ_I)=0 .

We take ẑ_I so as to coincide with z_I when δ X^+=0.
Assuming that ẑ_I is expanded in terms of the fluctuation
δ X^+ as

    ẑ_I=z_I+∑_n=1^∞δ^(n)ẑ_I ,

where δ^(n)z_I is at the n-th order in the
derivatives of δ X^+, in principle we can obtain δ^(n)z_I
if ∂^2ρ(z_I)0. Lower order examples
are given by

    δ^(1)ẑ_I    =     -2i/∂^2ρ∂δ X^+(z_I) ,
    
    δ^(2)ẑ_I    =     -4/(∂^2ρ)^2∂δ X^+∂^2δ X^+(z_I)+2∂^3ρ/(∂^2ρ)^3(∂δ X^+)^2(z_I) .

In general δ^(n)z_I becomes a polynomial of
the derivatives of δ X^+ 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. ẑ̅̂_I, which is
the antiholomorphic counterpart of ẑ_I, can be obtained
in the same way. 

The OPE of ẑ_I with the energy-momentum tensor T(z)
comes from the contractions of ∂ X^- in T(z)
with ∂^kδ X^+(z_I) in ẑ_I.
Taking the OPE of (<ref>) with T(z), we get

    1T(z)1ẑ_I∂^2X^+(ẑ_I)+1/(z-ẑ_I)^2∂ X^+(ẑ_I)+1/z-ẑ_I∂^2X^+(ẑ_I)=0 .

Using (<ref>) and the fact that ∂^2X^+(ẑ_I)=-i/2∂^2ρ(z_I)+⋯
is invertible perturbatively, we obtain

    1T(z)1ẑ_I∼-1/z-ẑ_I .

Expanding ẑ_I as in (<ref>), we can see that
the right hand side of (<ref>) involves poles of arbitrarily
high order at z=z_I. Only finite number of them are relevant
in the correlation functions (<ref>), (<ref>)
and (<ref>). 

With the OPE (<ref>) and its antiholomorphic version,
we can show the OPE's

    T(z)e^iα X^+(ẑ_I,ẑ̅̂_I)    ∼   regular ,
    
    T̅(z̅)e^iα X^+(ẑ_I,ẑ̅̂_I)    ∼   regular ,

for any constant α. Therefore e^iα X^+(ẑ_I,ẑ̅̂_I)
is a BRST invariant operator in the conformal gauge bosonic string
theory in noncritical dimensions. 

It is straightforward to define the operator valued supercoordinate
𝐳̂̃̂_I. We define 𝐳̂̃̂_I=(ẑ̃̂_I,θ̂̃̂_I)
to be the operator valued supercoordinate which satisfies

    ∂𝒳^+(𝐳̂̃̂_I)     =     0 ,
    ∂ D𝒳^+(𝐳̂̃̂_I)     =     0 .

Notice that 𝐳̂̃̂_I is the operator version
of the supercoordinate 𝐳̃_I defined in <cit.>
which is not superconformal, but it is sufficient for our purpose.
Similarly to the bosonic case, we decompose 𝒳^+(𝐳,𝐳̅)
as

    𝒳^+(𝐳,𝐳̅)=-i/2(ρ_s(𝐳)+ρ̅_s(𝐳̅))+δ𝒳^+(𝐳,𝐳̅) ,

where ρ_s,ρ̅_s are the supersymmetric version of
ρ,ρ̅ whose explicit form is given in (<ref>).
S_α in that equation is the one in (<ref>) or in
(<ref>) according to whether the spin structure of the
fermions is even or odd. Using this decomposition, we get ẑ̃̂_I,θ̂̃̂_I
as expansions around z̃_̃Ĩ, θ̃_I in terms
of the fluctuation δ𝒳^+ as

    ẑ̃̂_I    =    z̃_I+∑_n=1^∞δ^(n)ẑ̃̂_I ,
    
    θ̂̃̂_I    =    θ̃_I+∑_n=1^∞δ^(n)θ̂̃̂_I ,

assuming ∂^2ρ(𝐳̃_I)0.
For example,

    δ^(1)ẑ̃̂_I    =     -2i/∂^2ρ∂δ𝒳^+(𝐳̃_I) ,
    
    δ^(1)θ̂̃̂_I    =     -2i/∂^2ρ∂ Dδ𝒳^+(𝐳̃_I) .

We obtain the OPE's 

    2T(𝐳)2ẑ̃̂_I-2T(𝐳)2θ̂̃̂_Iθ̂̃̂_I   ∼    -θ-θ̂̃̂_I/𝐳-𝐳̂̃̂_I-1/(𝐳-𝐳̂̃̂_I)^2·D𝒳^+/2∂^2𝒳^+(𝐳̂̃̂_I) ,
    
    2T(𝐳)2θ̂̃̂_I   ∼    -1/2/𝐳-𝐳̂̃̂_I-θ-θ̂̃̂_I/(𝐳-𝐳̂̃̂_I)^3·D𝒳^+/∂^2𝒳^+(𝐳̂̃̂_I)
    
            -1/(𝐳-𝐳̂̃̂_I)^2·∂^2D𝒳^+D𝒳^+/2(∂^2𝒳^+)^2(𝐳̂̃̂_I) .

From these OPE's, we get 

    T(𝐳)e^iα𝒳^+(𝐳̂̃̂_I,𝐳̂̅̂̃̂̅̂_I)    ∼   regular ,
    T̅(𝐳̅)e^iα𝒳^+(𝐳̂̃̂_I,𝐳̂̅̂̃̂̅̂_I)    ∼   regular ,

for any constant α. Therefore e^iα𝒳^+(𝐳̂̃̂_I,𝐳̂̅̂̃̂̅̂_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/α_r𝒳^+(𝐳̂̃̂_I^(r),𝐳̂̃̂̅̂̃̂_I^(r)),






which can be replaced by

    e^-iQ^2/α_rX^+(z_I^(r),z̅_I^(r))

 in evaluating (<ref>), (<ref>) and
(<ref>). This expression can be used instead of the
complicated combination ∮_z_I^(r)dz/2π i∮_z̅_I^(r)dz̅/2π iS(z,Z_r)S̅(z̅,Z̅_r)
in <cit.>, which has a similar effect
.



§ 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 <cit.>

    ∫[dψ dψ̅dψ̅^†dψ^†]_ĝ_zz̅
    e^-Sψ^†(x_1)ψ̅^†(x̅_1)⋯ψ^†(x_n)ψ̅^†(x̅_n)ψ̅(y̅_n)ψ(y_n)⋯ψ̅(y̅_1)ψ(y_1)
            =(^'(-ĝ^zz̅∂_z∂_z̅)/ImΩ∫ d^2z√(ĝ))^-1/2
          ×ϑ[α_L](∑_i=1^n∫_P_0^x_iω-∑_i=1^n∫_P_0^y_iω)ϑ[α_R](∑_i=1^n∫_P_0^x_iω-∑_i=1^n∫_P_0^y_iω)^∗
          ×|∏_i<j[E(x_i,x_j)E(y_j,y_i)]/∏_i,jE(x_i,y_j)|^2 .

When both α_L and α_R correspond
to even spin structures, using the formula 

    ϑ(∑_i=1^n∫_P_0^x_iω-∑_i=1^n∫_P_0^y_iω-e)ϑ(e)^n-1∏_i<j[E(x_i,x_j)E(y_j,y_i)]/∏_i,jE(x_i,y_j)
            =(ϑ(∫_P_0^x_iω-∫_P_0^y_iω-e)/E(x_i,y_j)) ,

given in <cit.> for the case

    e_ν=-(Ωα^'+α^'')_ν ,

 it is straightforward to show that (<ref>) can be
transformed into

    ∫[dψ dψ̅dψ̅^†dψ^†]_ĝ_zz̅
    e^-Sψ^†(x_1)ψ̅^†(x̅_1)
      ⋯ψ^†(x_n)ψ̅^†(x̅_n)
    ψ̅(y̅_n)ψ(y_n)
      ⋯ψ̅(y̅_1)ψ(y_1)
    
        =    (^'(-ĝ^zz̅∂_z∂_z̅)/ImΩ∫ d^2z√(ĝ))^-1/2ϑ[α_L](0)ϑ[α_R](0)^∗[S_α_L(x_i,y_j)][S_α_R(x_i,y_j)^∗],

 where 

    S_α(z,w)=1/E(z,w)ϑ[α](∫_w^zω)/ϑ[α](0)

is the Szego kernel. The expression (<ref>) implies
that the partition function is given by 

    (Z^ψ[g_zz̅])^2=(^'(-g^zz̅∂_z∂_z̅)/ImΩ∫ d^2z√(g))^-1/2ϑ[α_L](0)ϑ[α_R](0)^∗ ,

and the propagators of the fermions are 

    1ψ^†(x)1ψ(y)     =     S_α_L(x,y) ,
    
    1ψ̅^†(x̅)1ψ̅(y̅)     =     S_α_R(x,y)^∗ .

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 (<ref>) for

    e_ν=-(Ωα_L^'+α_L^'')_ν-∫_q^pω_ν ,

in the limit q→ p, we get <cit.>

    ϑ[α_L]
       ( ∑_i=1^n∫_P_0^x_iω
              -∑_i=1^n∫_P_0^y_iω)
    ∏_i<j[E(x_i,x_j)E(y_j,y_i)]/∏_i,jE(x_i,y_j)
       =    ∫ dψ_0dψ_0^† [ψ_0^†h_α_L(x_i)
                   ψ_0h_α_L(y_j)
                 + 1/E(x_i,y_j)∑_ν∂_νϑ[α_L]
                              (∫_y_j^x_iω)
                           ω_ν(p)/∑_ν∂_νϑ[α_L]
                              ( 0 )
                            ω_ν(p)].

Here 

    h_α_L(z)=√(∑_ν∂_νϑ[α_L](0)ω_ν(z))

gives the zero mode of the fermion. Substituting (<ref>)
into the right hand side of (<ref>), we obtain

    ∫[dψ dψ̅dψ̅^†dψ^†]_ĝ_zz̅
    e^-Sψ^†(x_1)ψ̅^†(x̅_1)⋯ψ^†(x_n)ψ̅^†(x̅_n)ψ̅(y̅_n)ψ(y_n)⋯ψ̅(y̅_1)ψ(y_1)
            =(^'(-ĝ^zz̅∂_z∂_z̅)/ImΩ∫ d^2z√(ĝ))^-1/2
    
    ∫ dψ_0dψ_0^†[S_α_L(x_i,y_j)]ϑ[α_R](0)^∗[S_α_R(x_i,y_j)^∗],

where 

    S_α_L(x,y)=ψ_0^†h_α_L(x)ψ_0h_α_L(y)+1/E(x,y)∑_ν∂_νϑ[α_L](∫_y^xω)ω_ν(p)/∑_ν∂_νϑ[α_L](0)ω_ν(p) .

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. 






§ 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.



 §.§ 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

    S_super^±[ĝ_zz̅,𝒳^±]=S_free[ĝ_zz̅,𝒳^±]+d-10/8Γ_super[ĝ_zz̅, 2i𝒳^+] ,

where S_free[ĝ_zz̅,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 <cit.>

    ∫[d𝒳^+d𝒳^-]_ĝ_zz̅
      e^-S_super^±[ĝ_zz̅,
                               𝒳^±]∏_r=1^N e^-ip_r^+𝒳^-
                         (𝐙_r,𝐙̅_r)
     ∏_u=1^Me^-ip_u^-𝒳^+
                        (𝐰_u,𝐰̅_u)
    
            =∫[d𝒳^+d𝒳^-]_ĝ_zz̅e^-S_free[ĝ_zz̅,𝒳^±]∏_r=1^Ne^-ip_r^+𝒳^-(𝐙_r,𝐙̅_r) 
              × e^-d-10/8Γ_super[ĝ_zz̅, 2i𝒳^+]∏_u=1^Me^-ip_u^-𝒳^+
                     (𝐰_u,𝐰̅_u)
    
            =(2π)^2δ(∑_up_u^-)δ(∑_rp_r^+)
    
    
    (Z_super^𝒳[ĝ_zz̅])^2
          ×
    e^-d-10/8Γ_super[ĝ_zz̅, ρ_s+ρ̅_s] ∏_u
      e^-p_u^-ρ_s+ρ̅_s/2
          (𝐰_u,𝐰̅_u) .

Regarding the second and third lines as a correlation function of

    e^-d-10/8Γ_super[ĝ_zz̅, 2i𝒳^+]∏_u=1^M
       e^-ip_u^-𝒳^+ (𝐰_u,𝐰̅_u)

of the free theory with the source term

    ∏_r=1^Ne^-ip_r^+𝒳^-(𝐙_r,𝐙̅_r) ,

we can calculate it by replacing the 𝒳^+(𝐳,𝐳̅)
by its expectation value -i/2(ρ_s(𝐳)+ρ̅_s(𝐳̅))
and derive the fourth line. Here ρ_s,ρ̅_s are the
supersymmetric version of ρ,ρ̅ and expressed as

    ρ_s(𝐳)     =    ρ(z)-θ∑_rα_rΘ_rS_α_L(z,Z_r) ,
    ρ̅_s(𝐳̅)     =    ρ̅(z̅)-θ̅∑_rα_rΘ̅_rS_α_R(z̅,Z̅_r) ,

where S_α_L and S_α_R are
taken to be the Szego kernel (<ref>). The partition function
(Z_super^𝒳[ĝ_zz̅])^2
is described by using Z_X^±[ĝ_zz̅]
and Z_ψ^±[ ĝ_zz̅] 
in (<ref>) as

    (Z_super^𝒳[ĝ_zz̅])^2
    =
    
    
    
    
    
    
    
    
    
    
    
    
    Z_X^±[ ĝ_zz̅]
    Z_ψ^±[ ĝ_zz̅]  .

The explicit form of e^-d-10/8Γ_super[ĝ_zz̅, ρ_s+ρ̅_s]
can be found in <cit.>. 


In the case where α_L corresponds to an odd spin
structure, we can proceed in the same way. Since the correlation functions
of the free fermions are given in (<ref>) as an integral
over the zero modes ψ_0^±, we obtain

    ∫[d𝒳^+d𝒳^-]_ĝ_zz̅
     e^-S_super^±[ĝ_zz̅]∏_r=1^N e^-ip_r^+𝒳^-
                       (𝐙_r,𝐙̅_r)
    ∏_u=1^M e^-ip_u^-𝒳^+
                      (𝐰_u,𝐰̅_u)
        =    (2π)^2δ(∑_up_u^-)
       δ(∑_rp_r^+)
    
    
    
    
    Z_X^±[ĝ_zz̅]
    (^'(-ĝ^zz̅∂_z∂_z̅)/ImΩ∫ d^2z√(ĝ))^-1/2ϑ[α_R](0)^∗
          
     
     
     
    ×∫ dψ_0^+dψ_0^-∏_u e^-p_u^-ρ_s+ρ̅_s/2
                   (𝐰_u,𝐰̅_u)
     e^-d-10/8Γ_super[ĝ_zz̅, ρ_s+ρ̅_s] ,

where ρ_s,ρ̅_s in this formula are (<ref>)
with S_α_L given in (<ref>) and S_α_R
taken to be the Szego kernel (<ref>).

With the correlation function (<ref>), it is
straightforward to check the following properties of the energy-momentum
tensor:


  * T^𝒳^±(𝐳) is regular at z=z_I.

  * T^𝒳^±(𝐳) satisfies

    T^𝒳^±(𝐳)e^-ip_r^+𝒳^--ip_r^-𝒳^+(𝐙_r,𝐙̅_r)
       ∼   θ-Θ_r/(z-Z_r)^2(-p_r^+p_r^-)e^-ip_r^+𝒳^--ip_r^-𝒳^+(𝐙_r,𝐙̅_r)
    
            +1/𝐳-𝐙_r1/2 De^-ip_r^+𝒳^--ip_r^-𝒳^+
                        (𝐙_r,𝐙̅_r)
    
            +θ-Θ_r/z-Z_r∂ e^-ip_r^+𝒳^--ip_r^-𝒳^+
                    (𝐙_r,𝐙̅_r) .


  * The OPE between 𝒳^-'s is given by 

    D𝒳^-(𝐳)D𝒳^-(𝐳') 
    
            ∼-d-10/4
               ×  
    DD^'[θ-θ'/(𝐳-𝐳')^33D𝒳^+/(∂𝒳^+)^3(𝐳').
    
          +1/(𝐳-𝐳')^2(1/2(∂𝒳^+)^2+4∂ D𝒳^+D𝒳^+/(∂𝒳^+)^4)(𝐳')
    
          +θ-θ'/(𝐳-𝐳')^2(-∂ D𝒳^+/(∂𝒳^+)^3-5∂^2𝒳^+D𝒳^+/2(∂𝒳^+)^4)(𝐳')
    
          +1/𝐳-𝐳'(-∂^2𝒳^+/2(∂𝒳^+)^3+2∂^2D𝒳^+D𝒳^+/(∂𝒳^+)^4-8∂^2𝒳^+∂ D𝒳^+D𝒳^+/(∂𝒳^+)^5)(𝐳')
    
          +θ-θ'/𝐳-𝐳'(-∂^2D𝒳^+/2(∂𝒳^+)^3+3∂^2𝒳^+∂ D𝒳^+/2(∂𝒳^+)^4-∂^3𝒳^+D𝒳^+/2(∂𝒳^+)^4.
    
          ..+(∂^2𝒳^+)^2D𝒳^+/(∂𝒳^+)^5-∂^2D𝒳^+∂ D𝒳^+D𝒳^+/(∂𝒳^+)^5)(𝐳')],

and we can deduce that the energy momentum tensor T^𝒳^±(𝐳)
satisfies the OPE

    T^𝒳^±(𝐳)T^𝒳^±(𝐳^')
    
            ∼12-d/4(𝐳-𝐳^')^3+θ-θ^'/(𝐳-𝐳^')^23/2T^𝒳^±(𝐳^')+1/𝐳-𝐳^'1/2DT^𝒳^±(𝐳^')+θ-θ^'/𝐳-𝐳^'∂ T^𝒳^±(𝐳^'),

which corresponds to the super Virasoro algebra with the central charge
ĉ=12-d. It follows that combined with the transverse variables
X^i(𝐳,𝐳̅), the total central
charge of the system becomes ĉ=10. This implies that with
the ghost superfields B(𝐳) and C(𝐳)
defined as

    B(𝐳)=β(z)+θ b(z) ,
         C(𝐳)=c(z)+θγ(z) ,

it is possible to construct a nilpotent BRST charge

    Q_B=∮d𝐳/2π i[-C(T^𝒳^±-1/2D𝒳^i∂𝒳^i)+(C∂ C-1/4(DC)^2)B].


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. 


In the same way as in the even spin structure case <cit.>,
we can derive from (<ref>), 

    ∫[d𝒳^+d𝒳^-]_ĝ_zz̅e^-S_super^±[ĝ_zz̅]∏_r=1^Ne^-ip_r^+X^-(Z_r,Z̅_r)∏_s=1^Me^-ip_s^-X^+(w_s,w̅_s)
    
          ×ψ^+(u_1)⋯ψ^+(u_n)
      ψ^-(v_1)⋯ψ^-(v_m)
    
          ×ψ̅^+(ũ_1)⋯ψ̅^+(ũ_n)
      ψ̅^-(ṽ_1)⋯ψ̅^-(ṽ_m)
    
       =    (2π)^2δ(∑_sp_s^-)
                    δ(∑_rp_r^+)
    
    
    
    
    Z_X^±[ ĝ_zz̅]
    ∏_se^-p_s^-1/2(ρ+ρ̅)
                (w_s,w̅_s)
    e^-d-10/16Γ[σ;ĝ_zz̅]
          
        ×∫[dψ^+dψ^-dψ̅^+dψ̅^-]_ĝ_zz̅
       e^1/π∫ d^2z(ψ^-∂̅ψ^++ψ̅^-∂ψ̅^+)-S_int
          ×ψ^+(u_1)⋯ψ^+(u_n)
        ψ^-(v_1)⋯ψ^-(v_m)
    
          ×ψ̅^+(ũ_1)⋯ψ̅^+(ũ_n)
          ψ̅^-(ṽ_1)⋯ψ̅^-(ṽ_m) ,

where 

    S_int    =    d-10/8[-∑_r2/α_r∂ψ^+ψ^+/∂^2ρ(z_I^(r)).
    
          +∑_I{(5/3∂^4ρ/(∂^2ρ)^3-3(∂^3ρ)^2/(∂^2ρ)^4)∂ψ^+ψ^+-8/3∂^3ψ^+ψ^+/(∂^2ρ)^2.
    
          . +4∂^3ρ/(∂^2ρ)^3∂^2ψ^+ψ^++4/3∂^3ψ^+∂^2ψ^+∂ψ^+ψ^+/(∂^2ρ)^4}(z_I)
    
          .+c.c.] .

The path integral over ψ^±,ψ̅^± can be computed
by treating S_int perturbatively. Since S_int
involves only ψ^+, the perturbation series terminates at a
finite order. 



 §.§ A proof of equality of (<ref>) and (<ref>)


In this appendix, we show that (<ref>) is equal to (<ref>). 
In the case Q=0, this equality implies that  (<ref>) is equal to (<ref>). 
Proving this can be done by using 
a fermionic charge[This fermionic charge 
was used in <cit.>.]

    Q̂^'≡∮dz/2π i[-b/4∂ρ(iX_L^+-1/2ρ)(z)+β/2∂ρψ^+(z)] ,

and its antiholomorphic counterpart Q̂̅̂^'. Here
(iX_L^+-1/2ρ)(z)
is defined as

    (iX_L^+-1/2ρ)(z)≡∫_w_0^zdz^'(i∂ X^+-1/2∂ρ)(z^') ,

with a generic point w_0 on the surface. (iX_L^+-1/2ρ)(z)
thus defined is single valued on the surface in the correlation functions
we consider here because -i/2ρ coincides with the expectation
value of X_L^+ in the presence of 
the sources e^-ip_r^+X^-.
In order to use Q̂^', we need to rewrite the ghost
part of the correlation function. Inserting

    1=|∮_w_0dz/2π ib/∂ρ(z)∂ρ c(w_0)|^2

into (<ref>) and deforming the contours of the antighost
insertions, (<ref>) is transformed into 

    ∫[dX^μdψ^μdψ̅^μdbdb̅dcdc̅dβ dβ̅dγ dγ̅]_g_zz̅^Ae^-S^tot
              ×∂ρ c(w_0)∂̅ρ̅c̅(w̅_0)
              ×∏_j=1^g[(∮_α_jdz/∂ρb_zz+∮_α_jdz̅/∂̅ρ̅b_z̅z̅)(∮_β_jdz/∂ρb_zz+∮_β_jdz̅/∂̅ρ̅b_z̅z̅)]
              ×∏_I[∮_z_Idz/2π ib/∂ρ(z)X(z_I)∮_z̅_Idz̅/2π ib̅/∂ρ̅(z̅)X̅(z̅_I)]∏_re^-iQ^2/α_r𝒳^+(𝐳̂̃̂_I^(r),𝐳̂̃̂̅̂̃̂_I^(r))
              × V_1^(-2,-1)(Z_1,Z̅_1)V_2^(0,-1)(Z_2,Z̅_2)∏_r=3^N[V_r^(-1,-1)(Z_r,Z̅_r)] .

Here α_j and β_j are chosen so that they form a
canonical basis of the first homology group of the Riemann surface.
Using Q̂^', the operators inserted at z=z_I can
be expressed as 

    ∮_z_Idz/2π ib/∂ρ(z)X(z_I)^'∏_re^-iQ^2/α_r𝒳^+(𝐳̂̃̂_I^(r),𝐳̂̃̂̅̂̃̂_I^(r))
            =-∮_z_Idz/2π ib/∂ρ(z)e^ϕT_F^LC(z_I)^'∏_re^-iQ^2/α_r𝒳^+(𝐳̂̃̂_I^(r),𝐳̂̃̂̅̂̃̂_I^(r))
          -{Q̂^',∮_z_Idz/2π ib/∂ρ(z)∮_z_Idw/2π iA(w)/w-z_Ie^ϕ(z_I)^'∏_re^-iQ^2/α_r𝒳^+(𝐳̂̃̂_I^(r),𝐳̂̃̂̅̂̃̂_I^(r))}
          +1/4∮_z_Idz/2π ib/∂ρ(z)∮_z_Idw/2π i∂ρψ^-(w)/w-z_Ie^ϕ(z_I)^'∏_re^-iQ^2/α_r𝒳^+(𝐳̂̃̂_I^(r),𝐳̂̃̂̅̂̃̂_I^(r)) .

Here the prime in ^'∏ means that the product
is taken over those r which satisfy 

    z_I^(r)=z_I ,

and

    A(w)     =     -i∂ X^+∂ργ(w)-2∂(∂ρ c)ψ^-(w)
    
           -d-10/4i[(5(∂^2X^+)^2/4(∂ X^+)^3-∂^3X^+/2(∂ X^;)^2)(-2∂ργ)
    
    
    -2∂^2X^+/(∂ X^+)^2∂(-2∂ργ)
    . 
          . +∂^2(-2∂ργ)/∂ X^+-(-2∂ργ)∂ψ^+∂^2ψ^+/2(∂ X^+)^3](w) .

Substituting (<ref>) into (<ref>) and using the
commutators

    {Q̂^',Q̂^'}   =  0 ,
          {Q̂^',c(w_0)}  =  0 , 
          {Q̂^',b(z)}  =  0 , 
          [Q̂^',V_r^(p,-1)] =  0
        (p=-2,-1,0) ,
          {Q̂^',∮_z_Idz/2π ib/∂ρ(z)e^ϕT_F^LC(z_I)^'∏_re^-iQ^2/α_r𝒳^+(𝐳̂̃̂_I^(r),𝐳̂̃̂̅̂̃̂_I^(r))}   =  0 , 
          {Q̂^',∮_z_Idz/2π ib/∂ρ(z)∮_z_Idw/2π i∂ρψ^-(w)/w-z_Ie^ϕ(z_I)^'∏_re^-iQ^2/α_r𝒳^+(𝐳̂̃̂_I^(r),𝐳̂̃̂̅̂̃̂_I^(r))}   =  0 ,

we can show that (<ref>) is equal to 

    ∫[dX^μdψ^μdψ̅^μdbdb̅dcdc̅dβ dβ̅dγ dγ̅]_g_zz̅^Ae^-S^tot
              ×∂ρ c(w_0)∂̅ρ̅c̅(w̅_0)
              ×∏_j=1^g[(∮_α_jdz/∂ρb_zz+∮_α_jdz̅/∂̅ρ̅b_z̅z̅)(∮_β_jdz/∂ρb_zz+∮_β_jdz̅/∂̅ρ̅b_z̅z̅)]
              ×∏_I∮_z_Idz/2π ib/∂ρ(z)[-e^ϕT_F^LC(z_I)+1/4∮_z_Idw/2π i∂ρψ^-(w)/w-z_Ie^ϕ(z_I)]
              ×∏_I[∮_z̅_Idz̅/2π ib̅/∂ρ̅(z̅)X̅(z̅_I)]∏_re^-iQ^2/α_r𝒳^+(𝐳̂̃̂_I^(r),𝐳̂̃̂̅̂̃̂_I^(r))
              × V_1^(-2,-1)(Z_1,Z̅_1)V_2^(0,-1)(Z_2,Z̅_2)∏_r=3^N[V_r^(-1,-1)(Z_r,Z̅_r)] .

The correlation functions of ψ^± which appear in (<ref>)
can be calculated using (<ref>) treating S_int
perturbatively. 
Since ∂ρ(z_I)=0, 
∮_z_Idw/2π i∂ρψ^-(w)/w-z_I
vanishes unless ψ^-(w) becomes singular at w=z_I.
Hence if the ψ^+(Z_1) in 
V_1^(-2,-1)(Z_1,Z̅_1)
is contracted with the ψ^-(w) in 
∮_z_Idw/2π i∂ρψ^-(w)/w-z_I,
the contour integral over w vanishes. 
Therefore only the terms which involve contraction of the 
ψ^+(Z_1) in
V_1^(-2,-1)(Z_1,Z̅_1) with the
ψ^-(Z_2) in 
V_2^(0,-1)(Z_2,Z̅_2)
survive. 
The ψ^-(w) in 
∮_z_Idw/2π i∂ρψ^-(w)/w-z_I
should be contracted with ∂^nψ^+(z_I)
involved in 
e^-iQ^2/α_r𝒳^+(𝐳̂̃̂_I^(r),
             𝐳̂̃̂̅̂̃̂_I^(r))
or 
S_int but doing so induces another contraction

    ∮_z_Jdw/2π i1/w-z_J∂ρ1ψ^-(w)∂^n1ψ^+(z_I) ,

with z_J z_I, because 
e^-iQ^2/α_r𝒳^+( 𝐳̂̃̂_I^(r),
         𝐳̂̃̂̅̂̃̂_I^(r))
and S_int are Grassmann even. 
Hence we conclude that
∮_z_Idw/2π i∂ρψ^-(w)/w-z_I
in (<ref>) does not contribute to the path integral.
We can do the same thing for the antiholomorphic part and prove that
the X(z_I),X̅(z̅_I) which appear
in (<ref>) can be replaced by -e^ϕT_F^LC(z_I),-e^ϕ̅T̅_F^LC(z̅_I)
for all I. By deforming the contours of the antighost insertions
back, we can see that (<ref>) is equal to (<ref>). 


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