Multiloop amplitudes of lightcone gauge NSR string field theory in noncritical dimensions
Received: December
Multiloop amplitudes of lightcone gauge NSR string
Nobuyuki Ishibashi 0 1 2 4 5
Koichi Murakami 0 1 2 3 5
0 OtanoshikeNishi 2321 , Kushiro, Hokkaido 0840916 , Japan
1 Tsukuba , Ibaraki 3058571 , Japan
2 Faculty of Pure and Applied Sciences, University of Tsukuba
3 National Institute of Technology, Kushiro College
4 Center for Integrated Research in Fundamental Science and Engineering (CiRfSE)
5 Open Access , c The Authors
Feynman amplitudes of lightcone gauge superstring eld theory are illde ned because of various divergences. In a previous paper, one of the authors showed that taking the worldsheet theory to be the one in a linear dilaton background
String Field Theory; BRST Quantization; Conformal Field Models in String

Feynman i" (" > 0) and Q2 > 10 yields
nite amplitudes. In this paper, we apply this
iQX1 with
the analytic continuation Q ! 0.
Contents
1 Introduction
2 Superstring eld theory in linear dilaton background 2.1 2.2
Linear dilaton background
dilaton background
3 BRST invariant form of the amplitudes
The prescription
4.2 Q ! 0 limit of the lightcone gauge amplitudes
5 Conclusions and discussions
B Supersymmetric X
CFT
C.1 A formula for the superghosts
C.2 A formula for the reparametrization ghosts
D A proof of (3.9)
Introduction
14], using the lightcone gauge closed NSR superstring
eld theory, we have studied the
to the worldsheet theory.
theory in a linear dilaton background
iQX1 is considered so that the central charge
of lightcone gauge superstring
eld theory in the linear dilaton background are indeed
CFT constructed in [9] with the identi cation
cone gauge superstring
symmetric X
rstquantized
supersymmetric X
dices C and D.
Superstring eld theory in linear dilaton background
background constructed in [20]. The string
eld theory is given for Type II superstring
lar way.
from the rstquantized approach.
dilaton background
its fermionic partners 1
1 Z
; 1; g^zz =
dz ^ dzpg^ g^ab@aX1@bX1
2iQR^X1
transformations, are given as
dz ^ dzi
T X1 (z) =
(@X1)2
@ ln g^zz)@X1
(@ ln gzz)2 + @2 ln gzz ;
TFX1 (z) =
@X1 1 + Q(@
@ ln g^zz) 1
a free theory, we concentrate on the bosonic part. De ning
iQ ln(2gzz) ;
; N ) can be calculated on a Riemann
given as
= 2
X pr + 2Q(1
1 12Q2 [ ;gzAz] ZX gzz
24 A
e S[X1;gzz] Y eiprX~ 1 (Zr; Zr)
Y e prpsGA(Zr;Zs) Y
Arakelov metric (C.22), and
S X1; gab =
= ln gzz
[ ; gab] =
1 Z
dz ^ dzpg gab@aX1@bX1
2iQRX1 ;
dz ^ dzpg gab@a @b + 2R
of the linear dilaton conformal eld theory.
Notice that X~ 1 satis es
c = 1
12Q2
1 p2 + Qp =
(p + Q)2
@@X~ 1 = 0 ;
i@X~ 1(z) =
i@X~ 1(z) =
where n1 and
given as linear combinations of the Fock space states
hp1jp2i = 2
(p1 + p2 + 2Q) ;
n + 2Q n;0) ;
n + 2Q n;0) ;
where hpj,
plane as
are the BPZ conjugates of jpi, n1
; n1 respectively. On the sphere,
= 2
e S[X1;gzz] Y eiprX~ 1 (Zr; Zr)
action in the background.
theory with the variables
; i (i = 1;
where the action for X1, 1
eld theory with central charge
The string eld
c = 12
12Q2 :
t = x+ ;
= 2p+ :
and a function of
L0) j (t; )i = 0 ;
The action of the string eld theory is given by [7, 11]
S =
1 X Z 1
2 B
1 X Z 1 d
2 F
B1;B2;B3
B1;F2;F3
L0 + L0
L0 + L0
1 + Q2
1 + Q2
h V3j ( 1)i j ( 2)i j ( 3)i
denotes the BPZ conjugate of j (
string vertices and gs is the string coupling constant. P
) . The third and the fourth terms are the three
F denote the sums over
bosonic and fermionic string
interaction depicted in
can introduce a complex coordinate
whose real part
sponding to the rth external line so that
ds2 = d d ;
dilaton background
eld theory in linear
the vertex are given by the worldsheets depicted in
gure 2. Each term in the expansion
depicted in
gure 3.
moduli space of
as [28, 29]
(g) = (igs)2g 2+N C
[dT ][ d ][d ] F N(g) ;
[dT ][ d ][d ] =
2g 3+N
3g 3+N
I=1
A's denote the
can introduce a complex coordinate
in the same way as the complex coordinate (2.18) is
on the lightcone diagram
worldsheet, as
F (g) = (2 )
2g 2+N
2 (zI ) 2 TFLC (zI ) TFLC (zI )
path integral measure dXid i
= ln @ @
[ ; g^zz] =
ln g^zz ;
1 Z
dz ^ dzpg^ g^ab@a @b + 2R^
corresponds to the state
in the (NS,NS) sector, the lightcone vertex VrLC is given as
VrLC =
I dwr i@X~ i1 (wr)wr n1
I dwr j1 (wr)wr s1 12
I dwr i@X~ i1 (wr)wr n1
I dwr j1 (wr)wr s1 12
iQ i1 ln(2gzz) ;
local coordinate z on
X nk + X sl =
It is possible to calculate the right hand side of (2.23).
can be given as a function of
(z) =
r ln E(z; Zr)
r = 0 ;
is the period matrix.2
The base point P0 is arbitrary. There are 2g
2 + N zeros of @ and we denote them by
zI (I = 1;
obtain e
[ ; g^zz]. We can
Taking g^zz to be the Arakelov metric [27], e
in [13] to be
[ ;gzAz] for higher genus surfaces is calculated
[ ;gzAz] / e W Y e 2 Re N0r0r Y
2 X GA (zI ; zJ )
2 X GA (Zr; Zs) + 2 X GA (zI ; Zr)
+ 3 X ln 2gzAI zI ;
s ln E(Zr; Zs) +
! : (2.32)
interaction points and the degenerations of the surface
regularization of eld theory.
BRST invariant form of the amplitudes
function of the conformal gauge variables X ;
becomes
socalled X
= Y
reRe N0r0r e Q22 [ ;g^zz]
S (z; Zr)
ZX [g^zz]Z [g^zz]
S z; Zr VrDDF(Zr; Zr) :
Here X
S (z; w) is de ned to be
S(z; w)
(z) e i Qr2 (XL+(z) XL+(w)) ;
vertex operator given by
VrDDF(Zr; Zr) = Ai1n(r1)
i pr Npr+r X++ipirX~ i
(Zr; Zr) ;
Ai(rn) =
Bi(sr) =
product SSVrDDF is normal ordered as
where X~i is the super eld for X~ i; i; i,
is de ned in (B.5) and XL+ denotes the left
S (z; Zr) S z; Zr VrDDF Zr; Zr
w!Zr wl!imZr S (z; w) S (z; w) VrDDF Zr; Zr j
lim
3
2 (zI ) 2 e 21 [gzAz; ln j@ j2] ZX [gzAz]Z [gzAz]
TFLC (zI ) TFLC (zI )
S (z; Zr)
S z; Zr VrDDF(Zr; Zr) : (3.7)
{ 10 {
the amplitude (2.21) can be rewritten as
[dT ][d ][ d ]
dX d
dbdbdcdcd d d d gzAz e Stot
S (z; Zr)
bzz + "K
e TFLC (zI ) e TFLC (zI )i
VrDDF(Zr; Zr) : (3.8)
and the ghosts. It is
[dT ][d ][ d ]
dX d
dbdbdcdcd d d d gzAz e Stot
S (z; Zr)
bzz + "K
X (zI ) X (zI )
VrDDF(Zr; Zr) ; (3.9)
X (z) = c@
[dT ][d ][ d ]
dX d
dbdbdcdcd d d d gzAz e Stot
bzz + "K
S (z; Zr)
{ 11 {
X (zI + 2 ) X (zI + 2 )
VrDDF(Zr; Zr) :
S (z; Zr)
S (z; Zr)
@DC (z) e i Qr2 (XL+(z) XL+(Zr)) I
VrDDF(Zr; Zr)
VrDDF(Zr; Zr)
@DC (z) e i Qr2 (XR+(z) XR+(Zr))cce
VrDDF(Zr; Zr) ;
lim X (zI + 2 ) X (zI + 2 )
lim X (zI + 2 ) X (zI + 2 )
S (z; Zr)
VrDDF(Zr; Zr)
S (z; Zr) X (zI )
S z; Zr X (zI )
VrDDF(Zr; Zr) :
lim X (zI + 2 ) X (zI + 2 )
VrDDF(Zr; Zr)
VrD0DF(Zr0; Zr0)
S (z; Zr)
S (z; Zr0) X (zI )
S (z; Zr)
S (z; Zr)
S (z; Zr0)
S (z; Zr0)
S z; Zr0 X (zI )
{ 12 {
is proved in appendix D.
[dT ][d ][ d ]
dX d
dbdbdcdcd d d d
bzz + "K
VrDDF(Zr; Zr) :
X (zI ) X (zI )
VrDDF(Zr; Zr)cce
VrD0DF(Zr0 ; Zr0 ) :
su ers from the contact term divergences.
The amplitudes from the
rstquantized formalism
established.
by the method of these papers.
The prescription
K times
X is denoted by (m; a) with m 2 M; a 2
' : X ! M which maps (m; a) to m.
(m) and de ne a map
!n(m; s(m)) ;
where the integrand is schematically expressed as [24]
!n(m; z1;
Y (X(zi)
2 @(pggij ) bij dms
BRST invariant vertex operators, m1;
; mn are the coordinates of M and the subscript
detail in [26], it has been shown that
1. One can pick a dual triangulation
of M such that the map ' : X ! M has a local
section s over each of the codimension 0 polyhedron M
depicted in gure 4.
; VN are
2. The amplitude can be given as
!n(m; s (m)) + Avertical ;
the vertex operators, over @M
and their submanifolds which are called the vertical
segments, as long as the bad points are avoided.
4. The amplitude thus de ned is gauge invariant.
{ 14 {
S (z; Zr)
bzz + "K
X (zI ) X (zI )
VrDDF(Zr; Zr) :
Q ! 0 limit of the lightcone gauge amplitudes
A(Ng)(Q) =
!n(m; s(m)) ;
and we obtain
A(Ng)(Q) = A(Ng)SW(Q) ;
lim A(Ng)(Q) = A(Ng)SW(0) ;
{ 15 {
!n(m; s(m))
= dmZ1 ^ dm2 ^
dX d
dbdbdcdcd d d d
but may su er from the spurious singularities otherwise.
can de ne
A(Ng)SW(Q) =
X Z
!n(m; s0 (m)) + Avertical ;
Conclusions and discussions
the theory in a linear dilaton background
iQX1. The divergences of the amplitudes
c = 12
supersymmetric X
worldsheet theory by setting
Q2 in the action of the X
CFT given in (B.1).
extension elsewhere.
gauge closed superstring
have a gauge invariant string
eld theory to which our method here is applicable. The
Acknowledgments
{ 16 {
Let zz be
We note that
which follows from
The Arakelov metric on ,
is de ned so that its scalar curvature RA
2gAzz@@ ln gzAz satis es
and (15K05063) from MEXT.
!(z) :
dz ^ dz i zz = 1 ;
! ^ ! =
dsA2 = 2gzAzdzdz ;
gzAzRA =
de ned to satisfy4
@z@zGA(z; z; w; w) = 2
Let F (z; z; w; w) be the
which satis es
@z@z ln F (z; z; w; w) = 2
which can be given by
F (z; z; w; w) = exp
2 Im
GA(z; z; w; w) =
ln F (z; z; w; w)
4The delta function 2(z
w) is normalized by
w) = 1.
2gzAz = lim exp
GA(z; z; w; w)
Supersymmetric X
1 Z
Here the supercoordinate z is given by
the super eld X
is de ned as
z = (z; ) ;
supersymmetric X
CFT whose action is given by
Ssuper X ; g^zz =
+ DX
super X +; g^zz : (B.1)
(z; z) = X (z) + i
(z) + i
d (Rez) d (Imz) d d :
The interaction term
super is given by
super X +; g^zz =
(z; z) = ln
+ (z) =
1 Z
ln g^zz ;
which is the super Liouville action de ned for variable
with the background metric
T X (z) =
S(z; XL+) ;
where S(z; XL ) denotes the super Schwarzian derivative
S(z; XL ) =
{ 18 {
Y e ipr+X (Zr; Zr) Y e ips X + (ws; ws)
ZsXuper[g^zz] 2
the theory (B.1) with the identi cation
ZsXuper[g^zz] =
[dX ]g^zz exp
1 Z
d2zDX DX
Zr = (Zr; r) ;
ws = (ws; s) :
version of (z) in (2.30) which is de ned by
s (z) =
(z) + f (z) ;
r rS (z; Zr) ;
S (z; w) =
E (z; w)
all the external lines are in the NSNS sector and
is an even spin structure, S (z; w) is
equal to the socalled Szego kernel
0 + 00 ( 0; 00 2 Rg), given by
#[ ]( j ) =
2 Cg=(Zg +
Zg) . The right hand side of (B.8) can be calculated to be
Y e ps s +2 s (ws; ws) e d 810 super[ s+ s;g^zz] ; (B.15)
Y e ipr+X (Zr; Zr) Y e ips X + (ws; ws)
= (2 )
{ 19 {
super [ s + s; g^zz]) = exp
[ ; g^zz]
r =
I =
1 @f f
(zI(r) ) + c:c: ;
1 @3f @2f @f f
@f f +
(zI ) + c:c: ;
functions of the X
the correlation functions of fermions
, which is useful in appendix D. (B.15) can be
rewritten as5
= (2 )
Y e ipr+X +pr+( r
)(Zr; Zr) Y e ips X++ps ( s ++ s +)(ws; ws)
Y e ps 12 ( + )(ws; ws)e d 10
16
Z [g^zz]
)(Zr; Zr) Y eps ( s ++ s +)(ws; ws) ; (B.19)
Sint =
+ c:c: :
= (2 )2
the following identity
Y e ipr+X (Zr; Zr) Y e ips X+(ws; ws) + (u1)
Z [g^zz]
2 Z
+ (u~1)
+ (u~n)
Y e ps 12 ( + )(ws; ws)e d1610
e 1 R d2z( @ ++ @ +) Sint
+ (u~1)
+ (u~n)
(v~m) : (B.21)
4 @3 +@2 +@ + + )
CFT coincide
Let us consider the conformal eld theory with the action
1 Z
dz ^ dzpg brzc + brzc ;
2 Z or
We de ne =
1 to be
where the elds b; c are with conformal weight ( ; 0) ; (1
; 0) and b; c are their
antiholomorphic counterparts with conformal weight (0; ); (0; 1
). Here we consider the case
if b; c are Grassmann odd
1 if b; c are Grassmann even
{ 21 {
There exist local operators eq (z; z) q 2 Z2 , which satisfy
b (z) eq (w; w)
c (z) eq (w; w)
b (z) eq (w; w)
c (z) eq (w; w)
X qi =
Y e qi (zi; zi) ;
on a genus g Riemann surface. qi should satisfy
spin structure
000 . Namely, the elds b(z); c(z); b(z); c(z) transform as
; g) cycle once, and they transform as
c(z) ! e2 i 0 c(z) ;
b(z) ! e2 i 0 b(z) ;
c(z) ! e2 i 0 c(z) ;
b(z) ! e2 i 0 b(z) ;
c(z) ! e2 i 00 c(z) ;
b(z) ! e2 i 00 b(z) ;
c(z) ! e2 i 00 c(z) ;
b(z) ! e2 i 00 b(z) ;
Q e qi (zi; zi) is evaluated in [35] to be
i
gAzz@z@z
R d2zpgA
Ye qiqjGA(zi;zj)5 ;
d (q) =
q (q + 1
and the characteristics ab is de ned so that
( a + b) =
surface by 24 = K.
{ 22 {
A formula for the superghosts
Substituting (A.8), (A.9) and
e =
Therefore there exists a holomorphic g2 form
(g 1)z
where S is independent of z.
(z) has no zeros or poles, and it should transform as
(z) such that
cycles. These properties x
(z) and it should coincide with the
(z) in [30, 33] up to a
= 4
gAzz@z@z
R d2zpgA
2 exp
@z@z ln
Using (A.5), we can see
= 4
gAzz@z@z
R d2zpgA
(ej ) Y E (zi; zj )qiqj Y
{ 23 {
= 4
gAzz@z@z
R d2zpgA
2 exp 4 Im
2@@ ln gzAz =
X qi =
1) zz = 0 ;
1) = 0 ;
1)2 Im
with S; A independent of z, we get (C.17).
the determinant factor and e 3(2
1)2S. The determinant factor can also be recast into a
= 1; = 1. For arbitrary
and we get
(zi; zi)
gAzz@z@z
R d2zpgA jdet ! zi j
gAzz@z@z
R d2zpgA
Qi>j E (zi; zj ) Q
Qi E (zi; R)
e3S ; (C.21)
gAzz@z@z
R d2zpgA
Qi E (zi; R) (R) det ! zi
# 00 (ej ) Qi>j E (zi; zj ) Q
Therefore (C.17) can be rewritten as
Qi E (zi; R) (R) det ! zi
# 00 (ej ) Qi>j E (zi; zj ) Q
(ej ) Y E (zi; zj )qiqj Y
e ( 3(2 1)2+1)S :
gAzz@z@z
R d2zpgA
With vanishing central
morphic parts.
gAzz@z@z
R d2zpgA
#[ L] (0j ) #[ R] (0j ) :
The correlation function
be evaluated to be
X Z Zr
X Z Zr
I;r E (zI ; Zr)
QI>J E (zI ; zJ ) Qr>s E (Zr; Zs) Q
e 12S : (C.25)
meromorphic oneform @ (z) dz respectively,
2 + N ) are the zeros and the poles of the
{ 25 {
holds in the divisor sense. Therefore we obtain
On the other hand, (A.8), (A.9) and (C.15) imply
QI;r E (zI ; Zr)
QI>J E (zI ; zJ ) Qr>s E (Zr; Zs) Q
and from (2.31) we get
QI;r E (zI ; Zr)
QI>J E (zI ; zJ ) Qr>s E (Zr; Zs) Q
= exp 4
X GA (zI ; zJ ) + X GA (Zr; Zs)
X GA (zI ; Zr)5
= const.
A formula for the reparametrization ghosts
In [13] it was shown that the following identity holds:
re2 Re N0r0r e [gzAz; + ] ZX [gzAz]
dbdbdcdc gzAz e Sbc Y cc(Zr; Zr)
dz ^ dz i
K b + K b :(C.29)
6 + 2N )
de2 + N ) so that
Re (z1)
Re (z2)
Re (z2g 2+N ) ;
{ 26 {
TI
Re (zI+1)
Re (zI )
(I = 1; : : : ; 2g
3 + N ) :
d (Im ) (b
+ b ) =
There are 2g
3 + N insertions of this kind.
The twist
to the other. The antighost insertion should be
d (Im ) (b
b ) =
insertions of this kind.
= const:
[gzAz; + ] ZX [gzAz]
dbdbdcdc gzAz e Sbc Ycc(Zr; Zr)
bzz +"K
bzz : (C.35)
Here "K =
{ 27 {
A proof of (3.9)
fermionic charge
@ (z) c i@X+
@ (z) c i@X+
One can show that X (zI ) can be expressed as
X (zI ) =
(w) e (zI ) + b @ e2 +
(zI ) ; (D.2)
O =
+2Q2i
4 (@X+)3
2@2X+
(@X+)2
2 (@X+)2
and Q^ satis es the following identities:
hQ^; cce
VrDDF(Zr; Zr) = 0 ;
(w) e (zI ) = Q^; I
(w) e (zI ) = 0 ;
(zI ) = 0 ;
{ 28 {
Q^; e @
Q^; e @
= 0 :
6g 6+2N
The antighost insertions
bzz + "K
bzz is a product of the contour
integral of the type (C.32) becomes
dzi@X+
dzi@X+
the contour integrals (C.33), (C.34), we obtain
which vanishes because X+and
should be singlevalued. Hence Q^ commutes or
antican replace all the X (zI ) in the correlation functions by
(w) e (zI ) +
(zI ) ; (D.7)
Replacing X (zI ) by
(w) e (zI ) ;
{ 29 {
of (3.8) and the terms which involve
(w) e (zI ) ;
(w) e (zI ) ;
CFT part
of such terms are of the form
contributions from S; S; VrDDF EX
O+'s should be contracted with
's, which come from O (zI ). Therefore (D.11) with
(w) @k + (zJ ) ;
if it involves O
the form
for some integer k
vanish using (B.21). The contractions of
contributions from S; S; VrDDF
+'s from VrDDF do not contribute
The contractions of
+'s from S inevitably induce a factor of the form
(w) @k + (Zr) = 0 :
(w) @k + (z) = 0 ;
{ 30 {
with z
+'s from Sint necessarily induce a factor of the form
+'s. The contractions of
(w) @k + (zJ ) = 0 ;
side of (3.9) is equal to that of (3.8).
Open Access.
B 69 (1974) 77 [INSPIRE].
86 (1986) 163 [INSPIRE].
[INSPIRE].
[INSPIRE].
Phys. B 219 (1983) 437 [INSPIRE].
(1987) 1 [INSPIRE].
{ 31 {
Nucl. Phys. B 291 (1987) 557 [INSPIRE].
Phys. B 304 (1988) 108 [INSPIRE].
(1988) 559 [INSPIRE].
087 [arXiv:1508.05387] [INSPIRE].
[INSPIRE].
11 (2016) 050 [arXiv:1607.06500] [INSPIRE].
289 (1987) 227 [INSPIRE].
(1987) 167 [INSPIRE].
[1] S. Mandelstam , Interacting String Picture of the NeveuSchwarzRamond Model, Nucl . Phys.
[2] S. Mandelstam , Interacting String Picture of the Fermionic String , Prog. Theor. Phys. Suppl.
[3] S.J. Sin , Geometry of Super Light Cone Diagrams and Lorentz Invariance of Light Cone String Field Theory . 2. Closed NeveuSchwarz String, Nucl . Phys . B 313 ( 1989 ) 165 [4] M.B. Green and J.H. Schwarz , Superstring Interactions , Nucl. Phys . B 218 ( 1983 ) 43 [5] M.B. Green , J.H. Schwarz and L. Brink , Super eld Theory of Type II Superstrings, Nucl .
[6] D.J. Gross and V. Periwal , Heterotic String Light Cone Field Theory, Nucl . Phys . B 287 [7] Y. Baba , N. Ishibashi and K. Murakami , LightCone Gauge Superstring Field Theory and Dimensional Regularization , JHEP 10 ( 2009 ) 035 [arXiv:0906.3577] [INSPIRE].
[8] Y. Baba , N. Ishibashi and K. Murakami , LightCone Gauge String Field Theory in Noncritical Dimensions , JHEP 12 ( 2009 ) 010 [arXiv:0909.4675] [INSPIRE].
[9] Y. Baba , N. Ishibashi and K. Murakami , Lightcone Gauge NSR Strings in Noncritical [10] Y. Baba , N. Ishibashi and K. Murakami , Lightcone Gauge Superstring Field Theory and [11] N. Ishibashi and K. Murakami , Lightcone Gauge NSR Strings in Noncritical Dimensions II [12] N. Ishibashi and K. Murakami , Spacetime Fermions in Lightcone Gauge Superstring Field [13] N. Ishibashi and K. Murakami , Multiloop Amplitudes of Lightcone Gauge Bosonic String [19] C. Wendt , Scattering Amplitudes and Contact Interactions in Witten's Superstring Field [21] A. Sen , BV Master Action for Heterotic and Type II String Field Theories , JHEP 02 ( 2016 ) [22] A. Sen , Unitarity of Superstring Field Theory , JHEP 12 ( 2016 ) 115 [arXiv:1607.08244] [23] A. Sen , One Loop Mass Renormalization of Unstable Particles in Superstring Theory, JHEP [18] M.B. Green and N. Seiberg , Contact Interactions in Superstring Theory, Nucl. Phys . B 299 [24] A. Sen , O shell Amplitudes in Superstring Theory, Fortsch . Phys. 63 ( 2015 ) 149 [25] A. Sen , Gauge Invariant 1PI E ective Action for Superstring Field Theory , JHEP 06 ( 2015 ) [14] N. Ishibashi and K. Murakami , Worldsheet theory of lightcone gauge noncritical strings on higher genus Riemann surfaces , JHEP 06 ( 2016 ) 087 [arXiv:1603.08337] [INSPIRE].
[15] J. Greensite and F.R. Klinkhamer , New Interactions for Superstrings, Nucl. Phys . B 281 [16] J. Greensite and F.R. Klinkhamer , Contact Interactions in Closed Superstring Field Theory, [17] J. Greensite and F.R. Klinkhamer , Superstring Amplitudes and Contact Interactions , Nucl.
[26] A. Sen and E. Witten , Filling the gaps with PCO's , JHEP 09 ( 2015 ) 004 [27] S. Arakelov , Intersection Theory of Divisors on an Arithmetic Surface, Math. USSR Izv . 8 [28] E. D'Hoker and S.B. Giddings , Unitary of the Closed Bosonic Polyakov String, Nucl . Phys.
[29] K. Aoki , E. D'Hoker and D.H. Phong , Unitarity of Closed Superstring Perturbation Theory, [30] E. D'Hoker and D.H. Phong , The Geometry of String Perturbation Theory, Rev. Mod. Phys.
[31] S. Mandelstam , The Interacting String Picture and Functional Integration , lectures given at the Workshop on Uni ed String Theory, Santa Babara, CA U.S.A., Jul 29  Aug 16 1985 .
[32] L. AlvarezGaume , J.B. Bost , G.W. Moore , P.C. Nelson and C. Vafa , Bosonization on Higher Genus Riemann Surfaces, Commun. Math. Phys. 112 ( 1987 ) 503 [INSPIRE].
[33] E.P. Verlinde and H.L. Verlinde , Chiral Bosonization , Determinants and the String Partition [34] M.J. Dugan and H. Sonoda , Functional Determinants on Riemann Surfaces , Nucl. Phys . B [35] H. Sonoda , Conformal Field Theories With First Order Lagrangians, Phys. Lett . B 197 [36] J.J. Atick and A. Sen , Spin Field Correlators on an Arbitrary Genus Riemann Surface and Nonrenormalization Theorems in String Theories , Phys. Lett . B 186 ( 1987 ) 339 [INSPIRE].
[37] H. Sonoda , Functional Determinants on Punctured Riemann Surfaces and Their Application to String Theory, Nucl . Phys . B 294 ( 1987 ) 157 [INSPIRE].
[38] E. D'Hoker and D.H. Phong , Functional Determinants on Mandelstam Diagrams , Commun.