Construction of action for heterotic string field theory including the Ramond sector
Received: July
Construction of action for heterotic string eld theory including the Ramond sector
Keiyu Goto 0 1 3
Hiroshi Kunitomo 0 1 2
Open Access, c The Authors.
0 Kitashirakawa Oiwakecho , Sakyoku, Kyoto 6068502 , Japan
1 Komaba , Meguroku, Tokyo 1538902 , Japan
2 Center for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto University
3 Institute of Physics, The University of Tokyo
Extending the formulation for open superstring arXiv:1508.00366, we attempt to construct a complete action for heterotic string theory. The action is nonpolynomial in the Ramond string . Using a dual formulation in which the role of changed, the action is explicitly obtained at the quadratic and quartic order in gauge transformations.
including; the; String Field Theory; Superstrings and Heterotic Strings

Complete action for open superstring
eld theory
The NS sector of heterotic string
eld theory
Contents
1 Introduction 2 3 3.1
Basic ingredients
WZWlike action
Dual formulation
4 Inclusion of the Ramond sector Ramond string eld and restricted Hilbert space Perturbative construction 4.3
Fermion expansion
Cubic interaction in SR
Quartic interaction in SR
Quadratic in fermion
Quartic in fermion
Summary and discussion
A Construction of the dual gauge product
B Fourpoint amplitudes with external fermions
B.1 Propagators and vertices
B.2 Fourfermion amplitude
B.3 Twofermiontwoboson amplitude
Introduction
the A1=L
theory are interrelated by a partial gauge
xing [8]. In spite of this success, it had been
for a long time.
of the heterotic string eld theory.
introducing R string
from the constructed action.
Complete action for open superstring
eld theory
eld theory.
Sen [12, 13].
2See also [20, 21].
action for the NS sector. The original expression given in [3] is
S =
dt hA~t(t); A~Q(t)i;
is the zero mode of (z) and A~t and A~Q are the leftinvariant forms
A~t(t) = g 1(t)@tg(t);
A~Q(t) = g 1(t)Qg(t);
and its oneparameter extension
(t) are related
(0) = 0.
the role of
and Q is exchanged:
S =
dt hAt(t); QA (t)i;
They satisfy the relations
implying the operator XY is a projector:
XY X = X;
Y XY = Y;
(XY )2 = XY:
+ ( 0 + c0G) :
The former constraint imposes that
is in the small Hilbert space, and the latter restricts
the form of
expanded in the ghost zeromodes as
where At(t) and A (t) are the rightinvariant forms
At(t) = (@tg(t))g 1(t);
A (t) = ( g(t))g 1(t):
which the A
eld theory in which two operators
and Q do not appear symmetrically but act
di erently on the closed string products.
string eld
by the conditions3
= 0;
space at picture number
3=2 and
X =
( 0) G0 + 0( 0) b0;
Y =
string eld theory [23{25].
3In this paper we use the same symbol
to denote the string eld in the Ramond sector both for
the open superstring and for the heterotic string
eld. We will not confuse them since two cases never
appear simultaneously.
X = fQ; ( 0)g;
(x) is the Heaviside step function satisfying
introduce the following operator
which is more suitable for use in the large Hilbert
space [10]:
= 0 + ( ( 0) 0
0)P 3=2 + ( 0
0)P 1=2 ;
Hilbert space at picture number
operator F (t) as
F (t) =
1 + (D (t)
= 1 + X(
(D (t)
D (t)A
A (t)A + ( 1)AAA (t);
D (t)F (t) = F (t) ;
and thus the dressed Ramond string eld F (t)
with the Ramond string eld
by the constraints (2.5) is annihilated by D (t).
Now a complete gauge invariant action is given by
S =
dthAt(t); QA (t) + (F (t) )2i;
invariant under the gauge transformations [9]:
A = Q
+ fF ; F (fF ; g
= Q
+ X F
are gauge parameters in the NS sector and
is a gauge parameter in the
Ramond sector satisfying
= 0;
= 0:
The NS sector of heterotic string
eld theory
to the nilpotency 2 = 0.
(z) = @ (z)e
(z) = e (z) (z):
b0Vi = 0;
L0 Vi = 0;
(i = 1; 2);
hV1; V2i = hV1jc0 jV2i;
It satis es
in [21]
hQV1; V2i = ( 1)V1 hV1; QV2i;
h V1; V2i = ( 1)V1 hV1; V2i:
The nstring product carries ghost number
2n + 3 (and picture number 0). The string
and cyclic with respect to the inner product:
De ning [V ]
the L1relations:
0 =
[ [V (1); : : : ; V (m)]; V (m+1); : : : ; V (n)] :
The operator
acts as a derivation on the string products:
V1; : : : ; Vn =
It is useful to introduce new string products [
[Bm; V1;
If B satis es the MaurerCartan equation
QB + X
[Bn] = 0;
0 =
[ [V (1); : : : ; V (m)]B; V (m+1); : : : ; V (n)]B :
the shifted string products (3.14) satisfy the identical L
1 relation to (3.9):
charge, (QB)2 = 0, de ned by
[V ]B = QV +
[Bm; V ] :
WZWlike action
also satis es the closed string constraints
b0 Ve = 0 ;
L0 Ve = 0 :
operators @t and
as well as
act as derivations on the string products:
X[Ve1(t); : : : ; Ven(t)] =
X( 1)X(1+Ve1+ +Vek 1)[Ve1(t); : : : ; XVek(t); : : : ; Ven(t)];
QG(Ve ) + X
[G(Ve )n] = 0 :
X(Ve ), which we call
an associated eld, satisfying
QG(XG) = 0 :
@ G( Ve ) =
[G( Ve )m; Ve ] = QG( Ve )Ve ;
= 0, and set
= 1.
on (3.20), we have
We denote
t(Ve ) for
@t (Ve ) for simplicity. The associated eld
(Ve ) is
Grassmanneven and carries ghost number 2 and picture number
1. The associated elds and
X( Ve ) = XVe +
Ve ; X( Ve ) G( Ve );
with the initial condition,
X = 0 at
= 0, and set
= 1.
at order 2 [22].
Utilizing these functionals G and
X, a gaugeinvariant action can be written in the
WZWlike form:
SWZW =
dth t(t); G(t)i;
t(t); G(t) = @t
(t); G(t) ;
SWZW =
(Ve ); G(Ve )i;
G(Ve ) = 0;
= QG e +
X(Ve (t)) and G(t)
G(Ve (t)). One can show that the variation of the
integrand becomes a total derivative in t
and thus the variation of the action is given by
since Ve (0) = 0, and
is given by
Dual formulation
the role of
appendix A.
In the dual formulation, an L1structure starting with
plays a central role. Note
that, in the case of the open string, a set of products f ;
g satisfy the A1relations:
is nilpotent,
As a natural extension of f ;
g, we introduce a set of products satisfying L1relations,
which we call the dual sting products :
eld, and cyclic:
hV1; [V2;
5Note that
is invertible as a function of Ve . See also [3] and [28].
They satisfy the L
and picture number n
on the dual string products:
( 1) (fV g) [V (1); : : : ; V (k)] ; V (k+1); : : : ; V (n)
= 0;
Q V1; : : : ; Vn
+ X( 1)V1+ +Vk 1 V1; : : : ; QVk; : : : ; Vn
= 0:
In the dual formulation, we denote the NS string
eld as V , which is a Grassmannodd
closed string constraint:
b0 V = 0 ;
L0 V = 0 :
with ghost number 2 and picture number
1 satisfying the MaurerCartan equation dual
to (3.20):
acts as a derivation on the dual string
pro
X[V1; : : : ; Vn] =
X( 1)X(1+V1+ +Vk 1)[V1; : : : ; XVk; : : : ; Vn] :
[V1; V2;
[(G )m; V1; V2;
0 =
@ G ( V ) =
products as
(D )2V1 = 0;
= V +
[(G )m; V ]
1 relation:
The shifted dual string products satisfy the L
dual shifted two string products:
= 0 :
shifted operator D as the shifted onestring product [ ]G :
iteratively with BX = 0 at
= 0, and then set
= 0 ;
( 1)XYYX
D [V1; V2]G
[D V1; V2]G
( 1)V1 [V1; D V2]G :
The operator X = Q, @t, or
acts on the shifted dual products as
In particular,
Acting X on the MaurerCartan equation (3.37) we have
X[V1; : : : ; Vn]G
+ ( 1)X [XG ; V1; : : : ; Vn]G :
D XG (V ) = 0 :
XG (V ) = ( 1)XD BX(V );
carries ghost number 2 and picture number 0, and Bt (
B@t ) and B
are
Grassmanndi erential equations
@ BX( V ) = XV +
with Bt(t)
lated as
motion is given by
Bt(V (t)) and G (t)
G (V (t)). The variation of the action can be
calcuand the action is invariant under the gauge transformation
from the de nition (3.40).
using these functionals G (V ) and Bt(V ) by
S =
dt hBt(t); QG (t)i ;
S = hB (V ); QG (V )i ;
QG (V ) = 0 ;
B = Q
The gauge parameters
having ghost number 0 carry picture number 0 and 1,
2 = Q2 = 0, and
QG =
D BQ.
dual nstring products for n
3 themselves written as a BRST variation of some products
) which we call the dual gauge products:
; Vn] = Q(V1;
X( 1)V1+ +Vk 1 (V1;
; QVk;
and is commutative and cyclic:
hV1; (V2;
; V (n)) = ( 1) (fV g)(V1;
act as a derivation also on this product,
fV1; : : : ; Vng to
; Vn) =
X( 1)(V1+ +Vi 1)(V1;
It is useful again to de ne the shifted dual gauge products (
; Vn)G = X
((G )m; V1;
hV1; (V2;
; Vn)G ; Vn+1i:
; Vn]G = X
[(G )m;
= X
= Q(V1;
Q((G )m; V1;
m((G )
X( 1)V1+:::+Vk 1((G )m; V1;
X( 1)V1+ +Vk 1(V1;
; QVk;
; QVk;
(QG ; V1;
gauge product but satis es the relation
; Vn)G = X(V1;
; XVk;
; Vn)G + (XG ; V1;
Inclusion of the Ramond sector
Ramond string eld and restricted Hilbert space
constrained in the restricted Hilbert space,
= 0;
= ;
{ 12 {
which make the operator XY a projector:
XY X = X;
Y XY = Y ;
(XY )2 = XY:
1 =
XY Q 1 = XY QXY
1 = XY XQY
1 = XQY
1 = QXY
1 = Q 1
eld has the form
where G = G0 + 2b0 0.
= b0
= 0
= 0;
= b0
= 0
= 0;
picture number
The picture changing operators X and Y are de ned by
= L
= 0:
X =
( 0)G0 + 0( 0)b0;
Y =
3=2 and
action for the Ramond sector to be
which is invariant under the gauge transformation
hhA; Bii = hhAjc0 jBii:
S0 =
= Q :
{ 13 {
the picture number p.
The gauge parameter
also satis es the same constraints as :
= L
= 0;
= :
X is BRST trivial in the large Hilbert space [26]:
X = fQ; ( 0)g;
with the Heaviside step function
(x). More general operator
suitable in the large
Hilbert space is de ned by [10]
)P 3=2 + (
)P 1=2 ;
Then we generalize the operator X to the one given by
X = fQ; g ;
Hilbert space:
hhXV1; V2ii = hhV1; XV2ii:
Perturbative construction
can be expanded in powers of fermion:
For the NS sector, SNS
remaining part, SR
S(0), we adopt the dual WZWlike action de ned in (3.51). The
S =
{ 14 {
and interaction terms between two sectors.
We can further expand the action in the
coupling constant :
SNS = S(0) + S(0) + 2S(0) + O
0 1 2
The gauge transformations can also be expanded in
V =
V =
V (0) = Q ;
V (0) =
(1) = 0;
(1) = 0;
V =
V (2) = 0;
(1) = Q ;
is a gauge parameter in the NS sector,
is another gauge parameter in the NS sector, and
S(0) =
S(2) =
order by order in
by requiring the gauge invariance.
action (3.51):
included. Starting from the kinetic terms
S(0) =
S(0) =
4! hV; Q[V; ( V )2] i +
4! hV; Q[V; [V; V ] ] i:
{ 15 {
V (0) =
V (0) = [V;
[V; Q ] ;
V (0) =
V (0) =
[V; V; Q ] +
[V; [V; Q ]];
[V; V;
which keep the action (4.33) invariant at each order in :
( )(0)S(0) + ( )(0)S(0) = 0;
( )(00)S(0) + ( )(10)S(0) = 0;
1 0
candidate of cubic interaction term given by
S(2) =
(11)(= 1(0) ) requiring
the gauge invariances in this order
(# of R elds before transformation).
Cubic interaction in SR
Under the gauge transformation
(01) in (4.30), the variation of S1(2) is given by
(1) = 0 :
{ 16 {
( )(00)S(2) = 2 1hQ ; [V; ] i
1
= 2 1h ; [QV; ] i
= 2 1h
= 2 1h[ ;
; [Q V; ] i + 2 1h
] ; Q V i + 2 1h[ V;
0(0)S(2) + 1(0)S(2) + 1(2)S(0) = 0 :
1 0 0
gauge transformation
V (0) in (4.24) is calculated as
1hQ ; [ 2] i =
2 1h ; [ ; Q ] i =
This can be cancelled by ( )(10)S0(2) if we take
(1) =
V (0) in (4.27) is given by
; [ 2] i =
] i = 0;
because of
= 0, and so we have
where we used the fact that a relation,
h ; Bi = h
; Bi = h
holds for general string eld B since the parameter
is in the small Hilbert space. This
variation (4.44) can be canceled by ( )(12)S(0) + ( )(10)S(2) with
0 0
V (2) =
Quartic interaction in SR
(11), which is calculated as
4 12hX [ ; ] ; [V; ] i
4 12h[ ; ] ; X[ V; ] i :
h [ ; ] ; [Q V; ] i
h[ [ V; ] ; ] ; Q i
h[ ; [ ; ] ] ; Q V i
h [ ; ] ; [ V; Q ] i
hQ ; [ ; [ V; ] ] i
h[ V; [ ; ] ] ; Q i
Then we nd
terms with two Ramond strings as
S(2) =
] i + 2h Q ; [V; 2
= 2 2hQ ; [ V; 2
+ 2 2hQ ; [ ; [V; ] ] i
4 2h[ V; ; ] ; Q i
The variation of the second term in S(2)
can similarly be calculated as
3hQ ; [ ; [ V; ] ] i +
3h ; [V; [ Q ; ] ] i;
3hQ ; [ ; [V; ] ] i + 2 3hQ ; [ ; [ V; ] ] i: (4.52)
Therefore, in total, we have
2hQ ; [V; [ 2] ] i + (2 2 +
3)hQ ; [ ; [V; ] ] i
+ 2 3hQ ; [ ; [ V; ] ] i
From (4.49) and (4.53), we nd that the constants 1
3 should be chosen to be
1 =
2 =
3 =
Then we have
i + h[ V; [ ; ] ] ; Q
+ h[ [ V; ] ; ] ; Q
These terms can be cancelled by ( )
and ( )
if we choose
(2) =
(1) =
X [ V; ; ] + X [ V; [ ; ] ] + X [ [ V; ] ; ] :
Note that
= ( )
= ( )
transformation with the parameter
in this order holds:
. Thus the gauge invariance under
at this order
can easily be calculated as
] i =
; [V; [ 2] ] i =
and hence
The correction at this order is not necessary:
(2) = 0 ;
(1) = 0 :
h[QV; V ] ; ( )(12)V i =
] i = h
respectively, where we used a relation
The variation ( )(00)S2(2) is given by
Substituting the relation Q
= Q
, this can further be calculated as
hQ ; [V; [ V; ] ] i
hQ ; [ V; [V; ] ] i : (4.66)
hA; X Bi = h
A; X Bi = hh A; X Bii
= hhX A; Bii = h X A; Bi
= ( 1)AhX A; Bi :
= h[ V; ;
In total we have
( )(12)S1(0)+( )(10)S(2) + ( )(00)S(2)
1 2
V (2) =
(1) =
X [[V; V ] ;
holds at quadratic
order in both coupling constant and the Ramond elds:
( )(12)S(0) + ( )(10)S(2) + ( )(00)S(2) + ( )(22)S(0) + ( )(20)S(2) = 0 :
1 1 2 0 0
{ 19 {
V (2) = 0 ;
V (2) =
V (2) =
(1) = Q ;
(1) = X [ V;
(1) =
(3) =
Fermion expansion
V (0) =
V (0) =
V (0) =
(1) = 0 ;
(1) = 0 ;
(1) = 0 :
[V; V;
The gauge transformation with the gauge parameter
in the Ramond sector is given by
V =
] ; (4.110)
X [[V; V ] ;
The equations of motion are therefore given by
S(2n) =
E(0) + E(2) + E(4) +
E(1) + E(3) + E(5) +
= 0 ;
= 0 ;
for the NS and the Ramond string
elds, respectively. We can also expand the gauge
transformation in powers of the Ramond string eld as
B = B(0) + B(2) + B(4) +
requiring the gaugeinvariance at each order:
0 =
(2n 2k+1); Y E(2k 1)
0 = hB(0); E(0)i:
S(2) =
; Y (E(1))ii + B ; E(2)
; Y (Q
eld under the following gauge transformations at this order
B(2) =
(1) =
F 1(t) = 1 + (D (t)
) =
D (t) :
(1) so that the
0 =
(1); Y E(1)
ii + hB(2); E(0)i + hB(0); E(2)i:
by the same form of that for the open superstring (2.15):
S(2) =
where F (t) is the linear operator de ned by
F (t) =
1 + (D (t)
= 1 + X
(D (t)
We should note that this has the same form as (2.12) but D
de ned in (3.40) contains
variation of the action S(2) becomes
as summarized in the previous section.
Quadratic in fermion
Multiplying it by
from the left or by D from the right, we have
F 1(t) = F 1(t)D (t) =
D (t) ;
= D (t)F (t) ;
fD (t); F (t) g = 1 :
We further have
left and right and using
It is also shown
on the dual string products is given by
[X; F (t)]V1 =
F (t)[X; F 1(t)]F (t)V1
F (t)(X
X)(D (t)
)F (t)V1
F (t) [XG (t); F (t)V1]G (t) :
We also summarize the properties of F (t)
for later use. Since F (t)
= D (t)F (t) and
= 0, F (t)
is D (t)exact:
D (t)F (t)
+ F (t) D (t) = F (t)
+ F (t) D (t)
= F (t) 1 +
(D (t)
) = 1 :
= F (t)f ; g
= D (t)F (t)
Acting with QF (t) on
, (4.133) leads to
= F (t) Q
+ X F (t)
F (t) [QG (t); F (t) ]
= D (t)F (Q
+ X F (t) )
F (t) [QG (t); F (t) ]
, XF (t)
can be transformed into the
following form:
= F (t)X
= F (t)X
+ ( 1)X F (t) D (t)[BX(t); F (t) ]
+ ( 1)X [BX(t); F (t) ]
D (t)F (t) [BX(t); F (t) ]
Now let us consider the variation of S(2):
S(2) =
dt hBt(t); [F (t) ; F (t) ]
term, and obtain
For the second term, utilizing (4.137), we nd
Then the total variation is given by
= @t
; Y X F
= 0, we eventually nd
the integrand of the interaction term,
can be calculated as follows. Since [F ; F ]
G is D exact, we can use (3.49) for the rst
Bt; [ G ; F ; F ]
B ; [@tG ; F ; F ]
where we assumed that
@tB + [B ; Bt]G ; [F ; F ]
B ; [Bt; [F ; F ]G ]G :
[Bt; F ]G ; [B ; F ]
2 [Bt; F ]G ; [B ; F ]
2 B ; [F ; [Bt; F ]G ]G
D F [B ; F ]
Bt; [ G ; F ; F ]
Bt; [D B ; F ; F ]
D [B ; F ; F ]
G + [B ; [F ; F ]G ]G
2[F ; [B ; F ]G ]G
B ; [@tG ; F ; F ]
B ; [Bt; [F ; F ]G ]G
+ 2 B ; [F ; [Bt; F ]G ]G
S(2) =
; Y (Q
{ 25 {
E(1) = Q
+ X F
E(2) =
By requiring (4.122), let us determine
(1) and B(2) for each of gauge
transformations with the parameters ,
and . Let us rst consider the invariance under the
transformation with the parameter :
0 =
= D F E(1)
it can be calculated as
B(0); E(2) =
= h
B(2) =
(1) =
G + 2[F ; F [F ; ]G ]G
D [F ; ]G ; E(1)i :
X F
2[F ; F [F ; ]G ]G ;
The invariance under the transformation with the parameter
Since the second term is again known and calculated as
0 =
hB(0); E(2)i = hD
; E(2)i = h ; D E(2)i = 0 ;
we conclude that
B(2) = 0;
(1) = 0 :
Finally, for the invariance under the transformation with :
where we decomposed
0 =
(1); Y E(1)ii + B(2); E(0)
(1); Y E(1)ii + B(2); E(0) ;
(1) into the free part (4.30) and remaining:
(1) =
ii =
hhQ ; Y (Q
+ X F )ii =
= h[F ; F
The invariance (4.154) holds if we take
B(2) =
(1) =
X F D = X F : Thus, in total, the gauge transformation at this order becomes (4.127).
which has the same form as that of the open superstring
eld theory, and thus is its
0 =
(1); Y E(3)
(3); Y E(1)
+ B(0); E(4) + B(2); E(2) + B(4); E(0) ;
in which, in particular, we nd
hB(2); E(2)i 6= 0 :
(3) so that the equation (4.158) is satis ed.
we have
= Q(F ; F ; F )
3(F ; F ; QF )
hB(2); E(2)i =
; D [F ; F ; F ]G i
F ; [F ; F ; F ]G i
(QG ; F ; F ; F )
{ 27 {
and thus
6 hQF ; (F ; F ; F )
2 h(F ; F ; F )G ; QF i
6 h(F ; F ; F ; F )G ; QG i
6 hQF ; (F ; F ; F )
G i =
+ X F ); Y X F
D (F ; F ; F )
hB(2); E(2)i =
F ; Q(F ; F ; F )
F ; (F ; F ; QF )
F ; (QG ; F ; F ; F )
D (F ; F ; F )G ; Y (Q
under an arbitrary variation of
, where we used
satis es the constraint (4.1) and
therefore D F
= F
can integrate it, and obtain
= F
. Since the shifted dual gauge products are cyclic, we
S(4) =
F ; (F ; F ; F )
variation of the NS string eld, we have
S(4) =
F [ G ; F ]G ; (F ; F ; F )
B ; D [F ; F (F ; F ; F )G ]G
F ; ( G ; F ; F ; F )
B ; D (F ; F ; F ; F )
respectively, and we eventually have
hB(2); E(2)i =
D (F ; F ; F )
From this form of E(3), the action S(4) has to satisfy
; Y X F
D (F ; F ; F )
E(3) =
B(4) =
(3) =
X F D (F ; F ; F )
[F ; F (F ; F ; F )G ]G ;
X F
D (F ; F ; F )
D (F ; F ; F )G ; Y E(1)
(F ; F ; F ; F )
[F ; F (F ; F ; F )G ]G
[F (F ; F ; F )G ; F ]
(F ; F ; F ; F )
[F (F ; F ; F )G ; F ]
S(4) =
= D B .
Thus we obtain
E(4) =
{ 28 {
D (F ; F ; F ; F )
D [F ; F (F ; F ; F )G ]G :
Let us consider the invariance under the parameter
rst. The action is invariant if we
can determine B(4) and
(3) so that they satisfy
0 =
(3); Y (Q
; E(4) + B(4); QG :
However, since the second term vanishes,
we can consistently take
; E(4)i = h ; D E(4)i = 0 ;
B(4) = 0 ;
(3) = 0 :
that one can determine
(3) and B(4) so that the condition (4.120) at quartic order,
0 =
(1); Y E(3)
(3); Y E(1)
+ B(0); E(4) + B(2); E(2)
+ X F
and E(0) = QG , which
can be compensated by appropriately determining
(3) and B(4), respectively:
0 = B(0); E(4) + B(2); E(2)
useful to note that
= 0 ;
fQ; D g = 0 ;
Q(F ; : : : ; F )
= [F ; : : : ; F ]G :
Utilizing them, we have
Q[B1; : : : ; Bn]G
(1); Y E(3)
ii =
D (F ; F ; F )
[F ; ]G ; D F QF
D (F ; F ; F )
[F ; ]G ; D F [F ; F ; F ]
[F ; ]G ; QD F (F ; F ; F )
; [F ; [F ; F ; F ]G ]G
F [F ; ]G ; D [F ; F ; F ]
; [F ; QD F (F ; F ; F )G ]G
{ 29 {
where we used D F QD F
F D )QD F
one can show that the remaining two terms become
hB(0); E(4)i =
hB(2); E(2)i =
; [F ; QD F (F ; F ; F )G ]G
; D [F ; F ; F ; F ]
; [F ; F ; [F ; F ]G ]G
F [F ; ]G ; [F ; [F ; F ]G ]G
F [F ; ]G ; D [F ; F ; F ]G +3[F ; [F ; F ]G ]G
= 0 :
By picking up the terms with E(1) and E(0), the transformations
(3) and B(4) can be
explicitly determined as
B(4) =
[(F ; F ; F ; F )G ; ]
[[F ; F (F ; F ; F )G ]G ; ]
(F ; F ; F ; F ; D
(F ; F ; F ; F [F ; D
[F ; F (F ; F ; F ; D
[F (F ; F ; F )G ; F ; D
[F ; F (F ; F ; F [F ; D
[F ; F [F (F ; F ; F )G ; D
[F (F ; F ; F )G ; F [F ; D
X F D [F (F ; F ; F )G ; D
(3) =
X F D (F ; F ; F ; D
X F D (F [F ; D
Summary and discussion
Using the expansion in the number of the Ramond string
eld, we have constructed in
quartic order:
S =
dt Bt(t); QG (t) + [F (t) ; F (t) ]G (t)
F ; (F ; F ; F )G i + O( 6):
[(F ; F ; F ; F )G ; ]
[[F ; F (F ; F ; F )G ]G ; ]
(F ; F ; F ; F ; D
(F ; F ; F ; F [F ; D
[F ; F (F ; F ; F ; D
[F (F ; F ; F )G ; F ; D
[F ; F (F ; F ; F [F ; D
[F ; F [F (F ; F ; F )G ; D
[F (F ; F ; F )G ; F [F ; D
X F D (F ; F ; F ; D )
X F
D (F ; F ; F [F ; D
X F
D [F (F ; F ; F )G ; D
X F D [F ; ]
with the parameter ,
and with the parameter ,
= D
= O( 5);
= Q
2[F ; F [F ; ]G ]G
= Q
+ X F
X F D (F ; F ; F )
from this action are
ENS = QG +
(F ; F ; F ; F )
4[F ; F (F ; F ; F )G ]G
+ O( 6); (5.8)
ER = Q
+ X F
X F D (F ; F ; F )
at each
gauge transformation with the parameter
does not subject to change any more. One can
nd that the gauge transformations
are obtained by replacing elds
in the equations of motion with gauge parameters:7
E(2k+1) =
E(2k) = D B(2k);
E(2k+1) =
for k = 0; 1; 2 ;
for k = 0; 1 ;
for k = 1; 2 ;
for k = 0; 1 :
to complete an action to all orders.
form of the rstorder equations of motion obtained in [15]:
where Be = P1
( + Q)Be +
m! [Bem] = 0;
[Bem1] = 0 ;
D Be 1=2 = 0 ;
Be 1 = G ;
Be 1=2 = F
can be solved as
The next two with p =
QG +
QBe 1=2 + [Be0; Be 1=2]G +
{ 32 {
0). In the original formulation
simply become
which can be solved as
QBen=2 +
Ben(n=+24) =
= 0 ;
QG +
QF = 0 ; = 0 ; (5.20)
Be 1=2 = F
D F (F ; F ; F )
rede nition of the Ramond string eld
, we have to nd a way to reproduce the higher
for constructing a complete gauge invariant action.
Acknowledgments
Construction of the dual gauge product
a multilinear map dn : H^n
! H, where ^ is the symmetrized tensor product satisfying
{ 33 {
algebra S(H) = H^0
to S(H) itself called a coderivation. A coderivation
N ) = (dn ^ IN n)( 1 ^
(n+1) ^
(N) ; (A.1)
S(H), and it vanishes when acting on H^N<n. The graded
map [[bn; cm]] : H^n+m 1
! H which is de ned by
[[bn; cm]] = bn(cm ^ In 1)
( 1)deg(bn)deg(cm)cm(bn ^ Im 1) :
Then the L1relation can be written as
[[L; L]] = 0 ;
where L = L1 + L2 + L3 +
and Lk is a coderivation derived from the kstring product.
de ne a cohomomorphism bf : S(H) ! S(H0), which acts on
S(H) as
n) =
i n k1< <ki=n
^ fki ki 1 ( ki 1+1; : : : ; n) :
space, S(H) ! H, as
3 ^ 4 ^ 5 + : : : =
pLp+1, had
ghost number 1
by a similarity transformation of the BRST operator Q as
LNS( ) = Gb 1( )Q Gb( ) ;
Gb( ) = P exp
where [0]( ) = P1
p [p0+] 2, called gauge products, can be determined iteratively. The
(p + 2)gauge product [p0+] 2 carries ghost number
cohomomorphisms Gb( ) and Gb 1( ) satisfy
The L1relations are followed from the nilpotency of Q as
[[LNS( ); LNS( )]] = 2 LNS( )
@ Gb( ) = Gb( ) [0]( ) ;
@ Gb 1( ) =
= 2 Gb 1( )Q2 Gb( ) = 0 :
initial gauge product [0], whose explicit example is given in [7].
pLp+1 which provides a set of
the dual string products by
This nth dual product Ln carries ghost number 3
2n and picture number n
By construction, they satisfy the L1relation
L ( ) = Gb( ) Gb 1( ) :
[[L ( ); L ( )]] = 0 :
for example,
[V1; V2] = [V1; V2];
[V1; V2; V3] =
X0[V1; V2; V3]+[X0V1; V2; V3]+[V1; X0V2; V3]+[V1; V2; X0V3]
+ ( 1)V1 0[V1; [V2; V3]] ( 1)V1 [ 0V1; [V2; V3]]
+ [V1; [ 0V2; V3]]+( 1)V2 [V1; [V2; 0V3]]
considering Vi is either the NS string
eld or the Ramond string
eld, which preserves
the dual string products.
respect to
can be written as the commutator of Q and a product :
@2L ( ) = [[Q; ( )]] =
The dual gauge products can be read from
(V1; V2; : : : ; Vn) =
is related to the gauge products [0]( ) as
[[ ; [0]( )]] = L[1]( ) ;
and satis es L[1](
= 0) = LB, where L2B is a coderivation derived from the simple
two2
introduce a coderivation
[1]( ) = P1
n+3 derived from a set of intermediate gauge
products with de cit picture 1 [7]. It satis es the relation,
we can rewrite L as
The integrand in the second term becomes
{ 36 {
Then eventually the dual string products L can be written as
L ( ) =
00)[[Q; Gb( 00) [1]( 00) Gb 1( 00)]] :
00 d 0 and carried out 0integral. In this expression,
we obtain
@ L ( ) =
@2L ( ) =
d 00[[Q; Gb( 00) [1]( 00) Gb 1( 00)]] ;
[[Q; Gb( ) [1]( ) Gb 1( )]] :
Therefore we can de ne the product
by two gauge products [0] and [1] as
( ) =
G( ) [1]( )G 1( ) :
The cyclicity of
obtain the following expressions for the rst few orders:
3 =
4 =
5 =
Fourpoint amplitudes with external fermions
in this paper. We follow the notations and conventions in [29].
{ 37 {
S0 =
V = Q
= Q ;
b0+V = 0V = 0 ;
= 0 :
d2T ( 0b0 b0+) e T L0+ i L0
d2T (b0 b0+X ) e T L0+ i L0
R =
R =
= 0 ;
R( 0Y X ) =
are obtained
from (3.51) and (4.123) as
j ih j =
XY
This is invariant under the gauge transformations
which we x here by gauge conditions
be found as
expanding them in the coupling constant :
S(0) =
S(0) =
S(2) =
S(2) =
S(4) =
amplitudes including external fermions.
{ 38 {
threestring interaction (B.10) as
(ABjCD) = (
= 2
= 2
+
d2Tsh A B( 0b0 b0 ) C DiWs
+
d2Tshh A B(b0 b0 ) C DiiWs ;
coordinate on the NS propagator.9 The symbols A;
; D represent the wave functions
and the onshell condition Q
if necessary.
(ACjBD) = 2
(ADjBC) = 2
d2Tthh A C (b0 b0+) B DiiWt ;
+
d2Tuhh A D(b0 b0 ) B C iiWu :
which lls the gap in the moduli integration [18]:
(ABCD) = 2
d 1d 2hh(bC1bC2) A B C DiiW4 ;
ghost insertions, the details of which are given in [20].
AF 4 = AF 4
9These are denoted c and bc in [14].
rst quantized formulation.
Twofermiontwoboson amplitude
vertices (B.8) and (B.10):
AF 2B2
(ABjCD) =
d2Ts h A B( 0b0 b0+) (QVC ) ( VD) + ( VC ) (QVD) iWs
where we denoted H d
tor (B.5) and the vertex (B.10):
AF 2B2
(ACjBD) =
d2Tt h AVC (b0 b0+X ) BVDiWt :
as to be the correlation function of the same external vertices, A,
B, ((X0 VC ) ( VD) +
AF 2B2
(ACjBD) =
d2Tt hh A (X0 VC ) (b0 b0+) B ( VD)iiWt
+
+ hh A ( VC ) (b0 b0 ) B (X0 VD)iiWt
tices (B.11):
AF 2B2
(ABCD) =
+
d2Tu hh A (X0 VD)(b0 b0 ) B ( VC )iiWu
+
+ hh A ( VD)(b0 b0 ) B (X0 VC )iiWu
d 1d 2h( 0bC1bC2) A B (QVC ) ( VD) + ( VC ) (QVD) iW4
B VDiWth b0
A VDiWt Tt=0
A VC iWu Tu=0
d 1d 2hh(bC1bC2) A B (X0 VC ) ( VD) + ( VC ) (X0 VD) iiW4
AF 2B2
(ADjBC) =
d2Tu h AVD(b0 b0+X ) BVC iWu
Summing up all these contributions, one can
AF 2B2 = AF 2B2
+ hh A ( VC ) (b0 b0 ) B (X0 VD)iiWt
+
d2Tu hh A (X0 VD)(b0 b0 ) B ( VC )iiWu
+ hh A ( VD)(b0 b0 ) B (X0 VC )iiWu
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