Open superstring field theory on the restricted Hilbert space
HJE
eld theory on the restricted Hilbert space
Sebastian Konopka 0 1
Ivo Sachs 0 1
0 Theresienstra e 37 , D80333 Munchen , Germany
1 Arnold Sommerfeld Center for Theoretical Physics , LudwigMaximilians Universitat
It appears that the formulation of an action for the Ramond sector of open superstring eld theory requires to either restrict the Hilbert space for the Ramond sector or to introduce auxiliary elds with picture 3=2. The purpose of this note is to clarify the relation of the restricted Hilbert space with other approaches and to formulate open superstring eld theory entirely in the small Hilbert space.
String Field Theory; Superstrings and Heterotic Strings; BRST Quantization

Open superstring
1 Introduction 2 3
1
eld theory in the restricted Hilbert space
Introduction
can be remedied by smearing out the picture changing operator [2] (see also [3] for earlier
kinetic term in the Ramond (R) sector was not addressed in [10]. On the other hand,
in [11] and [12] the degeneracy of the Ramond kinetic term was avoided with the help of a
suitable restriction of the Ramond Hilbert space. Indeed, it was noticed [13] in the early
days of string
eld theory that Witten's theory propagates only a subset of constrained
string
elds [14]{[19]. This was subsequently related to the presence of an extra gauge
symmetry (not generated by the BRST charge) that can be xed to remove all elds that
do not satisfy the constraint [20] (see also [21]).
A gauge invariant action for the interacting theory was recently proposed in [12] (see
also [22]) with smeared picture changing operators and Ramond
elds in the restricted
Hilbert space. The above problem with cyclicity of the vertices was avoided by taking
{ 1 {
the the NS eld to live in the large Hilbert space akin to the Berkovits formulation. On
another front, in [11] a geometric approach, based on the decomposition of the supermoduli
space was outlined, which is formulated in the small Hilbert space with a constrained
Ramond sector. Furthermore, in [23] another geometric construction was proposed where
the restriction on the Ramond elds is substituted by the introduction of auxiliary elds.1
The purpose of this note is twofold. First we clarify the relation between the
restricted and unrestricted Ramond Hilbert spaces. In particular, we show explicitly that
the restrictions used in [12] and [11] are the same and furthermore that the cohomology
of the restricted Hilbert space is the same as that of the unrestricted space. The latter
result was previously obtained in [24].2 In the second part we propose a modi cation of
the construction [10] for the RNS vertices which is cyclic in the small, restricted Hilbert
space. Provided the picture changing operators used in [11, 12] can be de ned in a way
that is compatible with the interaction vertices, our construction immediately gives a
classical action for the open superstring in the small, restricted Hilbert space. More generally,
the vertices can be regarded as an algebraic construction of the interaction vertices of the
auxiliary eld construction of [23]. Then, invoking the results of [9] one concludes that the
resulting action reproduces the correct treelevel Smatrix.
2
Restricted Hilbert space
Let us start with the restricted Ramond Hilbert space spanned by vectors of the
form [12]{[21]
= 1j #i +
0 2j #i
( 1)j 1jG0 2j "i ;
where j #i = b0j "i, j j denotes the Grassman parity of , 0 is the zero mode of the
commuting superconformal ghost and G0 the (matter plus ghost) supercharge with the
0b0 contribution subtracted. More concretely, we decompose the BRST charge Q as
Q = c0L0 + b0M + 0G0 + 0K
0 b0 + Q~
2
where L0; M; G0; K; Q~ have no dependence on the ghost zero modes (see e.g. [21] for details).
Then, using that fQ~; G0g = 0 and G20 = L0 it is not hard to see that
Q
=
M (G0 2) + K( 2) + Q~( 1) j #i +
0 G0( 1) + Q~( 2) j #i
+ ( 1)j 1jG0 G0( 1) + Q~( 2) j "i :
According to [21], 2 can be gauged away completely.3 The closedness condition reduces to
with a residual gauge freedom
Q~ 1 = G0 1 = 0 ;
1 = Q~ ;
G0
= 0 :
1In fact, the proposals [11] and [23] were worked out for the closed type II superstring but the idea is
easily adapted to the open string.
2We would like to thank Y. Okawa for pointing out this reference to us.
3Notice however, that there are some subtleties when G0 2 = 0.
{ 2 {
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
Because the cohomology of Q is known to be isomorphic to the relative cohomology Hrel(Q)
calculated on on the subspace de ned by b0
A generic vector in this subspace is given by
=
=
0
Then, Q
= 0 reduces to
= 0 [24, 25] we consider this case.
j #i with
independent of 0 and c0.
~
Q
= G0
a cokernel.4 This leads to the condition on picture ( 12 ) states ,
This operator acts on picture ( 32 ) states and existence of Y0 implies that X0 cannot have
(2.6)
(2.7)
(2.8)
(2.9)
(2.10)
(2.11)
(2.12)
(2.13)
(2.14)
with general solution,
2
0
(1)
= 0
(0)
=
1 j #i + 0 1 j #i + 2 j "i + 0 2 j "i
(0)
(1)
where i(j) are independent of 0 and c0. Now requiring that the condition (2.8) is preserved
by Q implies that (21) = 0 and
2
(0) =
( 1)j (11)jG0 1
(1) and thus (2.8) and (2.1) de ne the
same invariant subspace. Finally we note that X0 is indeed no cokernel, i.e. every vector in
this subspace can be written as
= X0 ~, where ~ is an arbitrary string eld with picture
3
2
. This follows from the identities [
26
]
( 0) = j0;
( 0) = j0;
0( 0) =
j0;
3
2 ih0;
1
2 ih0;
1
2 ih1;
3
2
1
2
j
j
;
1
2
j + j1;
1
2 ih0;
1
2
j
~ = 1j #i + 2j "i with i = P
n (n) ( 0) we nd
0 i
1
n=0
where the index
12 resp.
32 denotes the picture and jn; 1
2 i =
0nj0; 12 i. Then, for
X0 ~ =
G0( (10))
( 1)j 2j (21)
(0)
j #i
( 1)j 2j 0 2 j #i + G0 2 j "i
(0)
where we have used that ( 0) ( 0) = j0; 32 ih0; 1 . We then see that X0 ~ is indeed of
2 j
the form (2.1) with
1 = G0( (10))
( 1)j 2j (21)
and
2 = ( 1)j 2j (20) :
0 and natural numbers nk and ml. This is not a problem for free string
eld theory but becomes an
issue in the presence of interaction vertices which generically do not preserve this de nition.
nk ml
k j i = l j i = 0 for l > 0 and
The vertices of open superstring eld theory can be written as
Cn( 1
;
; n) = !( 1; Mn 1( 2
;
; n 1));
(3.1)
where
denotes a combined string eld in the R and NSsector and Mn are string
nproducts. These products were constructed through a gauge transformation of the free
theory de ned by a hierarchy of gauge products on the large Hilbert space with each gauge
product obtained from lower order products by means of a contracting homotopy
for the
nilpotent operator 0, that enters in the bosonization of the superconformal ghost (z).
More precisely, we require the existence of an operator
such that [ 0; ] = 1. Upon
changing , the construction of [2] produces actions that are related by eld rede nitions,
so that any choice for is equally good. One additional condition on
is that the resulting
vertices should be nonsingular. In [2] a class of such good homotopies built out of
=
I dz
2 i
f (z) (z)
was proposed, where f (z) is required to be holomorphic in some annulus that contains the
unit circle and (z) enters in the bosonization of superconfomal ghost, (z) = @ (z)e
In [10] the homotopy for [ 0; ] was taken to be the same irrespective of whether the
string products de ning the string products have zero or one Ramond input. To illustrate
this we consider the string product
1
3
M2 =
fX; m2gP2<0> + Xm2P2<1> + m2P2<2>
where P2<n> is the projector on n Ramond inputs among the two inputs of m2 and m2 =
is Witten's string product. The picture changing operator, X is related to
through the
graded commutator, X = [Q; ]. Finally, fX; m2g is the graded anticommutator of X and
m2. For zero Ramond inputs M2 is cyclic with respect to the standard symplectic form by
construction since the combination fX; m2g sums over all possible insertions of a picture
changing operator (see [2] for details and notation). For vertices involving two Ramond
elds we have
(3.2)
(z).
(3.3)
!(N; M2(R; R)) = !(N; m2(R; R)) = !(R; m2(R; N ))
(3.4)
where N and R denote NS and R string elds respectively. At rst sight it looks as if M2
were not cyclic since there is an X missing in front of m2 on the right hand side of (3.4).
However, we will see in the end that this is exactly what we need, because of subtleties in
de ning a symplectic form on the Rstring elds.
Next, let us consider the 4vertex. First, we have from (3.3)
[M2; M2](R; R; R) = 2Xm2
m2(R; R; R) = 0
(3.5)
due to associativity of the star product (m2
m2 = 0). Thus, to this order the A1
consistency condition (or equivalently the BVequation) allows us to set M3(R; R; R) = 0.
{ 4 {
For two Ramond inputs we have
where
1
2
1
2
Then,
[M2; M2](R; N; R) = m2 Xm2(R; N; R) = [Q; [m2; 2]](R; N; R) ;
2 = m2P2<1> + f ; m2gP2<0> :
1
3
M3(R; N; R) = m3(R; N; R) ;
sistency condition, 12 [M2; M2] + [Q; M3] = 0, then xes M3 completely as
Since the gauge products n never have more than one Ramond input [10], the A1
conHJEP04(216)
where m3 = [m2; 2] and we have used associativity of m2. Associativity then also implies
that
M3(R; N; R) =
[m2; 2](R; N; R) = 0 and thus M3 is in the small Hilbert space.
Similarly, for one Ramond input
1
2
3 =
1
4
f ; m3gP3<0> + m3P3<1> :
[M2; M2](N; R; N ) = Xm2 Xm2(N; R; N ) =
[Q; [Xm2P2<1>; 2P2<1>]](N; R; N )
To continue we choose the homoptopy for
de ning the gauge product 3 as
1
2
1
2
=
[Q; [M2; 2P2<1>](N; R; N ) =
[Q; [M2; 2]](N; R; N ) : (3.8)
3(N; R; N ) = m3(N; R; N ) = m2
m2(N; R; N ) :
Using, associativity of m2 again we then nd
1
2
M3(N; R; N ) =
([M2; 2] + [Q; 3]) (N; R; N )
= M2<1> 2(N; R; N ) = Xm2<1> 2(N; R; N ) = Xm3P3<1>(N; R; N )
which is in the small Hilbert space. More generally, for a generic permutation of the
Rand N inputs
M3P3<1> = Xm2<1> 2P3<1> = Xm3P3<1>
holds. Thus, modulo the factor X that will be dealt with below, proving cyclicity of M3 is
reduced to show cyclicity of m3. Explicitly, we have
!(N1; M3(R1; N2; R2))
= !(N1; m3(R1; N2; R2))
= !L(N1; 0m2( m2(R1; N2); R2)) + !L(N1; 0m2(R1; m2(N2; R2))) ;
{ 5 {
(3.6)
(3.7)
(3.9)
(3.10)
(3.11)
(3.12)
(3.13)
where !L is the symplectic form evaluated in the large Hilbert space and which reproduces
the symplectic form, !, on the small Hilbert space upon insertion of the zero mode 0 [2].
Now, commuting 0 through to R1 and using cyclicity of m2 we get
Since is BPZeven we then have
Thus, m3 is cyclic with respect to the symplectic form !( ; ). In order to prove cyclicity to
arbitrary order we rst recall the recursion relations de ning the higher order products [10].
For zero or one Ramond input we have
Mn<+02=1> =
X[Mk+1; n k+2]Pn<+02=1> ;
M1 = Q
Similarly, for two adjacent Ramond inputs,
!(N1; M3(R1; R2; N2))
= !(N1; m2(R1; 2(R2; N2)))
!(N1; 2(m2(R1; R2); N2))
=
!L(N1; m2( 0R1; 2(N2; R2)))
!L(N1; 2(m2( 0R1; R2); N2)) :
Now, for the rst term we use cyclicity of m2 while for the second we use cyclicity of 2
for two Rinputs which gives
!(N1; M3(R1; R2; N2)) = !L( 0R1; m2( 2(R2; N2); N1)) + !L(m2( 0R1; R2); 2(N2; N1))
1
n + 1
n
k=0
mn+3 =
1
(3.15)
(3.16)
(3.17)
(3.18)
(3.19)
(3.20)
(3.21)
and for two Ramond inputs
where
with m2 = . Finally, the gauge products n are given by
Mn<+23> = mn+3Pn<+23> =
X[mk+2; n k+2]Pn<+23>
1
n + 1
n
k=0
1
n + 1
n
k=0
X[mk+2; n k+2]
n+2 =
n + 3 f ; mn+2gPn<+02> + mn+2Pn<+12> :
{ 6 {
) for all n. Indeed, upon inspection of (3.20), subject
to the homotopy (3.21), it is apparent that such a term would have to be of the form
P mn k+2mk+2 which vanishes due to the A1 condition [m; m] = 0. Furthermore, it
(n
1)Mn<+11> = X mn<1> 2 + mn<1>1 3 +
= (n
1)Xmn+1Pn<+11> :
(3.22)
To show this identity we proceed by induction. We have from (3.18)
HJEP04(216)
nMn<+11> = [Mn<1>; 2<1>] + [Mn<11>; 3<1>] +
+ [Q; n<+1>1]
+Mn<1> <0> + Mn<11> 3<0> +
2
[Q; mp<1>] together with the identity, [m; M ] = 0,
that is,
1
2
1
3
1
2
S =
!( ; Q )
!( ~; XQ ~) + !( ~; Q )
+ !( ; M2( ; )) + !( ; M3( ; ; )) +
[Q; n<+1>1] = Xmn<+1>1 +
[mn<1>; M2<1>] + [mn<1>1; M3<1>] +
+ M2<1>mn<0> + M3<1>mn<0>1 +
+ mn<1>M2<0> + mn<1>1M3<0> +
Upon substitution of (3.24) into (3.23) and using (3.21) as well as [m; m] = 0 the
result follows.
Thanks to (3.19) and (3.22) the problem of proving cyclicity of Mn is again reduced
to show cyclicity of mn. To prove cyclicity of mn+3, n
1, one then proceeds exactly as
in (3.13){(3.17) expressing mn+3 in terms of [mk+2; n k+2] and then using cyclicity of mq,
q
n + 2 as well as cyclicity of p, p
n + 2 for p NSinputs.
Let us now explain how these vertices lead to a gaugeinvariant action for the open
superstring in the small Hilbert space. Following [23] we write
1
4
{ 7 {
where,
=
+
string products Mn are given by
and ~ is an auxiliary Ramond string eld with picture ( 32 ). The higher
Mn = MnP <0> + mn(P <1> + P <2>)
which di ers from (3.3) by the ubiquitous factor X. To prove gauge invariance we use that
Mn is cyclic w.r.t. !. The standard proof of gaugeinvariance has to be modi ed as M is
not an A1algebra. However, M is an A1algebra and di ers from M in that it contains
an additional Xinsertion on Ramond outputs and contains no BRST operator Q. There
are three di erent types of gaugetransformations with odd parameters ,
and ~ having
picture 1, 12 and
3
2
.
(3.23)
(3.24)
(3.25)
(3.26)
Using antisymmetry of ! and cyclicity of Mn one arrives at the identities (n; k
2),
!( ; Mn
Mk) = !( ; Mn
Mk)
!( ; QMk) = !(Q ; Mk) = !(Q ; Mk);
!( ; Mn
Q) = !(Mn ; Q):
= !(Mn ; P1<0>
Mk + XP1<1>
Mk) = !(Mn ; Mk);
where
denotes the coderivation built from
as its 0string map and we suppressed the
string eld
. Explicitly, (3:29) reads as
!( ; Mn(Q ; : : : ; ) + Mn( ; Q ; : : : ; ) +
)
= !(Mn( ; ; : : : ; ) + Mn( ; ; : : : ; ) + : : : ; Q ):
HJEP04(216)
Summing over (3.27){(3.29) we obtain zero on the lefthand side due to the A1 relations,
while on the righthand side we nd,
0 = !( ; Q ) + !( ~; Q ) + X !((
+
); Mk( ; ; : : : ; ))
=
= S;
1
2
!( ; Q ) + !( ~; Q ) + X
!( ; Mk( ; ; : : : ; ))
!
!( ~; Q
)
De ne the transformation
,
(3.27)
(3.28)
(3.29)
(3.30)
(3.31)
(3.32)
(3.33)
(3.34)
(3.35)
(3.36)
where we used !( ~; Q
) =
12 !( ~; QX ~) in the last step. Consequently, the
transformations (3.30) and (3.31) are a bosonic gauge symmetry of the action. By replacing
with ~ in (3.27){(3.29) one veri es that the following transformation is a fermionic gauge
symmetry,
where X~ denotes the coderivation with 0string product X .
In order to derive the gauge transformations corresponding to the parameter , let
us recall that Mn and mn(P <0> + P <1>) give two commuting A1 structures [10].
Together with cyclicity of mn(P <0> + P <1>) w.r.t. ! one can then deduce that the following
transformations are a gauge symmetry of S, by imitating the previous derivation,
, ~ as
+
~ = Q
= X ~
{ 8 {
Notice that all gauge transformations preserve the constraint
form Q
with
not expressible in the form
= X
for some picture
= X ~ up to states of the
32 state .
Let us now comment on the applicability of our formalism to writing the proposal for
the superstring action [12] in the small Hilbert space. Assuming the constraint (2.8), we
can rewrite (3.25) without the need for the auxiliary eld ~ as
1
2
1
2
1
3
S =
!( ; Q ) +
!( ; Y Q ) +
!( ; M2( ; )) +
!( ; M3( ; ; )) +
(3.37)
1
4
where Y = c0 0( 0) is the inverse picture changing operator in the restricted Hilbert space.
The gauge transformation of this action agrees with that of (3.25) up to the contribution
coming from the kinetic term that is
S / !((X
X0)(m2( ; ) + m2( ; ) + m3( ; ;
+
)); Y Q )
(3.38)
Formally this term can be removed by replacing X by X0 (as well as
by
( 0)) in the
de nition of the higher string products Mn and the gauge products n when applied to
states containing one or two Ramond states, e.g. instead of (3.3) we take
and instead of (3.6) we take
1
3
M2 =
fX; m2gP2<0> + X0m2P2<1> + m2P2<2>
2 =
( 0)m2P2<1> +
f ; m2gP2<0> :
1
3
(3.39)
(3.40)
However, for this choice of homotopy to be well de ned, one needs that the mns are
compatible with the particular realisation of the picture ( 12 ) states in terms of the zero
modes 0 and 0 described in section 2.
Acknowledgments
We would like to thank Ted Erler and Barton Zwiebach for helpful discussions. I.S. would
like to thank the Center for the Fundamental Laws of Nature at Harvard University for
hospitality during the initial stages of this work. This work was supported by the DFG
Transregional Collaborative Research Centre TRR 33 and the DFG cluster of excellence
\Origin and Structure of the Universe".
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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