#### Complete action for open superstring field theory with cyclic A ∞ structure

HJE
Complete action for open superstring eld theory with cyclic A
Theodore Erler 0 1 3
Yuji Okawa 0 1 2
Tomoyuki Takezaki 0 1 2
0 Komaba , Meguro-ku, Tokyo 153-8902 , Japan
1 Theresienstrasse 37 , 80333 Munich , Germany
2 Institute of Physics, The University of Tokyo
3 Arnold Sommerfeld Center, Ludwig-Maximilians University
We construct a gauge invariant action for the Neveu-Schwarz and Ramond sectors of open superstring eld theory realizing a cyclic A1 structure, providing the rst complete and fully explicit solution to the classical Batalin-Vilkovisky master equation in superstring eld theory. We also demonstrate the equivalence of our action to the WessZumino-Witten-based construction of Kunitomo and one of the authors.
String Field Theory; Superstrings and Heterotic Strings
1 Introduction
Background The action 2 3
integration over the fermionic modulus in the Ramond propagator.1
A closely related
approach has recently been developed by Sen [10, 11], which has a somewhat simpler
worldsheet realization at the cost of introducing spurious free elds.
We are now in a position to complete the construction of all classical superstring
eld theories. The construction of [1] was realized by extending the Neveu-Schwarz (NS)
open superstring eld theory of Berkovits [12, 13] to include the Ramond (R) sector. The
Berkovits theory gives an elegant Wess-Zumino-Witten-like (WZW-like) action for the NS
sector in the large Hilbert space [14] and is a suitable starting point for the study of tachyon
condensation and classical solutions [15{22]. However, the question of recent interest is how
to construct other superstring eld theories and how to quantize them. In this capacity
1The construction of [1] is based on a very old idea for formulating the free action for the Ramond string
eld [2{8], which with the proper understanding is equivalent to the formulation of Witten [9]. However,
the construction of [1] gives the rst consistent nonlinear extension of this free action.
{ 1 {
the Berkovits formulation is not ideal, since it does not immediately generalize to type II
closed superstrings,2 and, despite some attempts [29{34], it is not known how to properly
de ne the gauge- xed path integral.
For this reason, in this paper we turn our attention to a di erent form of open string
eld theory which uses the small Hilbert space and realizes a cyclic A1 structure. The
construction of superstring eld theories based on A1 and L
1 algebras is attractive since all
forms of superstring eld theory can in principle be described in this language [35{38, 11].
In addition, the de nition of the gauge- xed path integral is straightforward thanks to
the close relation between homotopy algebras, Batalin-Vilkovisky quantization, and the
Feynman-graph decomposition of moduli spaces of Riemann surfaces (or their
supergeometrical extension3) [41{43].
Our construction of open superstring eld theory extends the NS open superstring
eld theory of [
36
] to include the Ramond sector, and the interactions are built from
Witten's open string star product dressed with picture changing insertions. Part of the
work for constructing this theory was done in [38], which gives classical equations of motion
describing the interactions between the NS and R sectors.
Our task is to modify the
equations of motion so that they can be derived from an action. This requires, speci cally,
that the equations of motion realize a cyclic A1 structure, where the notion of cyclicity
is provided by the inner products de ning the NS and R kinetic terms. Interestingly, the
action we
nd for the Ramond sector turns out to be identical to that of [1] after the
appropriate translation of NS degrees of freedom [44{46].
This paper is organized as follows. In section 2 we review the formulation of the
Ramond sector kinetic term used in [1] and the NS and Ramond equations of motion
described in [38]. In section 3 we construct an action by requiring compatibility of the
equations of motion with the bilinear form de ning the Ramond sector kinetic term. First
we describe the picture changing insertion which plays a central role in de ning the vertices.
Then we give an explicit discussion of the 2-string product, generalize to the higher string
products, and provide a proof that the resulting A1 structure is cyclic. We also describe
how the construction can be translated into the formulation of the Ramond kinetic term
used by Sen [10, 11]. In section 4 we relate our construction to the WZW-based formulation
developed by Kunitomo and one of the authors [1]. We end with some concluding remarks.
Note added.
While this paper was in preparation, we were informed of independent
work by Konopka and Sachs addressing the same problem. Their work should appear
concurrently [47]. See also [48] for related discussion.
2Some attempts to provide a WZW-like formulation of closed type II superstring eld theory are
described in [23, 24]. For heterotic string
eld theory a WZW-like formulation in the large Hilbert space is
well established [25], and its extension to the Ramond sector would be interesting to consider [26{28].
3The manner in which picture changing operators in the vertices implement integration over odd moduli
has not yet been made fully explicit, though the computation of the four-point amplitude in [39] has given
some preliminary insight. However, it follows from the computation of the S-matrix [40] that the
treeIn this section we review the Ramond kinetic term [1] and equations of motion [38]. To
describe compositions of string products and their interrelations in an e cient manner,
we will make extensive use of the coalgebra formalism. The coalgebra formalism expresses
string products in terms of coderivations or cohomomorphisms acting on the tensor algebra
generated from the open string state space He:
T He
He 0
He
He 2
He 3
: : : :
Coderivations will be denoted in boldface, and cohomomorphisms with a \hat" and in
coalgebra formalism works e ciently if we use a shifted grading of the open string
eld
called degree. The degree of a string eld A is de ned to be its Grassmann parity (A)
plus one:
deg(A) = (A) + 1
mod Z2:
For a detailed description of all the relevant de nitions, formulas, and the notational
conventions, see [45].
2.1
Ramond kinetic term
Let us start by summarizing what is needed to have a consistent open string eld theory
kinetic term from the perspective of an action realizing a cyclic A1 structure. We need
three things:
(A) A state space H, perhaps a subspace of the full CFT state space, which is closed
under the action of the BRST operator Q. The BRST cohomology at ghost number
1 computed in H reproduces the appropriate spectrum of open string states.
(B) A symplectic form ! on the state space H. This is a linear map from two copies of
the state space into complex numbers,
which is graded antisymmetric,
! : H
H ! C;
!(A; B) =
( 1)deg(A)deg(B)!(B; A);
and nondegenerate.
!(A; B)
h j
! A
whose ghost number adds up to 3.
We sometimes write the symplectic form as h!j, and write
B. We assume that ! is nonzero only when acting on states
(C) The BRST operator must be cyclic with respect to the symplectic form !:
Equivalently
where I is the identity operator on the state space.
!(QA; B) =
( 1)deg(A)!(A; QB):
h!j(Q
I + I
Q) = 0;
{ 3 {
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
If these three criteria are met, a string eld theory kinetic term can be written as
where
is a degree even and ghost number 1 dynamical string eld in H. Variation of
the action produces the expected equations of motion Q
= 0, and the action has the
linearized gauge invariance
where
2 H is degree odd and carries ghost number 0.
Let us see how this story applies to the NS and R sectors of the open superstring. We
consider the RNS formulation of the open superstring, described by a c = 15 matter
boundary superconformal eld theory tensored with the c =
15 ghost boundary superconformal
eld theory b; c; ; . The
system may be bosonized to the ; ; e system [14]. We will
write the eta zero mode 0 as . The state space of the open superstring is the direct sum
of an NS component HNS and a Ramond component HR:
to formulate the NS kinetic term, it will be useful to consider the large Hilbert space
symplectic form !L de ned in terms of the large Hilbert space BPZ inner product by
!L(A; B)
( 1)deg(A)hA; BiL;
We use He to denote the combined state space. Formulating the NS kinetic term requires
a subspace of HNS consisting of states at picture
1 and in the small Hilbert space. The
BRST operator preserves this subspace, and has the correct cohomology at ghost number
1. The symplectic form can be de ned by the small Hilbert space BPZ inner product (up
to a sign from the shifted grading):4
!S(A; B)
( 1)deg(A)hA; BiS;
where the subscript L denotes the large Hilbert space.
Zmatter, where Zmatter is the disk partition function in the matter boundary conformal eld theory. In the
Zmatter, with
the opposite sign.
{ 4 {
Now let us describe the Ramond kinetic term. The major technical problem in this
respect is de ning an appropriate symplectic form. For this purpose we introduce two
picture changing operators:
X
Y
ghost number 0 and picture 1, and Y is degree even and carries ghost number 0 and picture
1. Since these operators depend on
zero modes, they only act on states in the Ramond
sector. Moreover, it is clear that X should not act on states which are annihilated by 0
and Y should not act on states which are annihilated by 0. For this reason we will always
assume that X and Y act on states in the small Hilbert space at the following pictures:
X : small Hilbert space; picture
Y : small Hilbert space; picture
3=2;
1=2:
In particular, all pictures besides picture
3=2 either contain states annihilated by 0 or
are BPZ conjugate to pictures containing states annihilated by 0. Similarly, all pictures
besides picture
1=2 either contain states annihilated by
0 or are BPZ conjugate to
pictures containing states annihilated by 0
. Assuming X and Y act on the appropriate
picture as above, they satisfy
and are BPZ even:
XYX = X;
YXY = Y;
[Q; X] = 0;
h!SjX
I = h!SjI
X;
h!SjY
I = h!SjI
Y:
Note that (2.16) implies that the operator XY is a projector:
(2.13)
(2.14)
(2.15)
(2.16)
(2.17)
(2.18)
(2.19)
(2.20)
This projector selects a subspace
of Ramond states which satisfy
(XY)2 = XY:
HR
We will call this the restricted space. To ensure that the action of XY is well de ned, we
will assume that the restricted space only contains states in the small Hilbert space and
at picture
1=2. We claim that the restricted space allows for the de nition of a Ramond
kinetic term, and to see it, we check conditions (A), (B) and (C). First note that the
restricted space is preserved by the action of the BRST operator:
XYQA = XYQXYA = XYXQYA = XQYA = QXYA = QA;
A 2 HR
restricted:
(2.21)
{ 5 {
Moreover, the cohomology of Q computed in HR
restricted reproduces the correct physical
spectrum [49]. Therefore condition (A) is met. Next, we de ne a symplectic form on
graded antisymmetric. Nondegeneracy follows from the fact that YA = 0 implies A = 0
upon operating with X, and !S is nondegenerate on the subspace of Ramond states at
pictures
1=2 and
3=2. Therefore condition (B) is met. Finally, we have
!S(YA; QB) = !S(YA; QXYB)
degree even, carries ghost number 1 and picture
1=2, and satis es XY R =
R.
We can package the dynamical NS and R string elds together into a string eld:
HR
by
term as
(2.23)
(2.24)
(2.25)
(2.26)
which we call the \composite restricted space". In the NS sector, the space HNS
consists of states in the small Hilbert space at picture
1. In the Ramond sector, the space
restricted is de ned as above. We de ne a \composite symplectic form"
e =
NS +
R:
He
restricted = HNS
restricted
HR
restricted;
e
! : He
!
hej
restricted
restricted
He
! C
where, following notation to be introduced in a moment, h!Sj0j is nonzero only when
contracting two NS states, and h!Sj2j is nonzero only when contracting two Ramond states.
From the above discussion, it is clear that the composite restricted space together with the
composite symplectic form satisfy conditions (A), (B) and (C), so we can write the kinetic
which describes the free propagation of both the NS and R states.
S =
1
2 !e( e ; Q e );
{ 6 {
Now that we have a free action for the NS and R sectors, our task will be to add interactions.
The structure of interactions at the level of the equations of motion was described in [38].
It is helpful to review this before considering the action.
The equations of motion are characterized by a sequence of degree odd multi-string
products:
Mf1
Q; Mf2; Mf3; Mf4; : : : :
We call these \composite products" since they encapsulate the multiplication of both NS
and R states. We require three properties:
(I) The composite products satisfy the relations of an A1 algebra. Equivalently, if Mfn+1
is the coderivation corresponding to Mfn+1, the sum
M
f
Mf1 + Mf2 + Mf3 + Mf4 + : : :
de nes a nilpotent coderivation on the tensor algebra:5
(2.30)
(2.31)
(2.32)
(2.33)
(2.34)
(II) The composite products are de ned in the small Hilbert space. Equivalently, the
coderivation Mf commutes with the coderivation
representing the eta zero mode:
[Mf; Mf] = 0:
[ ; Mf] = 0:
(III) The composite products carry the required ghost and picture number so that the
equations of motion,
0 = Q e + Mf2( e ; e ) + Mf3( e ; e ; e ) + : : : ;
have an NS component at ghost number 2 and picture
1, and a Ramond component
at ghost number 2 and picture
1=2.
assume that
and picture
1=2.
Witten's associative star product:
When we write the equations of motion, the dynamical Ramond string eld does not have
to be in the restricted space. Formulating the equations of motion in the restricted space
is closely related to constructing the action, and will be described later. However, we still
R is in the small Hilbert space, is degree even, and carries ghost number 1
We will construct the composite products by placing picture changing insertions on
m2(A; B)
( 1)deg(A)A
B:
(2.35)
5Commutators of multi-string products are always graded with respect to degree [
36
]. Commutators of
string
elds, computed with the open string star product, are graded with respect to Grassmann parity.
three mutually commuting A1 structures. Though it is not important for the equations
of motion, we note that the star product is cyclic with respect to the small (and large)
Hilbert space symplectic form:
h!Sj(m2
I + I
The generalization to other forms of open string multiplication (for example, the star
product with \stubs" [37, 38]) is closely related to the generalization to heterotic and type
II superstring eld theories, and will be left for future work. The BRST operator, the eta
zero mode, and the star product satisfy
Similarly the eta zero mode is cyclic with respect to the large Hilbert space symplectic form.
Because
R carry di erent picture, the composite products Mfn+1 must
provide a di erent amount of picture depending on how many NS and R states are being
multiplied. To keep track of this, it will be useful to invoke the concept of Ramond
number. A multi-string product has Ramond number r if it is nonvanishing only when the
number of Ramond inputs minus the number of Ramond outputs is equal to r. We will
write the Ramond number of a product using a vertical slash followed by an index
indicating the Ramond number. For example, bmjr is an m-string product of Ramond number r.
The de nition of Ramond number implies that the product bmjr has the property
Any product can be written as a unique sum of products of de nite Ramond number:
bmjr r Ramond states = NS state;
bmjr r+1 Ramond states = R state;
bmjr otherwise = 0:
bm = bmj 1 + bmj0 + bmj1 + : : : + bmjm:
s
r= 1
{ 8 {
The Ramond number of bm is bounded between
1 and m since bm can have at most m
Ramond inputs and at most 1 Ramond output. Since Ramond number is conserved when
composing products, it is conserved when taking commutators of coderivations:
[bm; cn]js =
X [bmjr; cnjs r];
with the understanding that commutators in this sum vanish if the Ramond number exceeds
the number of inputs of the product. As an example of this identity, note that associativity
The components of the star product with odd Ramond number vanish identically.
We are now ready to describe the equations of motion constructed in [38]. The
composite products Mfn+2 have a component at Ramond number 0 and a component at Ramond
number 2:
which carry the following picture and ghost numbers:
Mfn+2 = Mn+2j0 + mn+2j2;
Mn+2j0 :
mn+2j2 :
picture n + 1; ghost number
picture n;
ghost number
n;
n:
The 1-string product M1j0 is identi ed with the BRST operator
M1j0
Q;
and m2j2 is the Ramond number 2 component of Witten's open string star product. We
also de ne bare products of odd degree and gauge products of even degree:
bare products mn+2j0 :
gauge products
n+2j0 :
picture n;
ghost number
picture n + 1; ghost number
n;
n
1:
(2.49)
(2.50)
The bare product m2j0 is the Ramond number zero component of Witten's open string
star product. We de ne generating functions
of the star product implies
where the star product is broken into components of de nite Ramond number as
X tnMn+1j0;
n=0
1
n=0
1
n=0
1
X tn
n=0
X tnmn+2j2;
X tnmn+2j0;
n+2j0;
{ 9 {
(2.41)
(2.42)
(2.43)
(2.44)
(2.45)
(2.46)
(2.47)
(2.48)
(2.51)
(2.52)
(2.53)
(2.54)
HJEP08(216)
Expanding in powers of t gives a recursive system of equations which determine higher
products in terms of sums of commutators of lower ones. A crucial step in solving this
system of equations concerns (2.58), which de nes the gauge product
the bare product mn+2j0. The solution of (2.58) requires a choice of contracting homotopy
of .6 This choice in uences the con guration of picture changing insertions which appear
in the vertices, and will determine whether or not the equations of motion can be derived
n+2j0 in terms of
from an action.
The products can be usefully characterized by the cohomomorphism
G^ (t)
P exp
ds j0(s) ;
Z t
0
where the path ordering is in sequence of increasing s from left to right. In particular, the
generating functions take the form
which are postulated to satisfy the di erential equations
d
dt
d
dt
d
dt
Also, using (2.58) and (2.62) it is straightforward to show that [44]
Here and in what follows, all objects are evaluated at t = 1 when the dependence on t is
not explicitly indicated. The coderivation representing the composite products is
From this expression it immediately follows that
because Q; m2 and
are mutually commuting A1 structures. Therefore the composite
products satisfy A1 relations and are in the small Hilbert space.
6In this context, a contracting homotopy for is a degree odd linear operator
acting on the vector
space of coderivations which satis es [ ;
[ ; D] = D for an arbitrary coderivation D.
Now we can bring the Ramond kinetic term and equations of motion together to de ne
1
1
1
S =
2 !e( e ; Q e ) +
3 !e( e ; Mf2( e ; e )) +
4 !e( e ; Mf3( e ; e ; e )) + : : : ;
an action:
of motion
symplectic form:
where e is the composite string eld and Mfn+1 are the composite products introduced in
subsection 2.2. Since we now consider the action, the dynamical Ramond string eld must
belong to the restricted space.
When we vary the action, it is assumed that we should reproduce the equations
However, this requires that the composite products are cyclic with respect to the composite
hej Mfn+1
!
I + I
Mfn+1 = 0
on He
Thus the composite products de ne a cyclic A1 algebra. Cyclicity does not follow
automatically from the construction of the equations of motion given in subsection 2.2, but
requires a special choice of picture changing insertions inside the vertices. More technically,
it requires a special choice of contracting homotopy for
in the solution of (2.58), and our
task is to nd it.
3.1
Picture changing insertion
The picture changing insertions in the action are de ned with the operator
(3.1)
(3.2)
(3.3)
(3.5)
which has the following properties:
1) ~ is a contracting homotopy for : [ ; ~] = 1,
2) ~ is BPZ even: h!Lj ~
3) [Q; ~] = X when acting on a Ramond state at picture
3=2 in the small Hilbert space,
4) ~2 = 0.
Property 1) is needed to de ne a contracting homotopy for
in the solution of (2.58).
Properties 2) and 3) will be needed in the proof of cyclicity. Property 4) will not be
essential for our purposes, but we would like to have it anyway.
A natural candidate for ~ is the operator
( 0) as used in [1], which in particular
satis es
~ : degree odd; ghost number
1; picture 1;
(3.4)
[Q; ( 0)] = X:
and obtain a nite result. Now suppose A = QA0 and B0 = QB. Then using the BPZ even
property of X gives
h ( 0)A; BiL = hA0; XBiL + ( 1) (A0)+1
h ( 0)A0; B0iL:
We have assumed that the left hand side is nite, and the second term on the right hand
side should be nite by the same assumption. However, this contradicts the fact that the
rst term on the right hand side can be in nite if B is annihilated by 0. Therefore, the
action of
( 0) in the large Hilbert space must generally be singular.
This causes problems with a direct attempt to identify
( 0) with the operator ~.
Nevertheless, it was shown in [1] that
However, in [1] it was assumed that
( 0) at least formally satis es properties 1)
4).
( 0) never acts on states annihilated by
0
. Here
we would like to provide a setting where this assumption is justi ed. First, note that (3.6)
implies that we can de ne operators ( 0) and
( 0) acting on the following states:
( 0) : large Hilbert space; picture
( 0) : large Hilbert space; picture
1=2;
1=2:
The operator
( 0) is well de ned since
maps from the large Hilbert space at picture
1=2 into the small Hilbert space at picture
3=2, after which we can act with
( 0). The
operator
( 0) is de ned by BPZ conjugation of ( 0) . Therefore we have
h!Lj
( 0)
when acting on states in the large Hilbert space at picture
1=2. We also have
However, we must be careful to avoid acting ( 0) on states annihilated by 0. This means
that
( 0) can only act \safely" on the states:
It may seem somewhat unnatural to require that
( 0) acts on the small Hilbert space,
since generically it maps into the large Hilbert space. Let us explain why this is necessary.
Suppose
( 0) could act on an arbitrary state A at picture
3=2 in the large Hilbert space.
Then we should be able to contract with a state B at picture
1=2,
( 0) : small Hilbert space; picture
3=2:
(3.6)
(3.7)
when acting in the large Hilbert space at picture
1=2. We can also say that
nilpotent in the sense that
( 0) +
which similarly holds on states in the large Hilbert space at picture
Having understood the limitations of
( 0), we can search for a more acceptable
alternative. For this purpose we introduce the operator [
39, 36
]
where the function f (z) is holomorphic in the vicinity of the unit circle. The function f (z)
can be chosen so that is BPZ even and commutes with
to give 1:
h!Lj
In addition 2 = 0. Therefore
realizes properties 1), 2) and 4), but it does not realize
property 3). Rather, the BRST variation gives the operator
which is not the same as X. This can be xed by de ning a \hybrid" operator between
HJEP08(216)
and
( 0):
~
+ ( ( 0)
)P 3=2 + (
( 0)
)P 1=2;
where Pn projects onto states at picture n. Note that ( 0) always appears here in allowed
combinations with
acting on allowed pictures. Note also that ~ reduces to
when acting
on NS states, as is appropriate for de ning the NS superstring eld theory [
36
]. It is also
clear that ~ is BPZ even, and so realizes property 2). To see that property 3) is realized,
let us de ne the picture changing operator
Note that in general Xe is di erent from X de ned in (2.13) and X de ned in (3.15).
However, Xe is identical to X when it acts on a state A in the small Hilbert space at
picture
3=2:
X
so property 3) is realized. Now let us con rm properties 1) and 4). Note
and compute
[ ; ~] = 1 +
h
h
+
+
+
+ (
= 1 + (
= 1;
Pn
= Pn+1;
( 0)
( 0)
( 0)
+
i
P 1=2
( 0)
i
In the second step we commuted all projectors to the right and dropped terms with a pair
of projections into incompatible pictures. Using 2 = 0 this further simpli es
( 0) P 3=2 +
( 0)
2
( 0)
( 0)
where
( 0) = 0 +
( 0)
(3.21)
(3.22)
(3.23)
(3.24)
(3.25)
HJEP08(216)
which vanishes as a consequence of (3.12). Therefore we have a de nition of the picture
changing insertion ~ with all necessary properties.
It is worth mentioning that X and
( 0) cannot be expressed in an elementary way in
terms of the local picture changing insertions X(z) and (z). Therefore, the computation
of correlation functions with X and
( 0) does not appear to be straightforward. However,
a recipe for computations with such operators was given in [50] in the context of
correlation functions, where they may be represented as formal integrals
X
Z
d
Z
d ~ e G0 ~ 0 ;
( 0)
Z
d ~
~
;
where
is an odd integration variable and ~ is an even integration variable. The key point
is that the integral over the even variable ~ should be understood algebraically, analogous
to the Berezin integral over the odd variable , rather than as an ordinary integral in the
sense of analysis. One di culty, however, is the appearance of a singular factor ~ 1 in the
integral for
( 0). This is related to the fact that
( 0) is an operator in the large Hilbert
space, and therefore its precise de nition must go slightly beyond the formalism of [50].
Here we give one prescription for dealing with this. We may express ( 0) in the form
Since the rst term is independent of z, we can write
Finally, we represent the integrand as the integral of a total derivative,
Z
d ~
~
+
I
dz 1 Z
jzj=1 2 i z
I
dz 1 Z
jzj=1 2 i z
d ~
1
~
d ~
as
e ~ (z)
~
e ~ 0 + e ~ (z) :
and 0 is the zero mode of the
ghost. The term
can be represented as an
algebraic integral
where
0 +
=
=
I
=
Z 1
0
dz 1 Z
jzj=1 2 i z
d ~
and taking the derivative with respect to t gives
=
dt
I
dz 1 Z
jzj=1 2 i z
d ~( 0
(z))e ~(t (z)+(1 t) 0):
Note that the problematic factor ~ 1 is canceled. The upshot is that we have de ned
( 0) as a sum of 0, which can be understood in the bosonized
system, and
, which
can be evaluated following [50]. To see how this de nition can be applied, note that the
computation of a typical open string eld theory vertex requires evaluating correlation
functions with multiple insertions of ( 0):
(1) (2) : : : (n);
(3.26)
(3.27)
(3.28)
(3.29)
(3.30)
(3.31)
(3.32)
(i) represent appropriate conformal transformations of
( 0). Writing
( 0) =
produces cross terms of the form
(1) (2) : : : (m) (m+1) (m+2) : : :
(n);
where (i) and
(i) represent appropriate conformal transformations of 0 and
,
respectively. Since ( (1))2 = 0, we can replace these insertions with
(1)( (2)
(1)) : : : ( (m)
(1)) (m+1) (m+2) : : :
(n):
We can now drop the factor (1), which only serves to saturate the
zero mode in the large
Hilbert space, and evaluate the remaining factors using
correlation functions as in [50].
An important question is whether our choice of picture changing insertions ~ and
Xe avoid contact divergences in vertices and amplitudes, as appear for example when we
use a local picture changing insertion in the cubic vertex [51]. In the NS sector such
divergences are absent since the picture changing insertions appear as holomorphic contour
integrals [
39, 36
]. In the Ramond sector, the picture changing insertions appear as
( 0)
and X; to our knowledge, such operators can only be divergent in the presence of a zero
mode of the path integral associated with 0
. We have taken some care to ensure that
( 0) and X act on states of pictures where such zero modes are absent, and therefore
the vertices are expected to be nite. Explicit calculations with similar operators will be
discussed in upcoming work [52], and no contact divergences appear.
We are ready to construct the products de ning the action. Let us start by expanding the
equations of motion out to second order in the string eld and in NS and R components:
0 = Q NS + M2j0( NS; NS) + m2j2( R; R) + : : : ;
0 = Q R + M2j0( NS; R) + M2j0( R; NS) + : : : :
(3.33)
(3.34)
In [
36
] the product of two NS states was de ned by
1
3
M2j0 =
Xm2j0 + m2j0(X
This de nition does not work for multiplying an NS and R state, since it does not multiply
into the restricted space in the Ramond sector. For this reason we take
M2j0 = Xm2j0
(multiplying NS and R state in He
restricted):
(3.36)
Because XYX = X, this product satis es XYM2j0 = M2j0 and therefore maps into the
restricted space. Note that this de nition of M2j0 di ers from [38], where it was assumed
that M2j0 multiplies two NS states and an NS and R state in the same way. To make
notation uniform it is helpful to write X and X together using the picture changing operator
Xe , so we de ne
Xe m2j0 + m2j0(Xe
I + I
Xe )
(0 Ramond inputs)
(1 Ramond input)
:
(3.37)
The full composite 2-product is then
Xe m2j0 + m2j0(Xe
I + I
Xe )
(0 Ramond inputs)
M2j0
Mf2
>
8> 1
>>< 3
:
>>>> Xe m2j0
>
8> 1
>>>> 3
>
>
>
<
Xe m2j0
>
>
>
>
>
:
>
>>> m2j2
(1 Ramond input)
:
(3.38)
(2 Ramond inputs)
Note that using Xe gives a de nition of the product M2j0 between arbitrary states in He.
Following the discussion of subsection 2.2, the product M2j0 should be derived from a gauge
2-product 2j0 and bare 2-product m2j0 satisfying the formulas
M2j0 = [Q; 2j0];
[ ; 2j0] = m2j0:
(3.39)
(3.40)
The last equation de nes 2j0 in terms of m2j0 with an appropriate choice of
contracting homotopy for . The choice of contracting homotopy which produces our preferred
de nition of M2j0 is realized by the following gauge 2-product:
2j0
>>< 3
8> 1 ~m2j0
>
>
This completes the de nition of the equations of motion up to second order.
Now we want to see that the equations of motion can be derived from an action. This
requires that the composite 2-product is cyclic:
h!ejI
Mf2 =
h!ejMf2
I on He
Note that cyclicity only needs to hold when the vertex is evaluated on the composite
restricted space, since this is the space of the dynamical string eld appearing in the action.
Outside this space the products will not be cyclic, and in fact the notion of cyclicity itself is
somewhat problematic since Y may act on a state of the wrong picture. The demonstration
of cyclicity goes slightly di erently depending on the arrangement of NS and R states in
the vertex. Let us discuss for example the case
h!ej(I
where R1; R2 are Ramond states and N1 is an NS state in Herestricted. Expanding into
components of de nite Ramond number, we have
h!ej(I
we obtain
Mf2)(R1
R2
N1) =
The product m2j2 drops out since it does not multiply a su cient number of Ramond
states, and h!Sj0j drops out since it contracts too many Ramond states. Plugging in (3.37)
(3.42)
(3.43)
N1)
(3.44)
(3.45)
h!ej(I
Mf2)(R1
R2
N1) = h!Sj2j(Y
= h!Sj2j(Y
= h!Sj2j(XY
= h!Sj2j(I
= h!Sj(I
Xe m2j0)(R1
Xm2j0)(R1
m2j0)(R1
m2j0)(R1
R2
R2
R2
N1)
N1)
N1)
R2
N1)
m2)(R1
R2
N1):
In the second step we noted that Xe acts on a state of picture
3=2 in the small Hilbert
space, and therefore can be replaced by X. In the third step we used that X is BPZ even
and in the fourth step we used the fact that R1 is in the restricted space. Finally we
dropped the Ramond number labels since in this context they are redundant. Note that
in these steps it is important to assume that the states are in Herestricted. Next consider
h!ej(Mf2
I)(R1
R2
N1) =
We therefore have
h!ej(Mf2
I + I
which vanishes because the open string star product is cyclic. The proof of cyclicity for
the other combinations R1
N1
R2 and N1
R1
R2 goes similarly. When all inputs are
NS states, cyclicity follows from the construction of the NS open superstring eld theory
in [
36
]. Therefore we have a cubic vertex consistent with a cyclic A1 structure.
Now let us discuss the generalization to higher string products. De ning the higher
products requires a choice of contracting homotopy for
in the solution of the equation
[ ; n+2j0] = mn+2j0:
The contracting homotopy we choose de nes the gauge products as follows:
(3.46)
(3.48)
:
(3.49)
(3.50)
(3.51)
n+2j0
1
8
>
>
>>< n+3
>
:
>>> ~mn+2j0
~mn+2j0
mn+2j0( ~
I n+1 + : : :+I n+1
~
)
(0 Ramond inputs)
(1 Ramond input)
It is not immediately obvious that this leads to a cyclic A1 structure. We will prove that
it does in the next subsection. For now, we demonstrate two important properties, which
follow from this de nition:
Mn+2j0 = Xe mn+2j0
mn+2j2 = 0
(1 Ramond input);
(3 Ramond inputs):
The rst equation generalizes (3.37), and implies that the interactions are consistent with
the projection onto the restricted space in the Ramond sector. The second equation
addresses a puzzle raised in [38] concerning the existence of cubic terms in the Ramond string
eld in the equations of motion. The existence of such terms is consistent with A1
relations, but is not compatible with the existence of an action since the equations of motion do
not possess quartic terms in the Ramond string eld. (Recall that Mfn has no component
with Ramond number 4.) Therefore, the fact that mn+2j2 vanishes with three Ramond
>>>> n + 3
>>>< Xe mn+2j0
>> mn+2j2
>:>>> 0
Mfn+2 =
operator
Ramond number:
We also de ne inputs is expected and in fact necessary to derive the equations of motion from an action. In total, then, we nd that the composite products appear as follows:
Xe mn+2j0 + mn+2j0(Xe I n+1 +: : :+I n+1
Xe ) (0 Ramond inputs)
The products mn+2j0 and mn+2j2 above are determined recursively by solving (2.56)
and (2.57) with our choice of gauge products (3.49).
To streamline the proof of (3.50) and (3.51), it will be useful to introduce the projection
which selects n-string states containing r Ramond factors (and therefore n
r NS factors).
This projector commutes in a simple way through coderivations of products with de nite
nr : T He ! T He;
mr+1 bnjs = bnjs ms++rn:
n =
n
X
r=0
r
n
;
Mn+2j0 n1+2 = Xemn+2j0 n1+2;
mn+2j2 n3+2 = 0;
11Mn+2j0 = Xe 11mn+2j0;
11mn+2j2 = 0:
11(Mj0
Q) = Xe 11mj0 ;
11mj2 = 0 :
which projects onto n-string states with an undetermined number of Ramond factors. With
these projectors we can express (3.50) and (3.51) in a more useful form using coderivations.
First we write
where Xe is the coderivation corresponding to Xe . Commuting the projectors through the
coderivations using (3.54) gives
Summing over n then implies
In the
rst equation we subtract Q since (3.60) only applies to the 2-string product
and higher.
To prove (3.60) and (3.61) it is helpful to rst derive the form of G^ 1 when it produces
one Ramond output:
To compute this, note that
11G^ 1
from the de nition of the path ordered exponential. Our choice of contracting homotopy
for
in the Ramond sector (3.49) implies
where ~ is the coderivation corresponding to ~. Plugging in gives
dt
d h 11G^ (t) 1i =
11 ~mj0(t)G^ (t) 1 =
11 ~G^ (t) 1m2j0;
where we used mj0(t) = G^ (t) 1m2j0G^ (t). Therefore we obtain
dt
d h 11G^ (t) 1i =
~h 11G^ (t) 1im2j0:
operator on the tensor algebra. This determines the solution to be
The solution is subject to the initial condition G^ (0) 1 = IT He, where IT He
is the identity
This satis es (3.66) since (m2j0)2 = 0 by (2.41). Setting t = 1 we have
11G^ (t) 1 = 11hIT He
t~m2j0 :
i
11G^ 1 = 11hIT He
~m2j0 :
i
This identity will play a central role in the following analysis, as it is the basis for our
proof of cyclicity and the relations (3.60) and (3.61), and it provides a crucial link to the
WZW-based theory in section 4. Note that expanding the path ordered exponential (2.59)
and integrating over the parameter in the generating function gives a general expression
for 11G^ 1
:
11G^ 1 = 11 IT He
2j0
1
2 3j0 +
1
2 2j0 2j0 + : : : :
This is substantially more elaborate than (3.68). With our choice of contracting homotopy
for , the higher order products in
11G^ 1 drop out, giving a closed form expression.
Now we are ready to prove (3.60) and (3.61). First note that the bare products with
one Ramond output simplify to
11mj0 = 11G^ 1m2j0G^
= 11m2j0G^ ;
(3.62)
(3.63)
(3.64)
(3.65)
(3.66)
(3.67)
(3.68)
(3.69)
(3.70)
since the second term in (3.68) cancels by associativity of the star product. Now consider
Mj0 with one Ramond output:
11Mj0 =
=
=
=
=
11G^ 1QG^
11hIT He
11hQG^
11QhIT He
11Q + Xe 11m2j0G^ :
~m2j0 QG^
Q~m2j0G^ + Xem2j0G^ i
~m2j0iG^ + Xe 11m2j0G^
This holds because the 2-string component of the state space cannot have three Ramond
From this we conclude
establishing (3.50). Next consider
11(Mj0
Q) = Xe 11mj0;
11mj2 =
11G^ 1m2j2G^
=
=
11hIT He
i
~m2j0 m2j2G^
11m2j2G^ + ~ 11m2j2m2j0G^ ;
where in the third line we used
from (2.42). Now note
factors. Therefore
which establishes (3.51).
3.4
Proof of cyclicity
m2j0m2j2 =
m2j2m2j0
11m2j2 = m2j2 23 = 0:
11mj2 = 0;
(3.71)
(3.72)
(3.73)
(3.74)
(3.75)
(3.76)
(3.77)
Having constructed the products, we are ready to demonstrate cyclicity:
h!ej(Mfn+1
I + I
Mfn+1) = 0
on He
restricted:
We will need to simplify this equation somewhat before we arrive at the key property
responsible for cyclicity and provide its proof. Note that the cyclicity of Mf1 = Q was
already demonstrated in subsection 2.1. When the vertex acts only on NS states, cyclicity
follows from the construction of the NS open superstring
eld theory in [
36
]. When the
vertex acts on one or three Ramond states, it vanishes identically since the symplectic form
and composite products do not carry odd Ramond number. When the vertex acts on four
or more Ramond states, it vanishes identically since the composite products vanish when
multiplying three or more Ramond states. Therefore, all that we need to show is that the
vertex is cyclic when it acts on two Ramond states:
h!ej(Mfn+2
I + I
Mfn+2) n2+3 = 0
on He
Expanding Mfn+2 into components of de nite Ramond number, this reads
h!Sj(mn+2j2
I + I
mn+2j2) n2+3
(3.78)
(3.79)
+ h!Sj(Y
I)(Mn+2j0
I + I
Mn+2j0) n2+3 = 0
on He
restricted:
In the rst term, both Ramond states must be channeled into the input of mn+2j2. In
the second term, the Ramond states split between the input of Mn+2j0 and the symplectic
form. This means that we can simplify the second term using (3.50):
In the rst step we used (3.50); in the second step we used the fact that Xe = X when
acting on a state in the small Hilbert space at picture
3=2; in the third step we used that
X and Y are BPZ even; in the fourth step we used that XY = 1 when acting on states in
the restricted space. Then the statement of cyclicity reduces to
h!Sj (mn+2j0 + mn+2j2)
I + I
Therefore mn+2j0 + mn+2j2 should be cyclic with respect to the small Hilbert space
symplectic form when the vertex acts on He
restricted including two Ramond states. Actually, we
wish to make a slightly stronger hypothesis: mn+2j0 + mn+2j2 is cyclic with respect to the
large Hilbert space symplectic when the vertex acts on the large Hilbert space including
two Ramond states:
h!Lj (mn+2j0 + mn+2j2)
I + I
This relation is the nontrivial property required for the proof of cyclicity. We will provide
a demonstration in a moment, but rst let us explain why (3.82) implies (3.81). The small
and large Hilbert space symplectic forms can be related by
h!Sj = h!Lj
I;
(3.83)
where
satis es [ ; ] = 1. The precise form of
is not important since its only role is
to saturate the
zero mode in the large Hilbert space CFT correlator. The left hand side
of (3.81) can be expressed as
I + I
2
n+3
= h!Lj(
I) (mn+2j0 + mn+2j2)
I + I
where for current purposes we assume that this equation acts on the small Hilbert space,
where
Moving
is equivalent to the identity since it acts on a state in the small Hilbert space.
to the left it will commute with
to give 1 and otherwise act on states in the
small Hilbert space to give zero. Thus we have
I + I
From this we can see that (3.82) implies (3.81) when operating on He
restricted.
We can proceed to prove (3.82) using the recursive de nition of the products. However,
the proof in this form requires consideration of several di erent cases depending on the
arrangement of NS and R inputs on the left hand side of (3.82). Earlier we encountered
similar inconvenience in the proof of cyclicity of Mf2 at the end of subsection 3.2. A more
e cient route to the proof uses the coalgebra formalism, and therefore it is useful to review
how the cyclicity is described in this language. An n-string product Dn is cyclic with respect
to a symplectic form ! if
h!j(Dn
I + I
Dn) = 0:
If we have a sequence of cyclic n-string products D0; D1; D2; : : : of the same degree, the
corresponding coderivation D = D0 + D1 + D2 + : : : will satisfy
We then say that the coderivation D is cyclic with respect to the symplectic form !. A
cohomomorphism H^ is cyclic with respect to ! if it satis es
An example of a cyclic cohomomorphism is
h!j 2D = 0:
h!j 2H^ = h!j 2:
H^ = P exp
ds h(s) ;
Z 1
0
where h(s) are a one-parameter family of degree even cyclic coderivations. To prove that
H^ in this form is cyclic, consider H^(u) obtained by replacing the lower limit s = 0 in the
path ordered exponential above with s = u. Taking the derivative with respect to u we nd
d
du h!j 2H^ (u) = h!j 2h(u)H^ (u) = 0:
This vanishes on the assumption that h(s) is cyclic. Therefore, the object h!j 2H^ (u) is
independent of u. Setting u = 0 and u = 1 reproduces (3.89). The construction of the NS
(3.87)
(3.88)
(3.89)
(3.90)
(3.91)
large Hilbert space symplectic form when acting on NS states. Therefore we have
Therefore G^ is cyclic in the large Hilbert space when acting on NS states.
the tensor product
into T He:
superstring eld theory [
36
] implies that the gauge products are cyclic with respect to the
and coproduct introduced in [45]. For this purpose we will need to think about \tensor
products" of tensor algebras, which we denote with the symbol 0 to avoid confusion with
Next it is helpful to recall a few things about the \triangle formalism" of the product
where at the extremes of summation 0 multiplies the identity of the tensor product 1T He.
and the coproduct 4 is a linear map from one copy of T He into two copies of T He:
(3.94)
(3.95)
(3.96)
4: T He
0 T He ! T He;
4 : T He ! T He
0 T He:
4A1
: : :
An =
: : :
Ak) 0 (Ak+1
: : :
An);
The coproduct is de ned by its action on tensor products of states:
Using (2.59) this also implies
with the understanding that nk vanishes if k > n.
Using coalgebra notation, the key equation (3.82) can be expressed as follows:
4D = (D
4H^ = (H^
0 H^ )4:
0 D)4;
m+n =4h
m
i
0 n 4:
r
k=0
mr+n =
X 4h r k 0 nki4;
m
h!Lj 22mj0 + 20mj2 = 0:
(3.92)
(3.93)
(3.97)
(3.98)
(3.99)
(3.100)
(3.101)
n
X(A1
k=0
A coderivation D and a cohomomorphism H^ satisfy the following compatibility conditions
with respect to the coproduct:
These are in fact the de ning properties of coderivations and cohomomorphisms. The
useful identity for our computations is
A generalization which also accounts for a projection onto r Ramond factors is
of the product and coproduct gives
= h!Lj
= h!Lj
= h!Lj 20m2j2G^ :
4h 1 0 11i4 mj0
1
The factor 11 in the two terms above can be written as
11 = 11G^ 1G^ = 11G^
h!Lj 22mj0 = h!Lj
The second term above can be simpli ed as follows:
h!Lj 22mj0 = h!Lj
0
11G^ +
11G^
11m2j0G^
0
11m2j0G^
where we used (3.68). Therefore we have
h!Lj 22mj0 = h!Lj
= h!Lj(I
coproduct:
To prove this, consider the second term on the left hand side:
In the second step we used the fact that G^ is cyclic with respect to !L when it only has
NS outputs, as in (3.93). Next consider the rst term of (3.101). Expressing 22 in terms
#
4:
#
#
4
0
0
11m2j0G^
11m2j0G^
(3.102)
(3.103)
(3.104)
(3.105)
(3.106)
#
4:
#
4
(3.107)
The form of mj0 with one Ramond output is given in (3.70). Plugging in gives
11mj0
11mj0
4h( 11m2j0G^ ) 0 11 + 11 0 ( 11m2j0G^ )i4:
11m2j0G^
11m2j0G^
which vanishes since ~ is BPZ even. With what is left we can disentangle the product and
"
"
= h!Lj
= h!Lj
= h!Lj 22m2j0G^ :
4h 1 0 11i4m2j0G^
1
11m2j0G^
11m2j0G^
#
4;
~ 0 I
Bringing the rst and second terms in (3.101) together therefore gives
Commuting the projectors past the star product in the two terms gives
h!Lj 22mj0 + 20mj2 = h!Lj 2(m2j0 + m2j2) 32G^
= h!Lj 2m2 32G^
= 0;
which vanishes since the star product is cyclic with respect to the large Hilbert space
HJEP08(216)
symplectic form. This completes the proof of cyclicity.
Relation to Sen's formulation
Here we would like to spell out the relation between our treatment of the Ramond sector
and the approach developed by Sen [10, 11]. The main advantage of Sen's approach is that
it utilizes simpler picture changing insertions, which may facilitate calculations. On the
other hand, the theory propagates spurious free elds and does not directly display a cyclic
A1 structure.
Sen's approach requires two dynamical string elds
Mf1
Q; Mf2; Mf3; Mf4; : : : ;
(3.109)
(3.110)
(3.111)
(3.112)
(3.114)
(3.115)
e =
e =
NS +
NS +
R;
R:
The NS elds
NS are in the small Hilbert space, degree even, and carry ghost
number 1 and picture
1. The Ramond elds
R and
R are in the small Hilbert space,
degree even, and at ghost number 1, but carry di erent pictures:
R carries picture
1=2
and
R carries picture
3=2. In this approach it is not necessary to assume that XY R =
R. The action takes the form
1
1
1
S =
2 !S( e ; GQ e )+!S( e ; Q e )+ 3 !S( e ; eb2( e ; e ))+ 4 !S( e ; eb3( e ; e ; e ))+: : : ; (3.113)
where ebn+2 are degree odd multi-string products which appropriately multiply NS and R
states, and the operator G is de ned by
G = I
G = X
For present purposes we can assume that the picture changing operator X is de ned as
in (3.15). In particular, G is BPZ even and [Q; G] = 0.
The action does not realize a cyclic A1 structure in the standard sense, but the
products ebn+2 satisfy a hierarchy of closely related algebraic identities. To describe them,
we introduce a sequence of degree odd multi-string products
Mfn+2
Gebn+2
(n = 0; 1; 2; : : :):
The relation to the composite products introduced earlier will be clear in a moment. The
rst few algebraic relations satis ed by the multi-string products are
0 = Qeb2(A; B) + eb2(QA; B) + ( 1)deg(A)eb2(A; QB);
0 = Qeb3(A; B; C) + eb3(QA; B; C) + ( 1)deg(A)eb3(A; QB; C)
+ ( 1)deg(A)+deg(B)eb3(A; B; QC) + eb2(Mf2(A; B); C) + ( 1)deg(A)eb2(A; Mf2(B; C));
HJEP08(216)
More abstractly, the full set of algebraic relations can be described using the coderivations
b
e
M
f
be2 + be3 + be4 + : : : ;
Q + Mf2 + Mf3 + Mf4 + : : : ;
1(Qbe + beMf) = 0:
In addition, gauge invariance requires that the products ebn+2 are cyclic with respect to the
small Hilbert space symplectic form:
Note that (3.116) implies
Multiplying (3.121) by G gives
h!Sj 2be = 0:
G 1be = 1(Mf
Q):
where
as
.
.. :
(3.116)
(3.117)
(3.118)
(3.119)
(3.120)
(3.121)
(3.122)
(3.123)
(3.124)
(3.125)
(3.126)
which implies that the products Mfn+1 satisfy A1 relations:
However, the products Mfn+1 are not required to be cyclic. Rather, cyclicity is realized by
the products ebn+2 which appear in the action. We will explain why this formulation leads
to a gauge invariant action in appendix A.
As suggested by the notation, it is natural to identify Mfn+1 with the composite
products constructed earlier. Indeed the composite products can be written in the form
0 = G 1(Qbe + beMf)
= 1 Q(Mf
= 1Mf2;
Q) + (Mf
Q)Mf
Mfn+2 = Ge ebn+2
(n = 0; 1; 2; : : :)
Then the composite products satisfy (3.116), where ebn+2 takes the form
Xmn+2j0 + mn+2j0(X
I n+1 + : : : + I n+1
X) (0 Ramond inputs)
ebn+2 = <
(1 Ramond input)
(2 Ramond inputs)
for some products ebn+2, where
Ge = I
Ge = Xe
This di ers from (3.116) only by the substitution of Xe with X. Therefore it is natural to
construct the products as before but replacing the picture changing insertion in (3.49) as
(3.127)
(3.128)
HJEP08(216)
(4.1)
(4.2)
subsection with the replacement of ~ with .
with the understanding that mn+2j0 and mn+2j2 are constructed out of
We can show that ebn+2 satis es (3.121) by pulling a factor of G out of the A1 relations for
Mfn+2.7 Furthermore, the cyclicity of ebn+2 follows from the proof of (3.82) in the previous
4
Relation to the WZW-based formulation
In this section we explain the relation between our construction to the WZW-based
formulation of [1]. The relation between the NS sectors was considered in [44{46], and our task
will be to extend this analysis to the Ramond sector.
The WZW-based theory uses an NS dynamical eld
which is Grassmann even, carries ghost and picture number zero, and lives in the large
Hilbert space (generically
bNS 6= 0). The dynamical Ramond eld
is the same kind of state as the Ramond eld
R from the A1 theory; it is Grassmann
odd, carries ghost number 1 and picture
1=2, and lives in the restricted space in the
Ramond sector. We will always denote objects in the WZW-based theory with a \hat" to
7Note that the products Mfn+1 satisfy A1 relations regardless of whether or not X has a kernel. This
can only be true if (3.121) holds regardless of whether X has a kernel. However, it is not di cult to
check (3.121) directly.
bNS;
b R;
The WZW-based action of [1] can be written as
The \potentials" are de ned by
1
2
Sb =
hY b R; Q b RiS
dt hAbt(t); QAb (t) + (Fb(t) b R)2iL:
The object Fb(t) is a linear operator acting on string elds, de ned by
where adAb (t) refers to the adjoint action of Ab (t):
distinguish from corresponding objects de ned in the A1 theory. To write the NS sector
of the action in WZW-like form, we introduce a one-parameter family of NS string elds
bNS(t); t 2 [0; 1], subject to the boundary conditions
bNS(0) = 0;
bNS(1) = bNS:
Ab (t)
Abt(t)
( ebNS(t))e bNS(t);
ebNS(t) e bNS(t):
All products of string elds are computed with the open string star product AB = A
B,
and all commutators of string
elds are graded with respect to Grassmann parity. The
WZW-based action only depends on the value of bNS(t) at t = 1. Variation of the action
produces the equations of motion [1]
(4.3)
(4.4)
(4.5)
(4.6)
(4.7)
(4.8)
(4.9)
(4.10)
(4.11)
Z 1
0
d
dt
I
Fb(t)
1
~adAb (t)
;
adAb (t)
0 = QAb + (Fb b R)2;
0 = QFb b R:
Unless the dependence on t is explicitly indicated, we will assume t = 1 here and in
The relation between these string
eld theories can be extracted by inspection of the
equations of motion [45]. The equations of motion of the A1 theory can be expressed in
what follows.
4.1
Field rede nition
the form
where
1
1
e
= 1
T He
+ e + e
e + e
e
e + : : :
0 = Mf
1
1
e
;
denotes the group-like element generated by e . Since
Mf = G^ 1(Q + m2j2)G^ ;
multiplying (4.10) by G^ gives
Let us look at the component of this equation with one NS output:
0 =
10(Q + m2j2)G^
= Q 10G^
= Q
1
10G^
0 = (Q + m2j2)G^
1
1
1
1
1
+ m2 22G^
+ m2
1
1
11G^
The component with one Ramond output is
(4.12)
(4.13)
(4.14)
(4.15)
(4.16)
(4.17)
(4.18)
(4.19)
(4.20)
(4.21)
(4.22)
:
1
1
1
e
1
R
R
R)2;
1
; 11G^
1
1
:
1
1
1
NS
;
NS
:
0 =
11(Q + m2j2)G^
= Q 11G^
= Q
1
11G^
+ m2 23G^
1
1
1
1
1
1
1
:
;
NS
NS
;
1
1
1
NS
NS
0 = QA + (F
0 = QF
R:
Further note that and de ne Therefore (4.14) and (4.15) reduce to
10G^
11G^
1
1
A
1
1
e
F
R
= 1G
1G
1G
1
1
These are the same as the equations of motion of the WZW-based theory, (4.8) and (4.9),
with the \hats" missing. It is therefore natural to suppose that the eld rede nition
between the theories is given by equating
In the NS sector, this only speci es the eld rede nition up to a gauge transformation of
the form
eb0NS = ebNS ev;
v = 0;
(4.23)
where v is a gauge parameter, since this transformation leaves Ab invariant. This ambiguity
can be removed by partial gauge xing [39, 44, 45], or by lifting the NS sector of the A1
theory to the large Hilbert space [46], as will be reviewed in the next subsection.
To further simplify the eld rede nition in the Ramond sector let us take a closer look
at F
R. Consider the expression:
Canceling G^ 1 and G^ and projecting onto the 1-string output gives
On the other hand, we can substitute (3.68) for 11G^ 1, obtaining
The rst term on the right hand side is F
R. By writing
we can show that the second term on the right hand side is
~m2j0 21G^
1
NS
R
1
1
NS
:
21 =4 h 1 0 10 + 10 0 11i4
1
11G^ 1G^
1
= 11(IT He
= 11G^
1
1
NS
1
NS
NS
11G^ 1G^
11G^ 1G^
R
R
1
1
1
1
1
1
R
1
1
1
NS
NS
NS
1
NS
1
1
NS
R
R
1
R
1
1
1
1
NS
1
1
NS
NS
= ~m2j0 F
A + A
F
R
8The coproduct 4 acts on a group-like element as [45]
A straightforward generalization gives the formulas
1
4 1 A
B
4 1 A
1
1
1
1
A
1
1
A
:
+
+
1
1
0 1
1 A
1
C
C
1
1 A
1
0 1 ;
where in the last step we switched from degree to Grassmann grading.8 Equating (4.25)
We use the rst formula in the derivation of (4.28), and later the second formula in the derivation of (4.61)
and the calculation of (A.7) from (A.8).
(4.24)
HJEP08(216)
(4.25)
(4.26)
(4.27)
(4.28)
(4.29)
(4.30)
(4.31)
and (4.26) then implies
This can be interpreted as a recursive formula for F
Plugging this formula into itself implies
This is the same formula which de nes Fb b R, but with the \hats" missing. Since the eld
rede nition in the NS sector implies Ab = A , the eld rede nition in the Ramond sector
The Ramond elds are equal; there is no eld rede nition between them. This was
anticipated in [1] and is not surprising for the following reason. Since the Ramond
elds have
identical kinetic terms, we can assume a eld rede nition relating them takes the form
b R =
R + X fe2( e ; e ) + fe3( e ; e ; e ) + : : : ;
(4.32)
(4.33)
(4.34)
(4.35)
(4.36)
(4.37)
(4.38)
(4.39)
4.2
Equivalence of the actions
Here we demonstrate that the eld rede nition given by (4.37) and (4.38) relates the
theories at the level of the action, not just the equations of motion. Following the analysis
of [44, 46], this can be demonstrated by expressing the A1 action in the same form as the
where fe2; fe3; : : : are string products and the factor of X is needed to ensure that both elds
live in the restricted space. Since the interaction vertices of both theories are built out of
Q; ~ and the open string star product, it is natural to assume that the eld rede nition can
be constructed from these operations. The (n + 2)-product in the eld rede nition fen+2
must carry ghost number
n
1. Therefore it must contain at least n + 1 insertions of
~, since no other operations carry negative ghost number. This implies that fen+2 carries
at least picture n + 1, and fen+2( e ; : : : ; e ) must have picture greater than or equal to
However, consistency of the eld rede nition requires that fen+2( e ; : : : ; e ) carries picture
3=2. Therefore fen+2 must vanish, and the Ramond elds are equal.
We therefore conclude that the eld rede nition between the A1 theory and
WZWbased theory is
up to a gauge transformation of the form (4.23). It is important to note that the proposed
eld rede nition is consistent with the assumption that
R are in the small Hilbert
space. In the Ramond sector this is obvious. In the NS sector it follows from the fact that
A and Ab satisfy
See [44, 45].
A
A
A = 0;
A
b
A
b
Ab = 0:
R = F
R
~[A ; F
F
R =
R + ~[A ; F
F
R =
I
1
~adA
Ab = A ;
b R =
R;
1
1
n+1
k=0 j=0
e
+ X !( R; Mfn+2( NS; : : : ; NS; R; NS; : : : ; NS))
k t{imzes
+ X
X !( NS; Mfn+2( NS; : : : ; NS; R; NS; : : : ; NS; R; NS; : : : ; NS)) :
k t{imzes
j t{imzes
Many terms in these sums are redundant. In fact, using cyclicity we can write the sum in
the second line as 2=(n + 1) times the double sum in the third line. Therefore we have
n + 3 !e( e ; Mfn+2( e ; : : : ; e )) =
n n k
k=0 j=0
e
1
n + 3 e
!( NS; Mfn+2( NS; : : : ; NS))
X !( NS; Mfn+2( NS; : : : ; NS; R; NS; : : : ; NS; R; NS; : : : ; NS)):
k t{imzes
j t{imzes
WZW-based action, including the contribution from the Ramond sector. Let us explain
how this is done.
The (n + 3)-string vertex in the A1 action is
1
n + 3 !e( e ; Mfn+2( e ; : : : ; e )):
Let us expand e into NS and R components. Since the composite products multiply at
most two Ramond states, the expanded vertex takes the form
n + 3 !e( e ; Mfn+2( e ; : : : ; e )) =
!( NS; Mfn+2( NS; : : : ; NS))
1
n + 3 e
0
(4.41)
#
(4.42)
(4.43)
(4.44)
Next we introduce a one-parameter family of NS string elds
the boundary conditions
NS(t); t 2 [0; 1] subject to
NS(0) = 0;
NS(1) =
NS:
The (n + 3)-string vertex can be written as the integral of a total derivative in t:
1
n n k
k=0 j=0
factors 1=(n + 3) and 1=(n + 1):
n + 3 !e( e ; Mfn+2( e ; : : : ; e )) =
X !S( _ NS(t); Mfn+2( NS(t); : : : ; NS(t); R; NS(t); : : : ; NS(t); R; NS(t); : : : ; NS(t))) :
k t{imzes
j t{imzes
1
1
X
n n k
n+1 k=0 j=0
e
n + 3 !e( e ; Mfn+2( e ; : : : ; e )) =
X!( NS(t); Mfn+2( NS(t); : : : ; NS(t); R; NS(t); : : : ; NS(t); R; NS(t); : : : ; NS(t))) :
k t{imzes
j t{imzes
Acting d=dt produces n + 3 terms with _ NS(t) = d NS(t)=dt in the rst line, and in the
second term it produces n+1 terms with _ NS(t). All of these terms are related by cyclicity,
and therefore we can bring _ NS(t) to the rst entry of the symplectic form and cancel the
dt
d
1
dt n + 3 e
!( NS(t); Mfn+2( NS(t); : : : ; NS(t)))
dt !S( _ NS(t); Mfn+2( NS(t); : : : ; NS(t)))
(4.45)
Summing over the vertices, the action can therefore be expressed as
1
1
2 !e( e ; Q e ) + X
1
1
1
n + 3 !e( e ; Mfn+2( e ; : : : e ))
dt !S _ NS(t); 1(Mj0
+ !S
_ NS(t); 1mj2 1
1
NS(t)
1
1
NS(t)
Q)
1
1
R
1
NS(t)
1
NS(t)
:
0
R
We can absorb the NS kinetic term into the integral over t, obtaining
S =
1
2 !S(Y R; Q R) +
0
dt !S
_ NS(t); 1Mj0 1
+ !S
_ NS(t); 1mj2 1
1
NS(t)
R
1
1
1
NS(t)
NS(t)
R
1
1
NS(t)
Because this form of the action was constructed from the integral of a total derivative, it
only depends on the value of
NS(t) at t = 1.
Next it will be helpful to reformulate the theory in the large Hilbert space. We replace
NS with a new NS string eld
NS in the large Hilbert space according to
NS =
NS:
simplify this expression using coderivations and group-like elements:
On the right hand side we replaced !e with !S since only NS states are contracted. We can
n + 3 !e( e ; Mfn+2( e ; : : : ; e )) =
+ !S
_ NS(t); 1mn+2j2 1
0
dt !S
_ NS(t); 1Mn+2j0 1
1
NS(t)
R
1
1
NS(t)
1
NS(t)
R
1
1
NS(t)
(4.46)
:
(4.47)
(4.48)
(4.49)
NS(t); t 2
(4.50)
:
(4.51)
1
S =
where
is in the large Hilbert space and
is in the small Hilbert space. Next we use the
identity [44, 46]
!(B; C) = !
1H
1
1
A
B
1
1
A
; 1H
1
1
A
C
1
1
A
;
(4.52)
The new
eld
NS is degree odd (because it is Grassmann even) and carries ghost and
picture number zero. We also introduce a corresponding family of string elds
[0; 1] such that
NS(t). Plugging into the action gives
S =
1
2 !S(Y R; Q R) +
0
dt !L
+ !L
_ NS(t); 1mj2 1
_ NS(t); 1Mj0 1
1
NS(t)
R
1
1
NS(t)
1
NS(t)
R
1
1
NS(t)
Here we replaced the small Hilbert space symplectic form with the large Hilbert space
symplectic form using the relation
!S(
; ) = !L( ; );
where B and C are string elds, A is a degree even string eld, and the cohomomorphism
H^ is cyclic with respect to !. In the current application we identify
A !
NS(t);
H ! G^ ;
^
! ! !L:
Note, in particular, that G^ is cyclic with respect to the large Hilbert space symplectic form
when it receives no Ramond inputs. Thus we can rewrite the action as follows:
(4.53)
(4.54)
(4.55)
1
NS(t)
(4.56)
; 1G^ Mj0 1
1
NS(t)
R
1
1
NS(t)
: (4.57)
(4.58)
(4.59)
S =
2 !S(Y R; Q R)
0
0
dt !L
dt !L
1G
1G
1
1
1
1
We can simplify the term with Mj0 by writing
The term with mj2 can also be simpli ed using
1
1
NS(t)
= mj2 1
1
1mj2 1
NS(t)
1
R
NS(t)
1
R
1
1
NS(t)
1
1
NS(t)
1G
1
NS(t)
1
R
1
1
1
1Mj0 1
NS(t) ;
1
1
NS(t) ; 1G
R
1
1
1
NS(t)
1
NS(t)
1Mj0 1
1
NS(t)
1
1
NS(t)
= Mj0 1
1
NS(t)
Therefore, we have
S =
2 !S(Y R; Q R)
1
0
Z 1
0
Now using
dt !L
dt !L
1G
1G
1
1
1
1
NS(t)
NS(t)
NS(t)
R
1
NS(t)
1
1
R
1
1
1
1
1
R
1
NS(t)
;
NS(t)
NS(t)
1
1G^ Mj0 = 1QG^ = Q 1G^ ;
1G^ mj2 = 1m2j2G^ = m2 22G^ ;
NS(t)
NS(t)
1
1
1
1
1G
1G
1
1
m2 22G^
1
A (t)
1G
1G
1
1
0
1
1
NS(t)
NS(t)
Using 22 =4 [ 11 0 11]4, one can show that
22G^
1
where
R
1
NS(t)
R
1
NS(t)
F (t) R
F (t) R ; (4.61)
Switching from degree to Grassmann grading, the action is therefore expressed as
F (t) R
1G
1
1
NS(t)
R
1
1
NS(t)
:
S =
h
Y R; Q RiS
dt hAt(t); QA (t) + (F (t) R)2iL;
we further obtain
S =
2 !S(Y R; Q R)
1
0
0
dt !L
dt !L
1
NS(t)
which is the eld rede nition anticipated in the previous subsection.
b R =
1
1
NS(t)
1
NS(t)
NS(t)
; Q 1G
NS(t)
NS(t)
: (4.60)
(4.65)
(4.66)
(4.67)
(4.68)
where following [44, 46] we de ne the potentials by
_ NS(t)
1
1
NS(t)
Thus the A1 action is expressed in the same form as (4.4) but with the \hats" missing.
Now we can show that the action of the A1 theory is related to the action of the
WZW-based theory by eld rede nition. We postulate that the two theories are related by
Equating the t-potentials provides an invertible map between
automatically equates the -potentials [46]:
NS(t) and bNS(t), and
Abt(t) = At(t);
b R =
R:
Ab (t) = A (t):
With these identi cations it is identically true that the actions (4.4) and (4.63) are equal.
Moreover, since the A1 action is only a function of
identi cation (4.66) is equivalent to
NS(t) =
NS(t) at t = 1, the
In this paper we have constructed the NS and R sectors of open superstring eld theory
realizing a cyclic A1 structure. This means, in particular, that we have an explicit solution
of the classical Batalin-Vilkovisky master equation,
fS; Sg = 0;
(5.1)
after relaxing the ghost number constraint on the NS and R string elds. Therefore, for
the purpose of tree level amplitudes we have a consistent de nition of the gauge- xed path
integral, and for the rst time we are prepared to consider quantum e ects in superstring
eld theory.
However, the absence of explicit closed string elds and the appearance of spurious
singularities at higher genus may make quantization subtle. Therefore it is desirable to
give a construction of superstring
eld theory realizing a more general decomposition of
the bosonic moduli space than is provided by the Witten vertex. This in turn is closely
related to the generalization to heterotic and type II closed superstring
eld theories.
The appropriate construction of NS actions and Ramond equations of motion is described
in [37, 38], and in principle all that is needed is to implement cyclicity. For example, in
the closely related open string eld theory with stubs [37, 38], it is not di cult to see that
the gauge products with one Ramond output and zero picture de cit should be de ned by
(n r+1)
(2r + 1 Ramond inputs);
(5.2)
so that the equations of motion are consistent with the projection onto the restricted
space in the Ramond sector. However, a full speci cation of the vertices requires many
additional gauge products of varying Ramond numbers and picture de cits. Solving the
entire recursive system of products consistent with cyclicity is a much more challenging
problem, which we hope to consider soon.
Acknowledgments
T.E. would like to thank S. Konopka and I. Sachs for discussion. The work of T.E. was
supported in part by the DFG Transregional Collaborative Research Centre TRR 33 and
the DFG cluster of excellence Origin and Structure of the Universe. The work of Y.O.
was supported in part by a Grant-in-Aid for Scienti c Research (B) No. 25287049 and
a Grant-in-Aid for Scienti c Research (C) No. 24540254 from the Japan Society for the
Promotion of Science (JSPS).
A
Gauge invariance in Sen's formulation
In Sen's formulation of the Ramond sector [10, 11], the action does not realize a cyclic A1
structure in the standard sense. Therefore it is worth explaining why it is gauge invariant.
where e and e are degree odd gauge parameters in the small Hilbert space, at ghost
the action is
HJEP08(216)
!S( e ; GQ e ) + !S( e ; Q e ) + !S( e ; Q e ) + !S
e ; 1be
:
1
1
The gauge parameter e immediately drops out since Qe always appears in the
symplectic form contracted with a BRST invariant state. Substituting the in nitesimal gauge
transformation then gives
The rst and second terms cancel upon using the BPZ even property of G and converting
be into Mf
Q. In the last term we replace 1Mf with 1Q + G 1be:
Next use the BPZ even property of G and again convert be into Mf
; GQ e
; Q e
1Mf
1Mf
1
1
1
1
e
e
1
1
1
1
; Q e
; 1be
1
; Q e
Qe +G 1be
1
; 1be
The in nitesimal gauge transformation can be written in the form
e = Qe + 1be
e = 1Mf
1
1
1
1
1
(A.1)
1
(A.4)
(A.6)
1
1
1
1
1
e
e
e
1
1
1
1
e
1
; 1Mf
1
1
e
e
1
1
e
1
1
1
+ !S
1
e
1
1
{ 38 {
Using cyclicity of be we can rewrite the second term as
1be
1
1
e
1
1
This follows from the relation
0 = h!Sj 2be
+ !S
1be
1be
1
1
1
1
S = !S
1be
1
1
e
1be
1
1be
1
1
S = !S
= !S
; 1Qbe
; Q e
+ !S
Qe; 1be
; 1(Mf
Q)
1be
1
1
= !S
e
; 1be
1
1
1
1Mf
1
1
1
1
1
e
1
Q:
; 1Mf
1Mf
1
1
1
1
1
e
variation of the action produces
1 0 1 4 and acting with the coproduct. Therefore the gauge
= !S
= 0;
; 1(Qbe + beMf)
+ !S
1
1
1Mf
1
1
1
which vanishes as a consequence of (3.121).
Open Access.
Attribution License (CC-BY 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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