Relating Berkovits and A ∞ superstring field theories; large Hilbert space perspective
HJE
superstring eld theories;
Theodore Erler 0 1 2
0 Theresienstrasse 37 , 80333 Munich , Germany
1 Arnold Sommerfeld Center, LudwigMaximilians University
2 Therefore, the eld strength F
We lift the dynamical eld of the A1 superstring eld theory to the large Hilbert space by introducing a gauge invariance associated with the eta zero mode. We then provide a eld rede nition which relates the lifted eld to the dynamical eld of Berkovits' superstring eld theory in the large Hilbert space. This generalizes the eld rede nition in the small Hilbert space described in earlier works, and gives some understanding of the relation between the gauge symmetries of the theories. It also provides a new perspective on the algebraic structure underlying gauge invariance of the WessZuminoWittenlike action.
String Field Theory; Superstrings and Heterotic Strings

Relating
Berkovits and
A
1 Introduction
Recap
3.1
3.2
3.3
3.4
4.1
4.2
4.3
A1 action in the large Hilbert space
Potentials and eld strengths
Variation of the action and gauge invariance
Higher potentials and the MaurerCartan equation
Little potentials
4
Field rede nition
From A1 to Berkovits
From Berkovits to A1
Okawa's approach
5
Mapping gauge invariances
A Some computations involving cyclicity
A.1 Proof of (3.13)
A.2 Proof of (3.78)
A.3 Proof of (3.126)
like open superstring eld theory of Berkovits [3, 4] and a new form of open superstring
eld theory based on A1 algebras [
5
].1 The rst issue one encounters in this regard is that
the A1 theory uses a string eld in the small Hilbert space, while the Berkovits theory uses
a string eld in the large Hilbert space. The large Hilbert space comes with an additional
gauge symmetry associated with the eta zero mode, on top of the usual gauge symmetry
associated with the BRST operator. In earlier works, this discrepancy was resolved by
xing the eta part of the gauge invariance [10], so that the remaining degrees of freedom of
the Berkovits theory could be described by a single string eld in the small Hilbert space.
Then one can compare the gauge xed eld of the Berkovits theory to the string eld of
the A1 theory.
Here we take a complementary approach. Instead of partially gauge xing the Berkovits
theory, we lift the eld of the A1 theory to the large Hilbert space, producing a new gauge
1For corresponding investigations in heterotic string eld theory [6, 7], see [8, 9].
{ 1 {
symmetry associated with the eta zero mode. This can be done as follows. One substitutes
the original string eld
Hilbert space according to
A in the A1 action with a new dynamical eld
A in the large
where
=
0 is the eta zero mode. The lifted A1 theory automatically possesses an
additional gauge symmetry
A =
A;
0A =
A +
A;
(1.1)
(1.2)
since the action only depends on
A in the combination
A. Therefore we can search
for a eld rede nition relating the lifted
eld
A to the string
eld
B of the Berkovits
theory. No gauge
xing is required. The eld rede nition we propose comes in the form
of a \Wilson line" which relates a path through
eld space in the Berkovits theory to
a path through
eld space in the lifted A1 theory. When the actions are expressed in
WessZuminoWittenlike form, this reduces their equivalence to an identity.
One advantage of this approach is that it gives a clearer understanding of the relation
between the gauge symmetries of the two theories.
This question is more di cult to
study in the small Hilbert space since the partially gauge xed Berkovits theory does not
exhibit a cyclic A1 structure.2
We show that the
gauge transformations of the two
theories map into each other, while the BRST gauge transformation in one theory maps
into a combination of BRST, , and trivial gauge transformations in the other. Another
consequence of our analysis is a new perspective on the algebraic structure underlying
the WessZuminoWittenlike (WZWlike) action. We show that the \potentials" which
appear in the WZWlike action are generally part of a hierarchy of higherform potentials
which together provide a solution to a certain MaurerCartan equation. MaurerCartan
gauge transformations implement eld rede nitions and relate equivalent realizations of
the WZWlike action. In this sense, the MaurerCartan equation plays a role in the large
Hilbert space somewhat analogous to the role of cyclic A1 algebras in the small Hilbert
space. These results may be a useful step towards a better understanding of the role
of the large Hilbert space in superstring eld theory, and in particular the problem of
quantization [11{15].
2
Recap
whose conventions we follow.
In this paper, string
This section contains a repository of de nitions and formulae that we will need in our
calculations. See earlier works for a more extended introduction to the formalism, especially [2],
elds are always elements of the NeveuSchwarz state space H
of an open superstring quantized in the RNS formalism, including bc and bosonized
superconformal ghosts ; ; e [16]. A string eld A is in the small Hilbert space if A = 0,
otherwise it is in the large Hilbert space. The degree of a string eld A, denoted deg(A),
2The full nonlinear gauge invariance of the partially gauge xed Berkovits theory has recently been
derived in [15], and it would be interesting to investigate the relation to the A1 gauge invariance.
{ 2 {
is de ned to be its Grassmann parity (A) plus one:
When using the degree grading, it is natural to work with a 2product m2 and a symplectic
form !L related, respectively, to Witten's open string star product and the BPZ inner
product by a sign:
deg(A) = (A) + 1 mod Z2:
m2(A; B) = ( 1)deg(A)A
B;
!L(A; B) = ( 1)deg(A)hA; BiL:
We will often drop the star when writing the star product, that is AB
A
B. The
subscript L denotes the BPZ inner product and symplectic form computed in the large
We will also encounter the symplectic form in the small Hilbert space
denoted !S(A; B). We will omit the subscript S or L for equations that hold for symplectic
forms in both the large and small Hilbert space. The de nition of the A1 theory requires
an operator built from the
ghost
I
dz
where f (z) is a function which is holomorphic in the neighborhood of the unit circle, is
BPZ even, and [ ; ] = 1.3 Using this operator, the symplectic form in the small Hilbert
space can be related to the symplectic form in the large Hilbert space by
where A and B are string elds in the small Hilbert space.
An nstring product cn(A1; : : : ; An) can be viewed as a linear map from n copies of
the state space H into one copy:
We write
cn : H
n
! H:
3In contrast to [2], in this paper we set the open string coupling constant to 1.
{ 3 {
cn(A1; : : : ; An) = cn A1
: : :
An;
where the right hand side is interpreted as the linear map cn acting on the tensor product
of states A1 : : :
An. The degree of cn is de ned to be the degree of its output minus the
sum of the degrees of its inputs. We consider the tensor algebra T H generated by taking
sums of tensor products of states:
T H = H
0
H
H
2
: : : :
1T H
A = A
1T H = A;
Here H
satisfying
0 consists of scalar multiples of the identity element of the tensor algebra 1T H,
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
(2.7)
(2.8)
(2.9)
n denotes the projection onto the nstring component of the tensor algebra and
I n is the identity operator on H
n
products of states as follows. If bk;m is a linear map H
`
n, then their tensor product is de ned to acts as
. The tensor product of linear maps acts on tensor
m
! H
k and c`;n is a linear map
: : :
Now suppose we have a list of degree even products H0; H1; H2; : : :. With this data we can
de ne an operator on the tensor algebra called a cohomomorphism
H^ = 0 + X
1
1
X
`=1 k1;:::;k`=0
(Hk1
: : :
Hk` ) k1:::+k` :
A typical example of a cohomomorphism is the identity operator on the tensor algebra IT H,
which is de ned by taking H1 = I and the remaining Hk to vanish. Given a degree even
string eld A, we can de ne an element of the tensor algebra called a grouplike element :
1
1
A
1T H + A + (A
A) + (A
A
A) + : : : :
In our calculations we will often need to act coderivations and cohomomorphisms on
grouplike elements. We note the formulas
product D1, a 2string product D2 and so on so that all products are either degree even or
degree odd. From this data we de ne an operator on the tensor algebra called a coderivation
D =
1
X
`;m;n=0
(I `
copies of the state space into complex numbers:
Likewise, a cohomomorphism H^ is cyclic if it satis es
which is graded antisymmetric upon interchange of its arguments:
!(A; B) =
( 1)deg(A)deg(B)!(B; A):
We write h j
! A
symplectic form ! if its coderivation cn satis es
B = !(A; B). An nstring product cn is cyclic with respect to the
This summarizes most of what we need from the coalgebra formalism. At the margins, a few
computations are helped by introducing the coproduct. We will review this in appendix A.
Now let's review the A1 superstring
eld theory. The dynamical string eld
A is
in the NeveuSchwarz (NS) sector,4 is degree even, lives in the small Hilbert space, and
carries ghost number 1 and picture number
1. The action is
1
1
1
multistring products in the small Hilbert space which satisfy the relations of a cyclic A1
algebra. The products Q; M2; M3; : : : de ne a coderivation
M = Q + M2 + M3 + : : : :
The statement that the products are in the small Hilbert space can be expressed by the
equation5
where
is the coderivation corresponding to the eta zero mode. The statement that the
products form an A1 algebra is expressed by the equation
In addition, the products form a cyclic A1 algebra because
4In this paper we only discuss the NS sector. The generalization to the Ramond sector [17{21] will be
considered in [22].
5Commutators of products and coderivations are always graded with respect to degree. Commutators of
string
elds, with the multiplication de ned by Witten's open string star product, are always graded with
respect to Grassmann parity.
h!j : H
2
! C:
h!j 2cn = 0:
{ 5 {
eld theory is that the coderivation M can be
related to Q using a similarity transformation in the large Hilbert space. The similarity
transformation is provided by an invertible, cyclic cohomomorphism G^
satisfying [
1, 5
]
h!Lj 2G^ = h!Lj 2
M = G^ 1QG^ ;
m2 = G^
G^ 1
;
(2.26)
(2.27)
(3.1)
(3.2)
(3.3)
(3.5)
(3.6)
A1 action in the large Hilbert space
In this section we reformulate the A1 superstring eld theory by replacing
A in the small
Hilbert space with a new dynamical eld
A in the large Hilbert space via the substitution
derivation follow [1], but we will make some re nements.
The nstring vertex in the A1 action takes the form
where m2 is the coderivation corresponding to the open string star product m2. The
construction of G^ requires the operator
in (2.4), and is described in [
5
].
dt 2 !S
n !S( A; Mn 1( A; : : : ; A)):
A(0) = 0;
as an integral of a total derivative with respect to t:
A(t); Q A(t) +
A(t); M2( A(t); A(t)) + : : : :
(3.4)
Acting the tderivative on the nstring vertex produces n terms containing a factor of
_ A(t) = d A(t)=dt. Since the vertices are cyclic, each of these terms are equal, canceling
the factor of 1=n. Using cyclicity to place _ A(t) in the rst entry of the symplectic form,
we can therefore write the action
which can be written more compactly as
1
3 !S
{ 6 {
_ A(t); 1M
1
1
A(t)
:
Since this form of the action was obtained from the integral of a total derivative, by
construction it only depends on the value of
A(t) at t = 1.
The next step is to lift to the large Hilbert space by making the substitution
A(t) is an interpolating 1parameter family of string elds subject to the boundary
and
A is the new dynamical string eld in the large Hilbert space. The new eld
degree odd (but Grassmann even) and carries ghost and picture number zero. The action
HJEP02(16)
where
conditions
becomes
(3.7)
(3.8)
A is
(3.9)
(3.10)
(3.11)
(3.12)
(3.13)
where the gauge parameter
A is degree even, ghost number
1 and picture number 1.
The in nitesimal BRST gauge transformation is
Q A = 1M
1
1
A
A
1
1
A
;
where the gauge parameter
A is degree even, ghost number
1 and picture 0. Acting
on both sides of this equation produces the standard A1 gauge transformation of
symplectic form is in the small Hilbert space. Therefore in the rst entry of the symplectic
form we can replace
with
+
= 1 at no cost. We can therefore write
dt !L
_ A(t); 1M
1
1
A(t)
;
where we replaced the small Hilbert space symplectic form by the large Hilbert space
symplectic form. The action only depends on the value of
A(t) at t = 1 (in fact, it only
depends on the value of
A(t) at t = 1), and has a new gauge invariance related to the
eta zero mode
A
!
1D1
; 1D2
1
1
A
= !
To simplify further, recall from [1] that a cyclic cohomomorphism H^ satis es the identity
{ 7 {
where A is a string eld and D1 and D2 are arbitrary coderivations. We provide another
proof of this identity in appendix A. Since the cohomomorphism G^ is cyclic with respect
to !L, this identity allows us to write
!L
string product. We can further write
This is the WessZuminoWittenlike action for the lifted A1 theory. Note that with this
de nition of At and A we will not be able to express the action in the standard WZWlike
form
1
2
SA 6=
A similar obstruction prevents the action of heterotic string eld theory from being
expressed in standard WZWlike form [6]. The action (3.20) turns out to be more general,
and will be the basis of our computations.
The above embedding of the A1 theory into the large Hilbert space is trivial  the
action only depends on the large Hilbert space eld in the combination
A. One might ask
whether there is a more interesting extension of the A1 theory to the large Hilbert space. In
this regard it is useful compare to the partially gauge xed Berkovits theory [2, 10], whose
extension to the large Hilbert space should apparently be Berkovits' open superstring eld
theory. If
B is the eld of the partially gauge xed Berkovits theory, we can attempt
to replace it with a eld
B in the large Hilbert space in the same way as we did for the
A1 theory:
B =
B ?
(3.22)
However, the resulting theory in the large Hilbert space is not Berkovits' superstring eld
theory. To get the Berkovits theory we must do something more re ned. First we start with
the potentials expressed in terms of the star product, , and
B. Then, for every instance
where
operates on
B, we should substitute
B, and for every other case we
1
A(t)
; Q 1G
^
1
A(t)
The action (3.12) therefore becomes
Now we de ne \potentials"
^
1G
1
At
A
1
A(t)
^
1G
^
1G
1
1
The potential At is degree odd but Grassmann even. The potential A is degree even but
Grassmann odd. Switching to the Grassmann grading the action is therefore expressed
0
1
1
should substitute
B. The resulting potentials de ne the WZWlike action for
Berkovits' superstring eld theory. In principle, we can follow the same procedure for the
A1 theory, producing a string
eld theory in the large Hilbert space with a nonlinear
gauge invariance. However, for the Berkovits theory this procedure has a special property:
it allows one to eliminate all instances of
from the partially gauge xed action, so that
the theory in the large Hilbert space can be expressed solely in terms of Q; and the star
product. In the A1 theory the same procedure does not eliminate all insertions of , since
in some cases
acts on products of elds rather than the eld itself. The upshot is that we
do not know of a nontrivial embedding of the A1 theory in the large Hilbert space which
is more \natural" than the trivial one. The trivial embedding is su cient for our purposes,
and will be the focus for the remainder of the paper.
3.1
Potentials and eld strengths
The structure of the WZWlike action can be understood in terms of the properties of
potentials and eld strengths. Suppose that we have a collection of graded derivations of
the open string star product which commute. We write the derivations @I with I ranging
over an index set, and they satisfy
where the commutator [; ] is graded with respect to Grassmann parity. In particular, if @I
eld AI with the same Grassmann parity, ghost and picture number. We refer to AI as
the potential corresponding to the derivation @I . The eld strength is de ned
FIJ
where (I) denotes the Grassmann parity of @I , and the commutator of string
elds is
computed with the star product and is graded with respect to Grassmann parity. The eld
strength is graded antisymmetric:
FIJ =
( 1) (I) (J)FJI :
Note that diagonal elements of the eld strength do not necessarily vanish. If the derivation
AI
AI );
(I) = 1 mod Z2:
It is useful to de ne a gauge covariant derivative
We have the identities
and
rI
[AI ; ]:
rJ AI + FIJ
[rI ; rJ ]
=
[FIJ ; ]:
{ 9 {
In particular, covariant derivatives commute if the associated eld strength vanishes.
(3.23)
(3.24)
(3.25)
(3.26)
(3.27)
(3.28)
(3.29)
When computing the variation of the WZWlike action we need four derivations of the
star product, with associated potentials:
The variational derivative
denotes an arbitrary variation of the interpolating eld
In the lifted A1 theory, the potentials are naturally de ned by
where in the last equation the string eld aQ(t) is de ned
aQ(t)
A comment about notation: the potentials, eld strengths, and related objects are functions
of the interpolating variable t. To avoid clutter, we will often leave this implicit. When
the dependence is not explicitly indicated, we will always assume that the interpolating
variable has been set equal to t. So, for example, A should be interpreted as A (t), and r
should be interpreted as r (t). The exception to this rule will be the dynamical string eld
and gauge parameters, where the dependence on t is always indicated except when t = 1.
The key property of the potentials (3.34){(3.37) is that the associated eld strengths
vanish along the
As we will review in a moment, this property implies the expected formula for the variation
of the WZWlike action. Therefore, the vanishing of these eld strengths is the basis for the
claim that the lifted A1 action can be written in WZWlike form. It is not di cult to show
that these eld strengths vanish by direct substitution of the provided de nitions [1], but
it will be helpful to give a slightly more general argument. Suppose the list derivations @I
includes and other derivations which we denote collectively as @i (with a lower case index):
F
F
= 0;
= 0;
Ft = 0;
FQ = 0:
For example, @i should include Q; d=dt and . We assume that the @is commute among
The string eld ai(t) will be called the little potential, and is de ned
where the coderivation Di is the image of @i under mapping with G^ :
ai(t) = ( 1)deg(i) Z t
= ( 1)deg(i) Z t
1Di
1
ds 1Di
d
ds
ds 1Di
1
1
1
A(t)
1
:
A(s)
1
A(s)
;
Now let us demonstrate that the eld strength Fi vanishes. Compute
and @i is the coderivation corresponding to @i. This de nition agrees with aQ(t) given
in (3.38), while in the t and
directions it simpli es to
since d=dt and
happen to commute with G^ . The little potential satis es the identity
and are derivations of the open string star product. By a
similar argument it follows that [Di; Dj ] = 0. Now act
on the little potential ai and
1
A(t)
1
where deg(i) = (i) is the degree of @i. To see that this identity holds, note that
commutes
r Ai = 1(
= 1G
^
= 1G
^
1
1
m2)G^
1
1
1
A(t)
A(t)
1
A(t)
ai(t)
ai(t)
ai(t)
1
1
1
1
1
A(t)
A(t)
;
:
(3.44)
(3.45)
(3.46)
(3.47)
(3.48)
(3.49)
(3.50)
(3.51)
(3.52)
(3.53)
(3.54)
(3.55)
r Ai = ( 1)deg(i)
^
1G
1
1G^ Di
1
1
1
A(t)
1
1
A(t)
hF t; QA iL =
hF t; r AQiL;
that F
vanishes is a di erent but straightforward computation, given in [1].
( 1) (i)
r Ai vanishes as claimed. The proof
large Hilbert space with the same quantum numbers as
A. The action is de ned by the
potentials
B (t)
B (t)
( e B(t))e
( e B(t))e
B(t);
B(t);
Bt(t)
BQ(t)
d
dt
e B(t) e
B(t);
(Qe B(t))e
B(t):
Bi(t)
B(t):
All eld strengths in the Berkovits theory vanish. By contrast, except along the
direction,
eld strengths in the lifted A1 theory do not vanish. However, they satisfy the Bianchi
identity:
rI FJK + ( 1) (I)( (J)+ (K))
rJ FKI + ( 1) (K)( (I)+ (J))
rK FIJ = 0:
Suppose we choose @I to be . Then the second two terms in this equation vanish, and we
nd that the nonvanishing eld strengths are covariantly constant in the
gauge invariance.
3.2
Variation of the action and gauge invariance
Consider the variation of the integrand of the WZWlike action:
hAt; QA iL = h At; QA iL + hAt; Q A iL;
= hrtA ; QA iL + hF t; QA iL + hQr A ; AtiL;
where in the second step we applied (3.28) to express the variation in terms of A . The
term with the eld strength actually vanishes because
(3.56)
(3.57)
(3.58)
(3.59)
(3.60)
(3.61)
(3.62)
(3.63)
(3.64)
(3.65)
and the eld strength is annihilated by r . Pulling Q and the r
o the A in the last
term of (3.64), we therefore have
hAt; QA iL = hrtA ; QA iL
hA ; r QAtiL:
Now apply (3.28) again in the second term to replace Q by a t derivative:
hAt; QA iL = hrtA ; QA iL
hA ; r (rtAQ + FQt)iL:
Again the eld strength does not contribute since it is annihilated by r . Moreover, we
HJEP02(16)
can commute r
and rt since Ft = 0. Therefore
hAt; QA iL = hrtA ; QA iL
hA ; rtr AQiL;
= hrtA ; QA iL + hA ; rtQA iL;
d
=
dt hA ; QA iL:
SA =
hA ; QA iLjt=1 :
Integrating t from 0 to 1, we obtain
This is the expected variation of a WZWlike action.
We would now like to use this formula to prove that the lifted A1 theory has the
expected gauge invariances. First consider the
gauge invariance (3.10). For this
transformation, the potential A takes the form
(3.66)
(3.67)
(3.68)
(3.69)
(3.70)
(3.71)
(3.72)
where for later reference we de ne
Plugging in we nd
A jt=1 = 1G
^
= 1G
^
= 1(
= r
1
1
1
1
A
m2)G^
A
1
Bjt=1 ;
B
^
1G
1
1
A
1
A
A
A
A
1
1
1
1
A
A
A
;
;
1
1
1
A
:
SA =
hr
B; QA iLjt=1 = 0;
1
A
;
since QA is annihilated by r . This proves that the action is invariant under this gauge
symmetry. Next consider the BRST gauge invariance (3.11). For this transformation, the
potential A takes the form
A Q jt=1 = 1G
^
where we substituted
h!Lj
^
1G
^
1G
1
1
^
1G
^
1G
1
1
1
1
1
A
1
A
QA jt=1 = 1G
^
A
1
1
^
1G
1
A
1
1
A
1M
1
^
1G
1
1M
1
1
1
1
A
A
A
1
1
1
A
;
1M
1M
1
1
1
A
1
A
1
;
1
;
A
A
(3.73)
(3.74)
:
(3.75)
where we have de ned
The Q B term vanishes when contracted with QA jt=1 since Q is nilpotent. Substituting
the
B term into the variation, while switching to the degree grading, we obtain the
1M
1
A
1
1
A
;
(3.77)
1
1
1M
A
1
A
A
1
1
1
1
1
1M
1
1
1
A
1
1
A
A
A
A
A
1
1
1
1
A
A
A
1
1
1
1
A
A
;
(3.76)
QSA = 0:
and the action contains the expected BRST gauge symmetry.
3.3
Higher potentials and the MaurerCartan equation
We would now like to get a better understanding of the nonvanishing eld strengths of the
lifted A1 theory. We already know that they are covariantly constant in the
direction.
But since F
vanishes, the covariant derivative r is nilpotent. In fact, it turns out that
the eld strengths are trivial in the r cohomology:
(3.78)
(3.79)
(3.80)
(3.81)
1
A(t)
(3.82)
More generally,
Aij (t)
1G^ ( 1)(deg(i)+1)(deg(j)+1)
+
+
1
1
1
1
A(t)
A(t)
where the sign is chosen for later convenience. The string eld Aij will be called the
2potential, and is given by the formula
and used graded antisymmetry of the symplectic form. Equation (3.76) can be further
simpli ed to
QSA =
We will prove this in appendix A. Note that we can drop the factor of G^ from (3.78) because
G^ is cyclic. Then the projector
2 acts directly on the object in parentheses. However,
the object in parentheses has no twostring component  it contains at minimum a tensor
product of three string elds. Therefore
Ft =
FtQ =
F Q =
FQQ =
r At ;
r AtQ;
r A Q;
r AQQ:
Fij = ( 1)( (i)+1) (j)+1
r Aij ;
1
1
1
1
A(t)
1
1
A(t)
A(t)
ai(t)
:
ai(t)
1
1
A(t)
1
1
A(t)
The string eld aij (t) will be called the little 2potential, and is de ned
Interestingly, the 2potential Aij looks like a twoindex generalization of the potential Ai
given in equation (3.44), and the little 2potential aij looks like a twoindex generalization
of the little potential ai given in equation (3.45). Computing r Aij reproduces the eld
strength Fij as a result of the identity
0 = aij (t) + ( 1)(deg(i)+1)deg(j)
1Di
+ ( 1)deg(j)+1 1Dj
1
1
A(t)
1
1
ai(t)
A(t)
1
1
which is a kind of twoindex generalization of the identity (3.48). The 2potentials have not
played a role so far since they do not appear in the action. But there are other expressions
for the action where 2potentials do appear. For example, if we try to express the action
in the standard WZWlike form (3.21), we nd an additional term proportional to the
2potential AtQ:
1
2
SA =
A similar generalization of the standard WZWlike action has also been discussed in the
context of heterotic string eld theory [6].
It turns out that the story does not end with 2potentials. By factoring r
Bianchi identity for the nonvanishing eld strengths, we learn that the 2potentials must
satisfy the identity:
riAjk + ( 1) (j)( (i)+ (k))+ (j)+ (k)rj Aki + ( 1) (i)( (j)+ (k))+ (i)+ (j)
rkAij
+ ( 1)( (i)+1)( (j)+ (k)+1)
r Aijk = 0;
where Aijk is a new object called the 3potential. Acting on this equation with the covariant
derivative ri and symmetrizing, one can further introduce a 4potential, and so on. To
clarify the structure of the hierarchy, it is helpful to invoke the language of di erential
forms.6 For each derivation @i we formally introduce a corresponding basis 1form:
6The author would like to thank S. Konopka for discussion which clari ed the nature of this hierarchy.
(3.83)
HJEP02(16)
(3.86)
(3.87)
Note that we do not introduce a 1form dual to . The basis 1forms carry Grassmann
parity and degree
and ghost and picture number
(dxi)
deg(dxi)
(i) + 1;
mod Z2;
gh(dxi)
1
picture(dxi)
1
We consider the algebra of \string eldvalued" di erential forms, de ned in the obvious
way by tensoring the wedge product with the open string star product, with the appropriate
signs from (anti)commutation of string elds and forms. We de ne the exterior derivative
1
2!
d
The exterior derivative is nilpotent and commutes with , since the @is commute among
themselves and with . Also, d acts as a derivation of the wedge/star product. Next, we
introduce di erential forms corresponding to the npotentials:7
A(0)
A ;
A(1)
dxiAi;
A(2)
dxi ^ dxj Aij ;
A(3)
dxi ^ dxj ^ dxkAijk; : : : :
1
3!
Note that A is a scalar, while Ai are components of a 1form. Therefore A
interpreted as the 0potential and Ai as the 1potential, and the pattern completes to
higher potentials. Accounting for the Grassmannality, ghost and picture number of the
basis 1forms, the npotentials and exterior derivative are Grassmann odd, carry ghost
number 1 and picture
1. On the basis of previous calculations it is easy to check that the
potentials satisfy
0 =
A(0)
A(0)
A(0);
0 = dA(0) + A(1)
0 = dA(1) + A(2)
0 = dA(2) + A(3)
The rst two identities are equivalent to the statement that the eld strengths vanish in the
direction. The third is equivalent to the statement that the nonvanishing eld strengths
are trivial in the r
cohomology, and the fourth is implied by factoring r
Bianchi identity. It is clear that these identities arise as components of a MaurerCartan
equation:
(d + )A
A
A = 0;
(3.96)
7The eld strength as de ned in (3.24) does not have the right symmetry properties in the odd directions
to de ne a 2form. This can be
xed by multiplying the eld strength by the appropriate sign, but we did
not bother since it would contradict the de nition used in previous papers. This is the origin of the sign
factor relating the 2potential and the eld strength in (3.81).
where A is given by the formal sum
A
A(0) + A(1) + A(2) + A(3) + : : : :
(3.97)
We will call A the multipotential. The MaurerCartan equation can be thought of as an
\equation of motion" which determines the potentials and their higherform descendants
as functionals of the interpolation
(t). Once we have solved this equation, we can write
down a WZWlike action. Actually, for this purpose we need to assume an additional
regularity condition: an arbitrary variation of the dynamical eld
produces an arbitrary
A at t = 1:
HJEP02(16)
Regularity Condition :
arbitrary
! arbitrary A jt=1:
(3.98)
This needs to be true, otherwise the variation of the WZWlike action (3.69) does not imply
the correct equations of motion. Thus, for example, A = 0 would not be a regular solution
for the purposes of de ning a WZWlike action. At least perturbatively, the potential B
of the Berkovits theory and the potential A of the lifted A1 theory are regular, since, to
leading order in the string eld, they are proportional to
B(t) and
A(t), respectively.
Berkovits' superstring eld theory provides a solution to the MaurerCartan equation
in the form
B = B + dxiBi:
Since the eld strengths vanish, the higher potentials can be set to zero. In the lifted
A1 theory the solution to the MaurerCartan equation is more interesting. We introduce
di erential forms corresponding to the little potentials:
dxi ^ dxj aij ;
a(3)
dxi ^ dxj ^ dxkaijk; : : : :
1
2!
1
3!
For convenience we have de ned the little 0potential to be a(0) =
A(t). All little
npotentials are degree even, ghost number 1 and picture
1 once we account for the basis
1forms. We de ne the little multipotential as the formal sum
and postulate a solution to the MaurerCartan equation in the form
a
a(0) + a(1) + a(2) + a(3) + : : : ;
A = 1G
^
1
1
a
:
Here the tensor algebra of \string eldvalued" di erential forms is de ned in the obvious
way by tensoring the wedge product of the basis 1forms with the tensor product of string
elds, with the appropriate signs from (anti)commutation of the string
elds and forms.
Note that (3.102) agrees with our earlier expressions for A ; Ai; Aij once we extract the
zero, one and two form components. The MaurerCartan equation for the multipotential
A translates into an equation for the little multipotential a:
1( + D)
1
1
a
= 0;
(3.99)
(3.100)
(3.101)
(3.102)
(3.103)
where D
dxiDi. Note that this produces (3.48) and (3.84) when we extract the 1form
and 2form components. The solution to this equation is not unique, but extrapolating
from the expressions for ai and aij , we propose a solution of the form
Z t
0
a(t) =
ds 1( + D)
1
1
a(s)
This de nes the little potentials and their higherform descendants recursively; the nform
component determines the little npotential in terms of products of little kpotentials for
k < n. To show that this formula solves (3.103), take the t derivative of the left hand side
of (3.103) and substitute
a
1
a
1
1
a
a
a
1
1
a
1
1
a(t)
:
Note that (D+ )2 vanishes since d and
are nilpotent and mutually commuting derivations
of the star product. Bringing the last two terms to the other side of the equation gives
0 =
d
dt 1( + D)
1
1( + D)
A
1
1
1
a
a
+ 1( + D)
1
1
1
a
1
a
Since 1( + D) 1 1 a vanishes at t = 0 (because
This is a rst order homogeneous di erential equation in the string eld 1( + D) 1 1 a .
A(t) vanishes at t = 0), it must vanish
for all t. Therefore (3.103) is satis ed, and (3.102) gives a solution to the MaurerCartan
equation.
One advantage of the MaurerCartan equation is that it gives a clearer understanding
of the symmetries implicit in the choice of potentials used to express the action in WZWlike
form. In particular, solutions can be modi ed by an in nitesimal \gauge transformation"
mcA = (d + )
is a sum of nform gauge parameters
(3.104)
(3.105)
1
1
1
1
a
;
1
1
a
a
:
(3.106)
1
1
a
(3.107)
(3.108)
(3.109)
In general
can be a functional of the interpolation
(t), and is Grassmann even, ghost
and picture number zero. Note that this \gauge transformation" alters the choice of the
potentials  not the dynamical string eld on which the potentials implicitly depend. It
is interesting to see how this transformation e ects the WZWlike action. For this purpose
it is enough to consider how the 0 and 1potentials transform:
mcA = r
(0);
mcAi = ri (0) + r
i
:
Following a similar computation as in the previous section, it is not di cult to show that
0
dthAt; QA iL = h
(0); QA iLjt=1:
From this we make two observations. First, while the 1form parameter
potentials, it does not change the action. Second, the 0form parameter
i alters the
(0) changes the
action in the same way as a variation of the eld. Thus the action after the transformation
is the same as the original action with the replacement
+
, where
is determined
!
by solving the equation
A =
(0):
Thus the MaurerCartan gauge transformation alters the WZWlike action at most by a
eld rede nition, and then only if the zeroform component of the transformation parameter
is nonvanishing. Furthermore, it is clear that a MaurerCartan gauge transformation of
the form
(0)jt=1 = Q B + r
B
will leave the action invariant. By equating this with A , this determines an in nitesimal
gauge transformation of the elds. In this sense, the MauerCartan equation plays a role
for the WZWlike action somewhat analogous to the role of cyclic A1 algebras in the A1
action: it is an algebraic structure that is covariant under eld rede nition and implies the
existence of a gauge invariant action.
The interpretation of the WZWlike action suggested by the MaurerCartan equation
is slightly di erent from the earlier interpretation in terms of potentials and eld strengths.
For example, from the MaurerCartan perspective A is a scalar, and so should not de ne
components of a eld strength. Nevertheless, the statement that the eld strengths vanish
in the
direction is a convenient way to characterize the essential information in the
MaurerCartan equation for the purposes of writing the action. We will translate back and
forth between both interpretations as convenient.
3.4
Little potentials
The action of the lifted A1 theory can be formulated in a di erent way using little
potentials. In (3.12) the lifted A1 action was expressed in the form
0
dt !L
(3.110)
(3.111)
(3.112)
(3.113)
Note that the little potentials satisfy
where the second equality follows from (3.48). Thus the A1 action can be expressed
This is reminiscent of a WZWlike action, but note that the little potentials do not come
from a solution to the MaurerCartan equation. Instead, the relevant equation for the little
HJEP02(16)
potentials is (3.103).
To make the description in terms of little potentials more familiar, we can de ne a few
objects by analogy with the potentials and eld strengths of the WZWlike action. First,
de ne the little potential in the
direction to be the little 0potential:
a
a(0) =
A(t):
fij
( 1)(deg(i)+1)deg(j)+1 aij :
Second, we de ne a \little eld strength:"
Unlike the WZWlike formulation, it does not make sense to de ne components of the little
eld strength in the
direction, for reasons that will be clear in a moment. The little eld
strength is graded antisymmetric,
;
0
dt !L(at; aQ):
Note that this last identity is not a special case of the previous one after choosing Di to
be
and setting fij to zero. This is why it doesn't make sense to de ne components of the
and constant in the
( 1)deg(i)deg(j)fji;
fij = 0:
Recall that the potentials and eld strengths satisfy (3.28)
Little potentials satisfy analogous identities which allow us to swap the index of a
coderivation with the index of the little potential. In particular, (3.84) can be expressed
1Di
1
1
a
aj
1
1
a
= ( 1)deg(i)deg(j)
1Dj
ai
1
1
a
+ fij ;
(3.122)
where switching i and j adds the little eld strength fij . Meanwhile (3.48) can be expressed
ai = ( 1)deg(i)
1Di
1
1
1
a
1
a
:
(3.115)
(3.116)
(3.117)
(3.118)
(3.119)
(3.120)
(3.121)
(3.123)
!L(at; aQ) = !L( a_ ; aQ) + !L a_ ; 1M
Using cyclicity of M one can show that
1
1
a
1
1
a
!L a_ ; 1M
a
1M
We prove this in appendix A. Therefore
!L(at; aQ) = !L( a_ ; aQ)
= !L( a_ ; aQ)
= !L( a_ ; aQ) + !L ( a_Q; a ) ;
d
=
dt !L(a ; aQ):
!L
!L
1M
1
d
dt 1M
1
1
a
1
a
Integrating from 0 to 1 gives the variation of the action
This implies that the equations of motion can be expressed
which using (3.123) is equivalent to
little eld strength in the
direction. Except for this small di erence, the formulation is
quite similar to that of the WZWlike action.
Now let's compute the variation of the action. Take the variation of the integrand:
(3.124)
!
;
(3.127)
(3.128)
(3.129)
(3.130)
(3.131)
1
1
a
1
1
a
a
a_
a_
; a
1
1
;
1
a
1
1
a
1
a
:
(3.125)
; a
:
(3.126)
; a
;
SA =
!L(a ; aQ)jt=1:
aQjt=1 = 0;
1M
1
= 0:
1
1
A
A
Q
1G
1
= 0:
!L(at; aQ) = !L( at; aQ) + !L(at;
= !L( a_ + f t; aQ) + !L at;
1M
1
1
a
1
1
a
+ f Q
where in the second step we used (3.122). Note that the identities (3.122) and (3.123)
simplify quite a bit in the case of the derivations
and d=dt since they commute with
the cohomomorphism G^ . We can drop the little eld strengths f t and f Q since they are
constant in the
direction (in fact f t already vanishes identically). Moving in the second
term onto the rst entry of the symplectic form and using (3.123) gives
This is the standard expression for the A1 equations of motion. By contrast the WZWlike
action gives the equations of motion in a \Berkovitslike" form
These two formulations are related by conjugation with the cohomomorphism G^ .
0
0
dt hAt; QA iL;
dt hBt; QB iL;
where the potentials Bt and B for the Berkovits theory were de ned in (3.60).
In earlier work, the relation between the Berkovits and A1 theories was described by
partially gauge xing the Berkovits theory to the small Hilbert space. In this context, the
eld rede nition between the theories was given by equating the respective
potentials [1]:
Unfortunately this condition is not enough to specify a eld rede nition between
B in the large Hilbert space. It leaves a residual ambiguity related to an overall
transformation. There are a number of ways to x this ambiguity. We will take a particular
approach which at rst seems orthogonal but has interesting implications. Instead of
equating the
potentials, we will equate the t potentials:
We now search for a eld rede nition relating the dynamical eld
theory to the dynamical eld
B of the Berkovits theory. This eld rede nition should
A of the lifted A1
transform the WZWlike action of the lifted A1 theory
into to the WZWlike action of the Berkovits theory
(4.1)
(4.2)
(4.3)
A and
gauge
(4.4)
(4.5)
(4.6)
(4.7)
A and
B at t = 1. In fact,
in a sense we will describe, this equation provides a eld rede nition between the entire
interpolations
Let us solve equation (4.4). Substituting Bt we obtain a di erential equation for
The solution is provided by a Wilson line
d
dt
e B(t) = At e B(t):
g(t2; t1) = P exp
ds At(s) ;
Z t2
t1
where P denotes the path ordered exponential in sequence of decreasing s. To emphasize
this, we have written g(t2; t1) so that the leftmost argument t2 is viewed as a later time
than the rightmost argument t1. Thus we have
e B(t) = g(t; 0) = P exp
ds At(s) ;
Z t
0
e B = g(1; 0) = P exp
ds At(s) :
0
This is our proposed eld rede nition between
A and
To make sense of this eld rede nition we must clarify an important point of
interpretation. At face value, the Wilson line (4.8) expresses
B as a function of the entire path
A(t), not only of
A at t = 1. Therefore we need to be more speci c about the meaning
of
A(t) when t 6= 1. We will assume
the dynamical eld
which is subject to boundary conditions
A(t) = fA(t; A);
fA(0; A) = 0;
fA(1; A) =
A(t) is given as some timedependent function of
HJEP02(16)
(4.8)
(4.9)
(4.10)
(4.11)
(4.12)
(4.13)
Using this, we can compute the proposed eld rede nition out to rst order in the elds.
To rst order, the t potential of the lifted A1 theory takes the form
At = f_1(t) A + higher orders:
Therefore, to rst order, the proposed eld rede nition takes the form
1 +
B + higher orders = 1 +
ds f_1(s) A + higher orders;
= 1 + f1(1)
f1(0)
A + higher orders:
(4.14)
Under this assumption, then indeed the Wilson line (4.8) expresses
B as a function of A.
However, for this to be a valid eld rede nition, it must be invertible. To simplify analysis
of this question, we will make an additional assumption about the interpolating function.
The interpolating function will have an expansion in powers of the string eld:
fA(t; A) = f0(t) + f1(t) A + f2(t) A
A + : : : ;
where the linear maps fn(t) : H
n ! H are nstring products. We will assume that the
zero string product in this expansion vanishes:
f0(t) = 0:
The boundary condition (4.10) implies f1(1) = I and f1(0) = 0. Therefore the proposed
eld rede nition is simply
B =
A + higher orders:
(4.15)
This is obviously invertible at rst order, and small corrections in higher powers of the
eld will not change this fact. Therefore, at least perturbatively, the Wilson line (4.8)
is a valid
eld rede nition between
A and
B. If we relax the assumption (4.12), the
proposed eld rede nition will map the vacuum con guration
A = 0 to a nite, pure
gauge con guration for
B. The question of invertibility in this case is more complicated,
and we will not consider it.
Now let's return to equation (4.7), which determines the Berkovits eld
B(t) when
t 6= 1. With a given choice of interpolating function for
A(t), equation (4.7) expresses
B(t) as a function of
A. In turn, we can express
A as a function
B by inverting the
eld rede nition, so in fact (4.7) gives an interpolating function for the Berkovits theory,
B(t) = fB(t; B);
which implicitly depends on the choice of interpolating function fA(t; A) of the lifted A1
theory. This is the sense that the equation At = Bt provides a \ eld rede nition" between
For the sake of being concrete, let us compute the eld rede nition between
A and
B
out to second order in the string
eld. Expanding e B and the path ordered exponential
to second order gives the expression
+ higher orders;
B =
A +
2 f1(s) A; f_1(s) A + 2 f_1(s) A; f1(s) A + f_1(s) A f1(s) A
where 2 is the gauge 2product of the A1 theory [
5
] and f1(t) is the 1string product in
the interpolating function (4.11). Let us restrict to a particular class of eld rede nitions
which can be written exclusively in terms of ; and the open string star product. This
implies that f1(t) can take the form
f1(t) = x(t)I + y(t) ;
where x(t); y(t) are numbervalued functions of t satisfying boundary conditions x(0) =
y(0) = 0 and x(1) = 1 and y(1) = 0. While there are an in nite number of possible choices
of x(t) and y(t), reparameterization invariance of the eld rede nition implies that most
choices are equivalent. By inspection of (4.17), it is clear that the only reparameterization
invariant quantity that can appear at second order is
C =
ds x(s)y_(s);
since the other possible combinations xx_ and yy_ are total derivatives which are xed by
boundary conditions. Explicitly, we nd that the eld rede nition takes the form
B =
A +
+ C
[
If we change the constant C, the eld rede nition will change by a term proportional to
+
1
3
[
1
3
1
2
1
2
A; A]:
(4.16)
1 2
(4.17)
(4.18)
(4.19)
(4.20)
(4.21)
One can check that this term is annihilated by , and therefore represents an
gauge
transformation. It is interesting to note that the eld rede nition is not completely arbitrary,
despite having a free parameter C. The most general eld rede nition at second order
constructed out of ; , and the star product actually has ve free parameters (only four if
we require that the elds are equal at linear order).
Perhaps the most signi cant appeal of this approach is that it gives the simplest
possible proof of equivalence of the actions. All we have to do is prove that the
are equal:
B = ( e B(t))e
B(t);
g(t; 0) g(t; 0) 1;
=
=
=
=
Z t
Z t
Z t
0
0
0
ds g(t; s)
ds g(t; s)
d
= A (t)g(t; 0)g(t; 0) 1;
= A :
At(s) g(s; 0) g(t; 0) 1;
d
A (s)
[A (s); At(s)] g(s; 0) g(t; 0) 1;
g(t; s)A (s)g(s; 0)
g(t; 0) 1;
Since At = Bt and A = B , the actions are identical.
According to the discussion of section 3.3, the eld rede nition must be equivalent
to a MaurerCartan gauge transformation of the potentials. Here we should note that
MaurerCartan gauge transformations do not transform the elds  rather they transform
the potentials, and therefore the action, while keeping the string eld
xed. But the net
e ect is equivalent to keeping the action
xed while transforming the string
eld. With
this understanding, the eld rede nition between the Berkovits and lifted A1 theories is
equivalent to a nite MaurerCartan gauge transformation
h
i
B0 = (d + )U U 1 + U BU 1
;
(4.22)
(4.23)
(4.24)
(4.25)
where B is the multipotential of the Berkovits theory and B0 is the transformed
multipotential, and the nite gauge parameter U is
Z t
0
U (t) = P exp
ds At[ B](s) e
B(t);
where At[ B](t) is the tpotential of the lifted A1 theory evaluated on the Berkovits string
eld. The net e ect of this transformation is to replace the grouplike element
parameterized by e B with the grouplike element parameterized as P exp hR0t ds At[ B](s)i. The
Berkovits action will then be replaced with the lifted A1 action
SB[ B] ! SA[ B];
after which we may as well rename
B as
A. Note that because U only has a zero form
component, the MaurerCartan gauge transformation does not generate expectation values
for the higher potentials. Therefore the transformed multipotential B0 gives a
representation of the WZWlike action for the lifted A1 theory where all eld strengths vanish.
So far we have described the eld rede nition between the Berkovits and lifted A1 theories
by equating the tpotentials. However, an equivalent characterization can be found by
equating the little tpotentials. This naturally leads to a formula for the eld rede nition
which inverts the Wilsonline of the previous section.
First we need to describe the little potentials of the Berkovits theory. They are de ned
implicitly by the formula
where the little multipotential b is a sum of little npotentials
B = 1G
^
1
1
b
b(0) + b(1) + b(2) + b(3) + : : : ;
which can be described using the basis 1forms dxi as
b(0) = b ; b(1) = dxibi; b(2) =
dxi ^ dxj bij ; b(3) =
dxi ^ dxj ^ dxk bijk; : : : : (4.28)
This is precisely analogous to (3.102) of the lifted A1 theory. We can invert (4.26) to
express b in terms of the multipotential of the Berkovits theory:
1
3!
Multiplying this equation by G^ 1 and projecting onto the 1string component of the tensor
algebra gives
Important special cases are
1
2!
1
1
B
b = 1G^ 1
=
1
= G^
1
1
1
1
B
B
B
B
1
1
;
1
1
1G^ 11 b ;
:
1
B
:
b = 1G^ 1
bt = 1G^ 1
;
;
:
(4.26)
(4.27)
(4.29)
(4.30)
(4.31)
(4.32)
(4.33)
(4.34)
(4.35)
Note that the higher little potentials of the Berkovits theory do not vanish, even though
the higher potentials do. The little multipotential satis es
by analogy with (3.103).
1( + D)
1
1
b
= 0;
hBt; QB i = !L
= !L
= !L
= !L bt; 1M
1G^ 1Bt
1G^ 1
1G^ 1
1
1
1
1
1
1
B
B
1
1
B
; 1G^ 1Q
1
1
1
1
1
B
B
1
B
; 1MG^ 1
; 1M
;
1
1
B
1
1
;
;
where we substituted the de nition of the little potentials. Next we use (4.35) to note that
Integrating t from 0 to 1 therefore expresses the Berkovits action in the form
This is the Berkovits action expressed in terms of little potentials.
Now it turns out that equating the tpotentials of the Berkovits and lifted A1 theories
is equivalent to equating the little t potentials. To see this, note that
1
A
At
1
1
A
=
1
1
B
Bt
1
1
B
:
where we used the fact that At = Bt implies A = B . The left hand side can be expanded
Next let us explain how to express the Berkovits action in terms of little potentials.
Consider the integrand
hBt; QB i = !L(Bt; QB );
= !L
1Bt
1
1
B
; 1Q
1
1
B
;
(4.36)
where Bt is the coderivation corresponding to Bt regarded as a zerostring product. Since
the cohomomorphism G^ 1 is cyclic, we can use (3.13) to write
1
0
Bt
Bt
;
1M
1
=
bQ:
SB =
dt !L(bt; bQ):
1
1
A
^
G
1
1
= G^
= G^
1
a
1
1
1
1
^
1G 1
1
^
1G 1
1
A(t)
1
at
a
at
1
1
a
1
A(t)
1
A(t)
;
1
1
=
a
:
1
B
1
at = bt:
A
At
1
Therefore
Multiplying this equation by G^ 1 and projecting onto the 1string component then implies
^
1G
1
1
A(t)
;
QQfA(t; A)
Q
Q
Q
s
3
T (t;0)
T (1;0) g(1;0)
A(t)
B(t)
+
k
Q
g(t;0)
Q
Q
fB(t; B) Q
Q
T (t2; t1)
ds bt(s):
and lifted A1 superstring eld theories.
Since at(t) = _ A(t) this equation is easily integrated to express
A in terms of
solution of this equation is de ned by the integral
Q
Z t2
t1
Z t
0
0
The interpolating eld of the lifted A1 theory is therefore,
and the eld rede nition from the Berkovits to the lifted A1 theory is
A(t) = T (t; 0) =
ds bt(s);
A = T (1; 0) =
ds bt(s):
which is the string eld we started with.
(4.44)
(4.45)
(4.46)
(4.47)
Similar to the previous section (but in reverse), this expresses
A as a function of an entire
path
B(t) in the Berkovits theory. For this to be a eld rede nition between
A and
B,
we assume that
B(t) is speci ed by an interpolating function fB(t; B) whose zerostring
product vanishes. Equation (4.45) determines the interpolating function fA(t; A) of the
lifted A1 theory in terms of the interpolating function fB( B; t) of the Berkovits theory.
We summarize the di erent elds, interpolations, and mappings between them in gure 1.
We should emphasize that this is the same as the Wilson line eld rede nition of the
previous section, but inverted. To see this, suppose we express the Berkovits interpolation
B(t) in terms of the lifted A1 interpolation
A(t) using the Wilson line (4.7). This allows
us to write Bt = At and B = A , and substituting into (4.45) gives
1
1
1
A (s)
1
At(s)
A(s)
;
;
A(t) =
ds 1G^ 1
=
=
=
Z t
0
Z t
Z t
0
0
ds 1G^ 1G^
ds _ A(s);
A(t);
Let us describe an unrelated proposal for the eld rede nition suggested by Y. Okawa.8
This approach does not consider the interpolations
B(t), but focuses on the
dynamical elds
A and
B at t = 1. The starting point is the condition
Since a jt=1 =
A, this gives
As mentioned before, this relation does not fully constrain the eld rede nition. However,
this can be remedied with a few additional choices. Equating the potentials is equivalent
to equating the little potentials:
Suppose we assume that the lifted A1
the above relation implies that the lifted A1
eld satis es
eld satis es the gauge condition
A = 0. Then
A = b jt=1
if
A = 0:
This is not quite a eld rede nition between
A and
B since it is not invertible  the eld
A always satis es the gauge condition
A = 0. We can x this by adding the closed
term
A(t) =
B(t) + b (t):
8The author would like to thank him for sharing this idea.
(4.48)
(4.49)
(4.50)
(4.51)
(4.52)
(4.53)
(4.54)
A(t)
(4.55)
Taking
of this expression implies a jt=1 = b jt=1, as desired. A nice property of this
eld rede nition is that it is compatible with the natural gauge xing to the small Hilbert
space. In particular, xing
theory, and viceversa:
1
3
A = 0 in the lifted A1 theory xes
B = 0 in the Berkovits
A = b jt=1:
1
A =
B + b jt=1:
A = 0
B = 0:
This is not generally true for the eld rede nition based on the Wilson line.
Expanding the eld rede nition to second order gives
B =
A +
[
A;
A]
[ A;
A] + higher orders:
Note that this is not a special case of (4.20) for some choice of the constant C. Therefore
the eld rede nition cannot be realized as the endpoint of a pair of interpolations
and
B(t) related by At = Bt. This makes the proof of equivalence of the actions less
direct. First let us generalize the eld rede nition to intermediate t by taking
With this identi cation the potentials are equal for intermediate t
but the tpotentials are not the same. Instead we have
A (t) = B (t);
At = B~t;
where B~t is de ned
1G
1
1
_ B(t) + b_
1
1
(4.58)
HJEP02(16)
(4.56)
(4.57)
(4.59)
(4.60)
(4.61)
(4.62)
(4.63)
which is not the same as Bt. Therefore applying the eld rede nition to the lifted A1
action gives
0
SB =
dt hB~t; QB iL:
This is not the Berkovits action as it is usually written. However, as noted in [1], B~t
usual Berkovits action. Another way to understand this observation is that (4.59) is an
expression of the Berkovits action using a nonstandard set of potentials, related to the
usual Berkovits potentials by a nite MaurerCartan gauge transformation
h
B~ = (d + )e
(1) i
e
(1) + e
Be
(1) ;
where the 1form gauge parameter (1) is
This MaurerCartan gauge transformation leaves B invariant, while it transforms Bt
into B~t:
We can compute the gauge parameter t as follows. Consider
where in the second step we traded b_ with bt. Now pull
out so it acts on the entire
Bt
Bt = 1G
= 1G
= 1(
1
1
1
1
1
1
b
bt) + bt
bt)
b
bt)
1
1
1
b
1
b
+ 1G
^
1
:
1
1
1
1
b
Bt;
1
b
Bt;
Bt;
b
t
1
1
b
Bt;
(4.64)
Bt = 1G
^
= 1G
1
1
1
1
b
_ B(t) + b_
_ B(t) +
b
t
(1) = dt t:
B~t = Bt + r
t
In the second step the second term cancels with Bt by the de nition of the little potentials.
What remains can be interpreted as r
t, where t is
t = 1G
1
1
bt)
:
(4.65)
Therefore relating the interpolations through (4.55) and performing a MaurerCartan gauge
transformation turns the WZWlike action of the lifted A1 theory into the standard
WZWlike action of the Berkovits theory.
It is interesting to contrast the variety of eld rede nitions we nd in the large Hilbert
space with the seeming uniqueness of the
eld rede nition found in the small Hilbert
space [1, 2]. The reason for this discrepancy is that the operations , ; m2 used to construct
the eld rede nition in the large Hilbert space can also implement
gauge transformations
of the eld rede nition. In the small Hilbert space the
gauge invariance is not present,
and
and m2 alone cannot implement interesting gauge transformations in the small
Hilbert space.
5
Mapping gauge invariances
the lifted A1
of (4.8) produces
eld
We will now use the eld rede nition to determine how the gauge symmetries of the lifted
Let us rst consider the Wilson line
eld rede nition (4.8). To get the information
we're after, we must compute the change of the Berkovits eld
B induced by a change in
A and/or the interpolating function fA(t; A). Taking the variation
0
e B =
dt g(1; t) At g(t; 0):
Using (3.28) we can switch the variation with a time derivative:
d
e B =
dt g(1; t) rtA + F t g(t; 0);
g(1; t)A g(t; 0) +
dt g(1; t)F t g(t; 0);
0
Z 1
0
= A jt=1e B
e B A jt=0 +
dt g(1; t)F t g(t; 0):
This is the expected formula for the variation of a Wilson line. We assume that A vanishes
at t = 0 because the interpolating function is required to satisfy the boundary condition
fA(0; A) = 0. Let us see what to do with the eld strength integrated along the curve.
We can express the eld strength in terms of the 2potential
0
dt g(1; t)F t g(t; 0) =
dt g(1; t) r At g(t; 0);
=
=
dt g(t; 0) 1
r At g(t; 0);
dt
g(t; 0) 1At g(t; 0) :
0
e B
e B
(5.1)
(5.2)
(5.3)
(5.4)
Therefore, the variation of the eld rede nition is
e B = A jt=1e B
e B
dt g(t; 0) 1At g(t; 0) ;
0
is constrained to vanish by boundary conditions, and (5.5) simpli es to
rede nition between
what we found in (4.21).
Now let us see how
Right multiplication of e B by an closed string eld is an in nitesimal gauge
transformation in the Berkovits theory. Therefore, a change of the interpolation only e ects the eld
A and
B by an
gauge transformation. This is consistent with
B responds to gauge transformations in the lifted A1 theory.
First consider the
gauge transformation
in (3.10). We obtain
0
e B = A jt=1e B
e B
g(t; 0) 1At g(t; 0) :
We can simplify the rst term using equation (3.71):
Plugging in we obtain
A jt=1 = r
Bjt=1;
= g(1; 0)
g(1; 0) 1
B g(1; 0) g(1; 0) 1:
e B =
e B
0
dt g(t; 0) 1At g(t; 0) :
HJEP02(16)
(5.5)
(5.6)
(5.7)
(5.8)
(5.9)
(5.11)
e B = e B
g(1; 0) 1
B g(1; 0)
dt g(t; 0) 1At g(t; 0) :
Therefore, the
gauge invariance of the lifted A1 theory maps into the
of the Berkovits theory. Now consider the BRST gauge transformation Q in (3.11). From
the computation of (3.73), we nd
0
Z 1
0
1M
1
1
A
= 0:
Qe B =
B e B
e B
dt g(t; 0) 1At Q g(t; 0) :
(5.10)
Left multiplication of e B by a BRST closed string eld is an in nitesimal gauge
transformation in the Berkovits theory. It follows from the computation at the end of subsection 3.2
that left multiplication of e B by
B is also a symmetry of the action, even though
B
is not BRST closed. However, note from (3.75) that
B vanishes when the equations of
motion are satis ed:
Therefore left multiplication by
B must represent a trivial gauge transformation [23].
The upshot is that the BRST gauge transformation of the lifted A1 theory maps into a
combination of a BRST gauge transformation, an
gauge transformation, and a trivial
gauge transformation in the Berkovits theory.
Now let's consider the reverse question, namely, how the gauge invariances of the
Berkovits theory map into those of the lifted A1 theory. For this purpose it is useful
to consider the inverse eld rede nition as described in section 4.2. Taking the variation
of (4.46), one
nds that a change of the Berkovits eld and/or interpolation changes the
lifted A1
eld through
A =
=
0
dt bt;
d
b
= b jt=1
0
b t ;
dt b t ;
where we used (4.35) to interchange
with a d=dt, which produces a term proportional to
the little 2potential. If we change the interpolating function of the Berkovits theory, the
boundary term at t = 1 drops out and what remains is an
gauge transformation of
The BRST and
gauge transformations of the Berkovits theory can be written
Qe B = Q B e B ;
e
B
Be B ;
Before
B and
B were de ned as functions of
A and the gauge parameters
A and
A
of the lifted A1 theory, but now we view them as independent variables de ning the gauge
parameters of the Berkovits theory. The potentials corresponding to these variations are
B Q jt=1 = Q B;
B jt=1 = r
Bjt=1:
Let us rst compute the little potential b jt=1 from (4.33):
B
1
B
B
B
(5.12)
(5.14)
(5.15)
(5.16)
(5.17)
b jt=1 =
Plugging this into (5.12), we nd
where
1G^ 1
1G^ 1
1G^ 1
1
1G^ 1
1
1
1
B
B
B
r
m2) 1
1
B
1
1
1
1
B
1
B
1
B
t=1
B
B
t=1
t=1
t=1
;
A =
A
dt b t ;
1G^ 1
1
1
B
1
1
B
t=1
Therefore the
gauge invariance of the Berkovits theory maps into the
of the lifted A1 theory. Note that (5.17) is the inverse of the formula (3.71) expressing
B as a function of
A. Therefore we have a \ eld rede nition" relating the
gauge
parameters in the two theories. Now consider the BRST gauge symmetry:
(5.18)
t=1
t=1
t=1
;
;
(5.19)
(5.20)
(5.21)
(5.22)
(5.23)
B
1
1
1
B
1
1
1
B
1
A
1
t=1
B
1
1
B
1
1
1
t=1
A
1
B
QB
B
B
B
1
1
1
1
QB
1 B t=1
dt b Qt ;
1
1
B
t=1
1G^ 1
1G^ 1
1
1G^ 1
= 1M
1
1
1G^ 1 1
B
QB
B
QB
QB
1
1
1
1 B t=1
1 B t=1
1 B t=1
1 B t=1
1
1
1
1
1G^ 1
B
B
B
1
Using
where
we therefore obtain
Q A =
A
A
A =
1G^ 1
1M
1
1G^ 1
1G^ 1
1
1
1
B
1
A
1
1
B
1
1
B
1
B
A
B
QB
1
1
B
Note that (5.21) is the inverse of the formula (3.74) expressing
B in terms of A. Also note
that A vanishes on shell, and so must represent a trivial gauge transformation. Therefore
the BRST gauge transformation of the Berkovits theory maps into a combination of a
BRST, an , and trivial gauge transformations in the lifted A1 theory. Similar conclusions
follow using the eld rede nition proposed by Okawa, since in this case the variation takes
the form
A = b jt=1 +
B
b jt=1 :
which, aside from unimportant di erences in the
closed term, is equivalent to (5.12).
Acknowledgments
The author would like to thank T. Takezaki and Y. Okawa for collaboration, and
S. Konopka and Y. Okawa for comments. This work was supported in parts by the DFG
Transregional Collaborative Research Centre TRR 33 and the DFG cluster of excellence
Origin and Structure of the Universe.
A
Some computations involving cyclicity
In this appendix we provide a few missing calculations referred to in the text, in particular
as pertains to cyclicity of the A1 products and cohomomorphism G^ . These calculations
are simpli ed with the help of the \triangle formalism" of the product and coproduct,
introduced in appendix A of [2]. Here we review this formalism and provide the missing
calculations in the text.
algebra into a pair of tensor algebras:
The tensor algebra has a coproduct, which is a coassociative linear map from the tensor
and grouplike elements satisfy
where we use the symbol 0 to distinguish from the tensor product used to construct T H.
The coproduct is coassociative
and acts on tensor products of states as
4A1
: : :
An =
: : :
Ak) 0 (Ak+1
: : :
An);
where at the extremes of summation 0 multiplies the identity of the tensor product 1T H.
Note that 1T H is not the identity with respect to 0. Coderivations and cohomomorphisms
satisfy
4 : T H ! T H
0 IT H)4 = (I 0 4)4;
n
X(A1
k=0
4D = (D
4H^ = (H^
0 IT H + IT H
0 H^ )4;
0 D)4;
4 1
1
A
1
1
A
0
1
1
A
0 T H ! T H;
m+n =4 h
m
0 n 4:
(A.1)
(A.3)
(A.4)
(A.5)
(A.6)
(A.7)
m+n
(A.8)
By taking variations we can derive the action of the coproduct on more general states. In
addition, the tensor algebra has a product
which operates by replacing the primed tensor product 0 with the ordinary tensor product
. The central formula of the triangle formalism is an expression for the projector
onto the (m + n)string component of the tensor algebra:
For further elaboration, see appendix A of [2].
h!j 2H
h!j 2H
1
1
1H
1H
1
1
1H
= h!j
+ h!j
+ h!j
Now let us revisit the derivation of (3.13), which is also featured in appendix A of [1]. A
cyclic cohomomorphism H^ satis es
Consider this formula acting on a particular element of the tensor algebra:
where A is degree even and B; C are arbitrary string elds. We nd
On the other hand, we can replace 2 =4 h
1 0 1 4 on the left hand side and act with
1
B
1
B
1
A
1 0 1 4H
( 1 0 1)(H^
1
1
1
A
1
A
A
B
1
1
1H
A
A
1
1
1
A
1
1
1
1H
1H
A
1
1
B
0
A
A
B
B
0
1
1H
1
A
1
1
1
1
A
1
1
1H
A
1
1
1
1
A
C
C
1
A
B
1
1
C
A
0
C
A
1
1
A
B
1
1
1
A
1
1
1
A
0
A
B
1
1
1
C
1H
B
1
1
1
1
1
1
1
A
A
= h!j 2
= h!jB
C
C
1
1
B
1
C
C
1
A
1H
B
1
1
1
1
A
A
B
1
C
1
1
A
B
1
1
1H
1
C
C
A
C
1
A
1
A
C
1
1H
1
1
A
1
1
1
1
1
A
A
A
1
1
A
A
(A.9)
(A.10)
1
;
(A.11)
C
(A.12)
The rst and last terms cancel by antisymmetry of the symplectic form. The second term,
however, remains. We have therefore shown
Now suppose that the string elds A and B happen to take the form
= !(A; B):
(A.13)
; 1H
1
1
A
1
1
A
1
1
A
C
A = 1D1
B = 1D2
for some coderivations D1 and D2. Plugging into the above formula then gives
1D1
; 1D2
Now let's prove the equivalence of equations (3.78) and (3.76). The left hand side of (3.78) is
1 0 1 4 and acting with the coproduct produces the expression
1
1
1H
which reproduces (3.13).
h!Lj 2G
A.2
A
1
1
A
1
h!Lj
B
1G
1
+ !L
1G
1
1
A
A
1
A
; 1G
A
1
1M
1M
1
1
1M
1
A
1
1
A
1
A
1M
1
A
A
A
1
1
1
1
1
1
1
1
1
A
1
1
1
1
1
A
A
A
A
1
0
1
1
A
1
1
1
1
A
A
1
1
1
1
A
1
1
A
A
1M
1
1
1
1
1M
1
0
1
1
1
1
1
1
1
1M
A
A
A
1
1
A
A
A
1
1
A
A
1
1M
1
1M
1M
1
1
1
1
A
1
A
A
; 1G
1
1
1
1
1
1
A
A
A
A
A
A
(A.14)
; (A.15)
1
1
1
1
0
1
1
1
1
1
1
(A.16)
(A.17)
A
A
A
A
1
1
1
1
1
1
1
1
A
A
A
A
0
1
1M
1
1M
1M
1M
1
1
1
A
1
1
1
The rst and last term simplify to
1
A
(A.18)
and they cancel by antisymmetry of the symplectic form. Meanwhile, the second and
third terms in (A.18) produce equation (3.76). This lls the missing steps between (3.76)
and (3.78).
Now let's prove (3.126):
a
1
1
0 = h!Lj
0 = h!Lj 2M
a
1
!L
1
a
1
1
a
1M
1
1
1
a
0 IT H +IT H
0
1
a
For this purpose consider the identity
which vanishes because M is cyclic with respect to the large Hilbert space symplectic form.
1 0 1 4 and acting with the coproduct gives
1
1 a
1
1 a
0
1
1 a
a
1
1 a
0 = h!Lj
+ h!Lj
+ h!Lja
+ h!Lj a_
1M
1
1M
1
1
a
1
1M
1M
a
1
1
1
1
a
a
1
a
1
1 a
1
a
1
1
1
a
1
1
1
a
a
1
a
a
1
1
a
a
1
a
1
1
1
a
a
Some cross terms drop out since 1 acts on the tensor product of two or more states. What
is left is
The second and third terms cancel out by antisymmetry of the symplectic form, while the
rst and last terms reproduce (3.126).
Open Access.
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
[1] T. Erler, Y. Okawa and T. Takezaki, A1 structure from the Berkovits formulation of open
superstring eld theory, arXiv:1505.01659 [INSPIRE].
[2] T. Erler, Relating Berkovits and A1 superstring eld theories; small Hilbert space
perspective, JHEP 10 (2015) 157 [arXiv:1505.02069] [INSPIRE].
[3] N. Berkovits, SuperPoincare invariant superstring eld theory, Nucl. Phys. B 450 (1995) 90
[Erratum ibid. B 459 (1996) 439] [hepth/9503099] [INSPIRE].
[4] N. Berkovits, A new approach to superstring eld theory, Fortsch. Phys. 48 (2000) 31
[hepth/9912121] [INSPIRE].
1
a
; a
:
(A.19)
1
a
1
1 a
a
(A.20)
1
1
0
1 a
1
a
(A.22)
(2014) 150 [arXiv:1312.2948] [INSPIRE].
08 (2014) 158 [arXiv:1403.0940] [INSPIRE].
theory, arXiv:1506.06657 [INSPIRE].
HJEP02(16)
superstring eld theory, arXiv:1512.03379 [INSPIRE].
formulation in open superstring eld theory, JHEP 03 (2014) 044 [arXiv:1312.1677]
xing, ghost structure and propagator, JHEP 03 (2012) 030 [arXiv:1201.1761]
[arXiv:1201.1769] [INSPIRE].
Superstring Field Theory, JHEP 04 (2012) 050 [arXiv:1201.1762] [INSPIRE].
formulation, Prog. Theor. Phys. Suppl. 188 (2011) 272 [arXiv:1201.1763] [INSPIRE].
Actions in the Witten Formulation and the Berkovits Formulation of Open Superstring Field
arXiv:1508.00366 [INSPIRE].
arXiv:1508.05387 [INSPIRE].
[5] T. Erler , S. Konopka and I. Sachs , Resolving Witten`s superstring eld theory , JHEP 04 [6] N. Berkovits , Y. Okawa and B. Zwiebach , WZWlike action for heterotic string eld theory , [7] T. Erler , S. Konopka and I. Sachs, NSNS Sector of Closed Superstring Field Theory , JHEP [11] M. Kroyter , Y. Okawa , M. Schnabl , S. Torii and B. Zwiebach , Open superstring eld theory