Marginal Solutions for the Superstring
August 14, 2018
Marginal Solutions for the Superstring
Theodore Erler
Harish-Chandra Research Institute
Chhatnag Road, Jhunsi, Allahabad 211019, India
E-mail:terler@mri.ernet.in
Abstract
We construct a class of analytic solutions of WZW-type open superstring field theory describ-
ing marginal deformations of a reference D-brane background. The deformations we consider
are generated by on-shell vertex operators with vanishing operator products. The superstring
solution exhibits an intriguing duality with the corresponding marginal solution of the bosonic
string. In particular, the superstring problem is “dual” to the problem of re-expressing the
bosonic marginal solution in pure gauge form. This represents the first nonsingular analytic
solution of open superstring field theory.
http://arxiv.org/abs/0704.0930v2
Contents
1 Introduction 1
2 Bosonic Solution 2
3 Superstring Solution 5
4 Pure Gauge for Bosonic Solution 10
5 Conclusion 12
A B0,L0 with Split Strings 13
B Unitary eΦ 15
1 Introduction
Following the breakthrough analytic solution of Schnabl[1], our analytic understanding of open
string field theory (OSFT) has seen remarkable progress[2, 3, 4, 5, 6, 7, 8]. So far most work has
focused on the open bosonic string, but clearly it is also important to consider the superstring.
This is not just because superstrings are ultimately the theory of interest, but because there
are important physical questions, especially the holographic encryption of closed string physics
in OSFT, which may be difficult to decipher in the bosonic case[9].
Ideally, the first goal should be to find an analytic solution of superstring field theory1 on a
non-BPS brane describing the endpoint of tachyon condensation, i.e. the closed string vacuum.
However, the construction of this solution is will likely be subtle—indeed, Schnabl’s solution for
the bosonic vacuum is very close to being pure gauge[1, 2]. Thus, it may be useful to consider a
simpler problem first: constructing solutions describing marginal deformations of a (non)BPS D-
brane. Marginal deformations correspond to a one-parameter family of open string backgrounds
obtained by adding a conformal boundary interaction to the worldsheet action—for example,
turning on a Wilson line on a brane by adding the boundary term Aµ
dt∂Xµ(t) to the
worldsheet action. Such backgrounds were studied numerically for the bosonic string in ref.[11]
and for the superstring in ref.[12]. Recently, Schnabl[13] and Kiermaier et al[14] found analytic
solutions for marginal deformations in bosonic OSFT2. The solutions bear striking resemblance
1In this paper we will work with the Berkovits WZW-type superstring field theory[10].
2For previous efforts to construct such solutions analytically in bosonic and super OSFT, see refs.[15, 16].
to Schnabl’s vacuum solution, but are simpler in the sense that they are manifestly nontrivial
and can be constructed systematically with a judicious choice of gauge.
In this note, we construct solutions of super OSFT describing marginal deformations gen-
erated by on-shell vertex operators with vanishing operator products (in either the 0 or −1
picture). As was found in ref.[13, 14] such deformations are technically simpler since they
allow for solutions in Schnabl’s gauge, B0Φ = 0—though probably more general marginal solu-
tions can be obtained once the analogous problem is understood for the bosonic string, either
by adding counterterms as described in ref.[14] or by employing a “pseudo-Schnabl gauge” as
suggested in ref.[13]. The superstring solution exhibits a remarkable duality with its bosonic
counterpart: it formally represents a re-expression of the bosonic solution in pure gauge form.
It would be very interesting if this duality generalized to other solutions.
This paper is organized as follows. In section 2 we briefly review the bosonic marginal
solution in the split string formalism[2, 8, 17], which we will prove convenient for many com-
putations. In section 3 we consider the superstring, motivating the solution as analogous to
constructing an explicit pure gauge form for the bosonic marginal solution. This strategy
quickly gives a very simple expression for the complete analytic solution of super OSFT. In sec-
tion 4 we consider the dual problem: finding a pure gauge expression for the bosonic marginal
deformation describing a constant, light-like gauge field on a non-compact brane. Though quite
analogous to the superstring, this problem is slightly more complex. Nevertheless we are able
to find an analytic solution. We end with some conclusions.
While this note was in preparation, we learned of the independent solution by Yuji Okawa[18].
His paper should appear concurrently.
2 Bosonic Solution
Let us begin by reviewing the bosonic marginal solution[13, 14] in the language of the split
string formalism[2, 8, 17], which is a useful shorthand for many calculations. The first step in
this approach is to find a subalgebra of the open string star algebra, closed under the action of
the BRST operator, in which we hope to find an analytic solution. For the bosonic marginal
solution the subalgebra is generated by three string fields K,B and J :
K = Grassmann even, gh# = 0
B = Grassmann odd, gh# = −1
J = Grassmann odd, gh# = 1 (2.1)
satisfying the identities,
[K,B] = 0 B2 = J2 = 0 (2.2)
dK = 0 dJ = 0 dB = K (2.3)
where d = QB is the BRST operator and the products above are open string star products (we
will mostly omit the ∗ in this paper). The relevant explicit definitions of K,B, J are3,
K = −π
(K1)L|I〉 K1 = L1 + L−1
B = −
(B1)L|I〉 B1 = b1 + b−1
J = J(1)|I〉 (2.4)
where |I〉 is the identity string field and the subscript L denotes taking the left half of the
corresponding charge4. The operator J(z) is a dimension zero primary generating the marginal
trajectory. It takes the form,
J(z) = cO(z) (2.5)
where O is a dimension one matter primary with nonsingular OPE with itself. This is crucial
for guaranteeing that the square of the field J vanishes, as in eq.(2.2). With these preliminaries,
the marginal solution for the bosonic string is:
Ψ = λFJ
1− λB F 2−1
F (2.6)
where λ parameterizes the marginal trajectory and F = eK/2 = Ω1/2 is the square root of the
SL(2,R) vacuum (a wedge state). To linear order in λ the solution is,
Ψ = λFJF + ... = λJ(0)|Ω〉+ ... (2.7)
which is the nontrivial element of the BRST cohomology generating the marginal trajectory.
Let us prove that eq.2.6 satisfies the equations of motion. Using the identities Eqs.(2.2,2.3),
dΨ = −λFJd
1− λB F 2−1
= −λFJ
1− λB F 2−1
F 2 − 1
1− λB F 2−1
= −λ2FJ 1
1− λB F 2−1
(F 2 − 1)J 1
1− λB F 2−1
F (2.8)
3We may generalize the construction by considering other projector frames[4, 7, 8] or by allowing the field F
in eq.(2.6) to be an arbitrary function of K[2, 8]. Such generalizations do not add much to the current discussion
so we will stick with the definitions presented here.
4“Left” means integrating the current counter-clockwise on the positive half of the unit circle. This convention
differs by a sign from ref.[8] but agrees with ref.[4].
Notice the (F 2−1)J factor in the middle. Since J2 = 0, the −1)J term vanishes when multiplied
with the Js to the left—thus the necessity of marginal operators with nonsingular OPE. This
leaves,
dΨ = −λ2FJ 1
1− λB F 2−1
1− λB F 2−1
F = −Ψ2 (2.9)
i.e. the bosonic equations of motion are satisfied.
The solution has a power series expansion in λ:
λnΨn (2.10)
where,
Ψn = FJ
F 2 − 1
F (2.11)
To make contact with the expressions of refs.[13, 14], note the relation,
F 2 − 1
dtΩt (2.12)
To prove this, recall Ωt = etK and calculate5,
dtΩt =
etK = eK − 1 = F 2 − 1 (2.13)
Using this and the mapping between the split string notation and conformal field theory de-
scribed in ref.[8], the Ψns can be written as CFT correlators on the cylinder:
〈Ψn, χ〉 =
dt1...
dtn−1 〈J(tn−1 + ...+ t1 + 1)B...J(t1 + 1)BJ(1) fS ◦ χ(0)〉Ctn−1+...+t1+2
(2.14)
where fS(z) =
tan−1 z is the sliver conformal map, and in this context B is the insertion
∫ −i∞
b(z) to be integrated parallel to the axis of the cylinder in between the J insertions on
either side. This matches the expressions found in refs.[13, 14].
In passing, we mention that this solution was originally constructed systematically by using
the equations of motion to recursively determine the Ψns in Schnabl gauge. If desired, it is also
possible to perform such calculations in split string language; we offer some sample calculations
in appendix A.
5Note that, in general, the inverse of K is not well defined. However, when operating on F 2 − 1 it is. This
is why we cannot simply use F 2/K in the solution in place of F
, which would naively give a solution even
for marginal operators with singular OPEs.
3 Superstring Solution
Let us now consider the superstring. The marginal deformation is generated by a −1 picture
vertex operator,
e−φcO(z) (3.1)
where O(z) is a dimension 1
superconformal matter primary. We will use Berkovits’s WZW-
type superstring field theory[10]6, in which case the string field is given by multiplying the −1
picture vertex operator by the ξ ghost:
X(z) = ξe−φcO(z) (3.2)
This corresponds to a solution of the linearized Berkovits equations of motion,
η0QB (λX(0)|Ω〉) = 0 (3.3)
since η0 eats the ξ and the −1 picture vertex operator is in the BRST cohomology. We will
also find it useful to consider the 0 picture vertex operator,
J(z) = QB ·X(z) = cG−1/2 · O(z)− eφηO(z) (3.4)
A complimentary way of seeing the linearized equations of motion are satisfied is to note that
J(z) is in the small Hilbert space. As with the bosonic string, it is very helpful to assume that
X(z) and J(z) have vanishing OPEs:
J(z)X(w) = lim
J(z)J(w) = lim
X(z)X(w) = 0 (3.5)
We mention two examples of such deformations. The simplest is the light-like Wilson line
O(z) = ψ+(z) (α′ = 1), where
X(z) = ξe−φcψ+(z)
J(z) = i
2c∂X+(z)− eφηψ+(z) (3.6)
There is also a “rolling tachyon” marginal deformation[22] O(z) = σ1eX
2(z) on a non-BPS
brane. The corresponding vertex operators are,
X(z) = σ1ξe
−φceX
J(z) = σ2(cψ
0 − ieφη)eX0/
2(z) (3.7)
6See refs.[19, 20, 21] for nice reviews.
The Pauli matrices σ1, σ2, σ3 are “internal” Chan-Paton factors[23, 24], necessary to accom-
modate non-BPS GSO(−) states into the Berkovits framework. Though we will not write it
explicitly, in this context it is important to remember that the BRST operator and the eta zero
mode are carrying a factor of σ3 (thus the presence iσ2 = σ3σ1 in the above expression for J).
We mention that both X(0)|Ω〉 and J(0)|Ω〉 are in Schnabl gauge and annihilated by L0.
Let us describe the subalgebra relevant for finding the marginal solution. It consists of the
products of four string fields, K,B,X, J7:
K = Grassmann even, gh# = 0
B = Grassmann odd, gh# = −1
X = Grassmann even, gh# = 0
J = Grassmann odd, gh# = 1 (3.8)
All four of these have vanishing picture number. K and B are the same fields encountered
earlier in eq.(2.4); X and J are defined,
X = X(1)|I〉 J = J(1)|I〉 (3.9)
with X(z), J(z) as in Eqs.(3.2,3.4). We have the identities,
[K,B] = 0 B2 = 0 X2 = J2 = XJ = JX = 0 (3.10)
where the third set follows because the corresponding vertex operators have vanishing OPEs.
The algebra is closed under the action of the BRST operator:
dB = K dK = 0
dX = J dJ = 0 (3.11)
Note that the eta zero mode d̄ ≡ η0 annihilates K,B and J ,
d̄K = d̄B = d̄J = 0 (3.12)
since they live in the small Hilbert space. However, it does not annihilate X , and the algebra is
not closed under d̄. Though it is not a priori obvious that the K,B,X, J algebra is rich enough
to encapsulate the marginal solution, we will quickly see that it is.
7Note that for for a GSO(−) deformation the Grassmann assignments of X, J are opposite. Still, as far
as the solution is concerned X is even and J is odd because QB, η0 carry a σ3 which anticommutes with the
internal Chan-Paton matrices of the vertex operators.
We seek a one parameter family of solutions of the super OSFT equations of motion,
e−ΦdeΦ
= 0 (3.13)
where Φ is a Grassmann even, ghost and picture number zero string field which to linear order
in the marginal parameter takes the form,
Φ = λFXF + ... (3.14)
There are many strategies one could take to solve this equation, but before describing our
particular approach it is worth mentioning the “obvious” method: fixing Φ in Schnabl gauge
and attempting a perturbative solution, as in refs.[13, 14]:
λnΦn Φ1 = FXF (3.15)
At second order8, the Schnabl gauge solution is actually fairly simple:
F 2 − 1
JF + FJB
F 2 − 1
(3.16)
and seems quite similar to the bosonic solution. At third order, however, we found an extremely
complicated expression (though still within the K,B,X, J subalgebra). It seems doubtful that
a closed form solution for Φ in Schnabl gauge can be obtained.
Since the Schnabl gauge construction appears complicated, we are lead to consider another
approach. To motivate our particular strategy, we make two observations: First, the combi-
nation e−ΦdeΦ which enters the superstring equations of motion also happens to be a pure
gauge configuration from the perspective of bosonic OSFT. Second, there is a basic similarity
between the K,B, J algebra for the bosonic marginal solution and the K,B, J,X algebra for
the superstring. The main difference of course is the presence of X for the superstring, whose
BRST variation gives J . If such a field were present for the bosonic string, the bosonic marginal
solution would be pure gauge because J would be trivial in the BRST cohomology. With this
motivation, we are lead to consider the equation
e−ΦdeΦ = λFJ
1− λB F 2−1
F (3.17)
From the bosonic string perspective, this equation represents an expression of the bosonic
marginal solution in a form which is pure gauge. From the superstring perspective, this is a
8Explicitly, if we plug eq.(3.15) into the equations of motion, we find a recursive set of equations of the form
d̄dΦn = d̄Fn−1[Φ], where Fn−1[Φ] depends on Φ1, ...,Φn−1. The Schnabl gauge solution is obtained by writing
Fn−1[Φ].
partially gauge fixed form of the equations of motion, since the expression on the right hand
side is in the small Hilbert space.
Let us now solve this equation. It will turn out to be simpler to solve for the group element
g = eΦ; we make a perturbative ansatz,
g = eΦ = 1 +
λngn g1 = Φ1 = FXF (3.18)
Expanding out eq.(3.17) to second order gives,
dg2 = FJB
F 2 − 1
JF + g1dg1
= FJB
F 2 − 1
JF + FXF 2JF (3.19)
As it turns out, this equation is solved by the second order Schnabl gauge solution eq.(3.16):
g2 = Φ2 +
Φ21 =
F 2 − 1
JF + FJB
F 2 − 1
XF + FXF 2XF
(3.20)
but there is a simpler solution:
g2 = FXB
F 2 − 1
JF (3.21)
Using this form of g2 we can proceed to third order—remarkably, the solution is practically just
as simple:
g3 = FX
F 2 − 1
F (3.22)
This leads to an ansatz for the full solution:
eΦ = 1 + λFX
1− λB F 2−1
F (3.23)
To check this, calculate:
deΦ = λFJ
1− λB F 2−1
F + λFXd
1− λB F 2−1
= λFJ
1− λB F 2−1
F + λFX
1− λB F 2−1
F 2 − 1
1− λB F 2−1
= λFJ
1− λB F 2−1
F + λ2FX
1− λB F 2−1
1− λB F 2−1
1 + λFX
1− λB F 2−1
1− λB F 2−1
= eΦλFJ
1− λB F 2−1
F (3.24)
Therefore, eq.(3.23) is indeed a complete solution to the super OSFT equations of motion!
Note, however, that it is not quite a solution to the pure gauge problem of the bosonic string.
In particular, in step three we needed to assume XJ = 0—something we would not expect
to hold in the bosonic context. We will give the solution to the bosonic problem in the next
section.
Let us make a few comments about this solution. First, though the string field Φ itself is
not in Schnabl gauge, the nontrivial part of the group element eΦ is—this is not difficult to see,
but we offer one explanation in appendix A. The second comment is related to the string field
reality condition. In super OSFT, the natural reality condition is that Φ should be “imaginary”
in the following sense:
〈Φ, χ〉 = −〈Φ|χ〉 (3.25)
where 〈Φ| is the Hermitian dual of |Φ〉 and χ is any test state. In split string notation we can
write this,
Φ† = −Φ (3.26)
where † is an anti-involution on the star algebra, formally completely analogous to Hermitian
conjugation of operators. With this reality condition, the group element should be unitary:
g† = g−1
Using,
K† = K B† = B J† = J X† = −X (3.27)
it is not difficult to see that the analytic solution eΦ is not unitary9. However, it is possible to
obtain a unitary solution by a simple gauge transformation of eq.(3.23); we explain details in
appendix B.
Let us take the opportunity to express the solution in a few other forms which may be more
convenient for explicit computations. Following the usual prescription we may express the gns
as correlation functions on the cylinder:
〈gn, χ〉 =
dt1...
dtn−1 〈X(tn−1 + ...+ t1 + 1)BJ(tn−2 + ..+ t1 + 1)...BJ(1) fS ◦ χ(0)〉Ctn−1+...+t1+2
= (−1)n
dt1...
dtn−1 〈X(L+ 1)[O′(ℓn−2 + 1)...O′(ℓ1 + 1)]BJ(1) fS ◦ χ(0)〉CL+2 (3.28)
In the second line we manipulated the multiple B insertions, simplifying the vertex operators
and obtaining a single B insertion to the right; we introduced the length parameters[14]:
tk L = ℓn−1 (3.29)
9By contrast, the Schnabl gauge construction automatically gives an imaginary Φ and unitary eΦ.
and defined O′(z) = G− 1
· O(z) (times a σ3 for GSO(−) deformations). We may also express
the solution in the operator formalism of Schnabl[1]:
|gn〉 =
(−1)nO+1
dt1...
dtn−1ÛL+2 f
S ◦ (ξe
−φO(L/2))Õ′(yn−2)...Õ′(y1)
Õ′(−L
)[B+c̃(L
)c̃(−L
)− c̃(L
)− c̃(−L
)] + f−1S ◦ (ηe
φO(−L
))[B+c̃(L
) + 1]
(3.30)
where yi = ℓi −L/2 and[6] Ûr =
. Also we have used f−1S to define the tilde to hide
some factors of π
. The expression is somewhat more complicated than the bosonic solution
since the vertex operator J(z) has a piece without a c ghost, so in the bc CFT the solution has
a component not proportional to Schnabl’s ψn[1].
4 Pure Gauge for Bosonic Solution
In the last section, we found a solution for the superstring by analogy with the pure gauge
problem of the bosonic string; but we did not solve the latter. The scenario we have in mind is
a constant, lightlike gauge field on a non-compact D-brane. Since there is no flux and no way
to wind a Wilson loop, such a field configuration should be pure gauge. From the string field
theory viewpoint, this is reflected by the fact that the marginal vertex operator becomes BRST
trivial in the noncompact limit,
ic∂X+(z) = QB · 2iX+(z) (4.1)
Of course, on a compact manifold the operator X+(z) is not globally defined so the marginal
deformation is nontrivial.
Translating to split string language, we consider an algebra generated by four fieldsK,B,X, J ,
where K,B are defined as before and,
X = 2iX+(1)|I〉 J = ic∂X+(1)|I〉 (4.2)
These have the same Grassmann and ghost number assignments as eq.(3.8). We have the
algebraic relations,
[K,B] = 0 B2 = 0 J2 = 0 [X, J ] = 0 (4.3)
Note the difference from the superstring case: the products of X with itself and with J , though
well defined (the OPEs are nonsingular), are nonvanishing. However, we still have
dB = K dK = 0
dX = J dJ = 0 (4.4)
with the second set implying that J is trivial in the BRST cohomology.
We now want to solve eq.(3.17) assuming this slightly more general set of algebraic relations.
Playing around a little bit, the solution we found is,
eΛ = 1 + λFuλ(X)
1− λB F 2−1
F (4.5)
where,
uλ(X) =
eλX − 1
(4.6)
The relevant identity satisfied by this particular combination is,
duλ = J(λuλ + 1) (4.7)
Let us prove that this gives a pure gauge expression for the bosonic marginal solution:
deΛ = λFduλ
1− λB F 2−1
F + λFuλ
1− λB F 2−1
F 2 − 1
1− λB F 2−1
= λFJ(λuλ + 1)
1− λB F 2−1
F + λ2Fuλ
1− λB F 2−1
(F 2 − 1)J
1− λB F 2−1
Now we come to the critical difference from the superstring. Note the −1)J piece in the middle
of the second term. Before it vanished when multiplied by X, J to the left. This time it
contributes because XJ 6= 0; still, the Js in the denominator of the factor to the left get killed
because J2 = 0. Thus we have,
deΛ = λFJ(λuλ + 1)
1− λB F 2−1
F + λ2Fuλ
1− λB F 2−1
1− λB F 2−1
−λ2FuλJ
1− λB F 2−1
F (4.8)
where the third term comes from the −1)J piece. Note the cancellation. We get,
deΛ = λFJ
1− λB F 2−1
F + λ2Fuλ
1− λB F 2−1
1− λB F 2−1
1 + λFuλ
1− λB F 2−1
1− λB F 2−1
= eΛλFJ
1− λB F 2−1
F (4.9)
thus we have a pure gauge expression for the marginal solution.
To further emphasize the duality with the superstring, note that for the pure gauge problem
the role of the eta zero mode is played by the lightcone derivative:
d̄ ∼ d
(4.10)
In particular we have solved the equation,
e−ΛdeΛ
= 0 (4.11)
Though there are many pure gauge trajectories generated by FXF , only a trajectory which
in addition satisfies this equation will be a well-defined, nontrivial solution once spacetime is
compactified.
5 Conclusion
In this note, we have constructed analytic solutions of open superstring field theory describing
marginal deformations generated by vertex operators with vanishing operator products. We
have not attempted to perform any detailed calculations with these solutions, though such
calculations are certainly possible. The really important questions about marginal solutions—
such as mapping out the relation between CFT and OSFT marginal parameters, obtaining
analytic solutions for vertex operators with singular OPEs, or proving Sen’s rolling tachyon
conjectures[22]—require more work even for the bosonic string. Hopefully progress will translate
directly to the superstring.
For us, the main motivation was the hope that marginal solutions could give us a hint
about how to construct the vacuum for the open superstring. Indeed, for the bosonic string
the marginal and vacuum solutions are closely related: To get the vacuum solution (up to the
ψN piece), one simply replaces J with d(Bc) = cKBc and takes the limit λ→ ∞10. Perhaps a
similar trick will work for the superstring.
The author would like to thank A. Sen and D. Gross for conversations, and A. Bagchi for
early collaboration. The author also thanks Y. Okawa for correspondence which motivated
discovery of the unitary analytic solution presented in appendix B. This work was supported
in part by the National Science Foundation under Grant No.NSF PHY05-51164 and by the
Department of Atomic Energy, Government of India.
10The λ used here and the λ parameterizing the pure gauge solutions of Schnabl[1] are related by λ(Schnabl) =
A B0,L0 with Split Strings
In many analytic computations in OSFT it is useful to invoke the operators B0,L0 and their
cousins[1, 4]. To avoid unnecessary transcriptions of notation, it is nice to accommodate these
types of operations in the split string formalism.
We begin by defining the fields,
L = (L0)L|I〉 L∗ = (L∗0)L|I〉 (A.1)
and their b-ghost counterparts B,B∗. We can split the operators L0,L∗0 into left/right halves
non-anomalously because the corresponding vector fields vanish at the midpoint[4]. The fields
L,L∗ satisfy the familiar special projector algebra,
[L,L∗] = L+ L∗ (A.2)
Following ref.[4] we may define even/odd combinations,
L+ = L+ L∗ = −K L− = L − L∗ (A.3)
where K is the field introduced before. . Note that we have,
L0 ·Ψ = LΨ+ΨL∗
B0 ·Ψ = BΨ+ (−1)ΨΨB∗ (A.4)
We can use similar formulas to describe the many related operators introduced in ref.[4]
Let us now describe a few convenient facts. Let J(z) be a vertex operator for a state J(0)|Ω〉
in Schnabl gauge, and let J = J(1)|I〉 be its corresponding field. Then,
[B−, J ] = 0 (A.5)
where [, ] is the graded commutator. A similar result [L−, J ] = 0 holds if J(0)|Ω〉 is killed by
L0. We also have the useful formulas,
LF = 1
FL− FL∗ = −1
L−F [L−,Ωγ ] = 2γKΩγ (A.6)
The third equation is a special case of,
[L−, G(K)] = 2KG′(K) (A.7)
with similar formulas involving B,B∗. Of course, these equations are well-known consequences
of the Lie algebra eq.(A.2).
As an application, let us prove the identity,
J1(0)|Ω〉 ∗ J2(0)|Ω〉 = (−1)J1FJ1B
F 2 − 1
J2F (A.8)
where J1, J2(0)|Ω〉 are killed by B0,L0. This expression occurs when constructing the marginal
solution (bosonic or superstring) in Schnabl gauge. The direct approach is to compute L−10
on the left hand side in split string notation; the resulting derivation is fairly reminiscent of
ref.[14]. Instead, we will multiply this equation by L0 and prove that both sides are equal. The
left hand side gives,
B0 · FJ1F 2J2F = BFJ1F 2J2F + (−1)J1+J2FJ1F 2J2FB∗
(−1)J1FJ1[B−, F 2]J2F
= (−1)J1FJ1BF 2J2F (A.9)
The right hand side gives,
L0 · FJ1B
F 2 − 1
J2F = LFJ1B
F 2 − 1
J2F + FJ1B
F 2 − 1
J2FL∗
L−, B
F 2 − 1
= FJ1B
F 2 − 1
J2F +
L−, F
2 − 1
J2F (A.10)
Focus on the commutator:
F 2 − 1
= [L−, F 2]
+ (F 2 − 1)
= 2F 2 − 2F
2 − 1
(A.11)
where we used eq.(A.7). This computation is a somewhat formal because the inverse of K is
not generally well defined, but it can be checked using the integral representation eq.(2.12).
Plugging the commutator back in, the F
terms cancel and we are left with,
L0 · FJ1B
F 2 − 1
J2F = FJ1BF
2J2F (A.12)
which after multiplying by (−1)J1 establishes the result.
Before concluding, we mention that any state of the form,
FJ1BG2(K)J2 ... BGn(K)JnF (A.13)
with [B−, Ji] = 0, is in Schnabl gauge. The proof follows at once upon noting,
[B−, BG(K)] = −2B2G′(K) = 0 (A.14)
so the entire expression between the F s commutes with B−. This is one way of seeing that the
nontrivial part of the group element eΦ − 1 for the superstring solution is in Schnabl gauge.
B Unitary eΦ
The analytic solution eq.(3.23) is very simple, but it has the disadvantage of not satisfying the
standard reality condition, i.e. eΦ is not unitary and Φ is not imaginary. Presumably there
is an infinite dimensional array of marginal solutions which do satisfy the reality condition,
and some may have analytic descriptions. In this appendix we give one construction which
is particularly closely related to our solution eq.(3.23). For a very interesting and completely
different solution, we refer the reader to an upcoming paper by Okawa[25].
Our strategy will be to find a finite gauge transformation of g in eq.(3.23) yielding a unitary
solution. The transformation is,
U = V g (B.1)
where V is some string field of the form,
V = 1 + dv (B.2)
with v carrying ghost number −1. A little thought reveals a natural candidate for V :
(B.3)
where g† is the conjugate of eq.(3.23):
g† = 1− λF 1
1− JλB F 2−1
XF (B.4)
and we use the Hermitian definition of the square root. Intuitively, this is just taking the
original solution and dividing by its “norm.” More explicitly, if we define,
gg† = 1 + T
T = λFX
1− λB F 2−1
F − λF 1
1− JλB F 2−1
−λ2FX 1
1− λB F 2−1
1− JλB F 2−1
XF (B.5)
then the required gauge transformation is given by the formal sum,
T n (B.6)
This proposal must be subject to two consistency checks. First, of course, is that the field
U is actually unitary. The proof is straightforward:
UU † =
= gg†
U †U = g†
g = g†(g†)−1g−1g = 1 (B.7)
The second check is that V is a gauge transformation of the form eq.(B.2). This follows if the
field T is BRST exact, T = du, since then we can write (for example),
V = 1 + d
uT n−1
(B.8)
A little guesswork reveals the following BRST exact expression for T :
T = d
1− λB F 2−1
F 2 − 1
(B.9)
This establishes not only that U is an analytic solution, but (perhaps more importantly) that
the simpler expression g is in the same gauge orbit with a solution satisfying the physical reality
condition. This leaves no question as to the physical viability of our original analytic solution
eq.(3.23).
As usual, the unitary solution U can be defined explicitly in terms of cylinder correlators by
expanding eq.(B.1) as a power series in λ. Unfortunately this is somewhat tedious because the
implicit dependence on λ in eq.(B.1) is complicated. As an expansion for the imaginary field
Φ, the first two orders agree with the Schnabl gauge solution (as they must11), while at third
order we find:
F 2 − 1
F 2 − 1
JF + FJB
F 2 − 1
F 2 − 1
FXF 2JB
F 2 − 1
+ FJB
F 2 − 1
F 2 − 1
JF 2XF + F 2XB
F 2 − 1
(FXF )3 (B.10)
This expression is much simpler than the Schnabl gauge solution at third order, which involves
intricate constrained and entangled integrals over moduli separating vertex operator insertions.
11The reality condition fixes the form of the second order solution uniquely within the K,B, J,X subalgebra.
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http://arxiv.org/abs/hep-th/0002211
	Introduction
	Bosonic Solution
	Superstring Solution
	Pure Gauge for Bosonic Solution
	Conclusion
	B0,L0 with Split Strings
	Unitary e
