Comments on real tachyon vacuum solution without square roots
HJE
Comments on real tachyon vacuum solution without square roots
E. Aldo Arroyo 0 1
0 Santo Andre , 09210170 S~ao Paulo, SP , Brazil
1 Centro de Ci
2 encias Naturais e Humanas, Universidade Federal do ABC
We analyze the consistency of a recently proposed real tachyon vacuum solution without square roots in open bosonic string eld theory. We show that the equation of motion contracted with the solution itself is satis ed. Additionally, by expanding the solution in the basis of the curly L0 and the traditional L0 eigenstates, we evaluate numerically the vacuum energy and obtain a result in agreement with Sen's conjecture.
String Field Theory; Tachyon Condensation

1 Introduction
2
3
4
5
6
Conservation laws and the two point vertex in the sliver frame
Curly L0 level expansion analysis of the real solution
L0 level expansion analysis of the real solution
Summary and discussion
Introduction
ErSch = c(1 + K)Bc
1
1 + K
;
{ 1 {
In open string eld theory [1], we say that a string eld
is real if obeys the following
reality condition
z =
;
where the double dagger denotes a composition of Hermitian and BPZ conjugation
introduced in Gaberdiel and Zwiebach's seminal work [2].
Analytic tachyon vacuum solutions that satisfy the above reality condition (1.1) exist
in the literature [3, 4], however they carry some technical complications. For instance,
Schnabl's original solution is real, but has some subtleties, the solution contains a singular,
projectorlike state known as the phantom term [5].
Solutions without the phantom term, known as simple solutions or ErlerSchnabl's type
solutions have been proposed [6{10], but they often fail to satisfy the reality condition. By
performing a gauge transformation over a nonreal simple solution, a real phantomless
solution has been constructed in reference [6]. However, as noted in reference [11], the cost
of having this real solution is the introduction of somewhat awkward square roots.
It would be desirable to have a solution that is both real and simple, namely without
square roots and phantom terms. This is precisely the issue that has been studied in a
recent paper [11], where the author has presented an alternative prescription to obtain a
real solution from a nonreal one which does not make use of a similarity transformation.
Basically, it has been shown that given a tachyon vacuum solution
together with its
corresponding homotopy operator A [12{14], the string eld de ned by
= Re( ) +
Im( ) A Im( ) is a real solution for the tachyon vacuum.
Applying this prescription for the case of the nonreal ErlerSchnabl's tachyon vacuum
solution [6]
the corresponding real solution [11] has been constructed
where the QBexact terms are given by
=
1
For this real solution the corresponding energy has been computed and shown that the
value is in agreement with the value predicted by Sen's conjecture [15, 16].
HJEP01(28)6
Nevertheless, for the evaluation of the energy, the equation of motion contracted with
the solution itself was simply assumed to be satis ed. In this paper, we compute the
cubic term of the action for the real solution (1.3) and discuss the validity of the previous
assumption. Additionally, by expanding the solution in the basis of curly L0 eigenstates,
we evaluate the energy numerically and obtain a result in agreement with Sen's conjecture.
Since the numerical evaluation of the energy by means of the curly L0 level expansion of the
solution is not a trivial task, in order to automate the computations of relevant correlation
functions de ned on the sliver frame, we have developed conservation laws.
This paper is organized as follows. In section 2, we evaluate the cubic term of the
action for the real solution and test the validity of the equation of motion when contracted
with the solution itself. In section 3, in order to automate the computations involved in the
numerical evaluation of the energy associated with the solution, we developed conservation
laws for operators de ned on the sliver frame. In sections 4, and 5, we compute the energy
by means of the curly L0 and the standard Virasoro L0 level expansion of the solution and
after using Pade approximants we show that the numerical results obtained for the energy
are in agreement with Sen's conjecture. In section 6, a summary and further directions of
exploration are given.
2
Computing the cubic term for the real solution
In reference [11], a new real solution for the tachyon vacuum has been proposed. This
solution in the KBc subalgebra [17, 18] takes the form
=
1
1
real solution (2.1). Recall that these terms were not necessary in the evaluation of the
kinetic term. The QBexact terms in (2.1) are given by
1
2
], after a lengthy algebraic manipulations, we arrive to
experience with other solutions [9, 18{20] that this assumption is not a trivial one. In
general, a priori there is no justi cation for assuming the validity of
without an explicit calculation. Therefore the cubic term of the action must be evaluated.
The computation of the kinetic term has been already done in reference [11] given as
a result
tr[ QB
+
Thus, for equation (2.3) to be valid, we must show that
HJEP01(28)6
by means of the following basic correlators
tr ce t1K ce t2K ce t3K
t2
(t1 + t2 + t3) 3 sin t1+t2+t3
t3
sin t1+t2+t3
tr Be t1K ce t2K ce t3K ce t4K c
s2(t2 + t3 + t4) sin
s2(t3 + t4) sin
3
3
+
2
s t34 sin
t3
s
t4
s
sin
t2
s
where s = t1 + t2 + t3 + t4.
1
16
1
=
=
(2.3)
(2.4)
(2.5)
(2.6)
(2.8)
3
16
cKc
c
To compute correlators containing the B string eld, we proceed in the same manner.
As an illustration, let us explicitly evaluate the correlator trhB (1+1K)2 cKc (1+1K)2 cKci. The
integral representation of this correlator is given by
Z 1
0
dt1dt2 t1t2e t1 t2 @s1;s2 trhBe t1K ce s1K ce t2K ce s2K ci
s1=s2=0
Using the correlator (2.9), from equation (2.12) we obtain
Performing the change of variables t1 ! uv, t2 ! u
the above double integral (2.13), we get
uv, R01 dt1dt2 ! R01 du R01 dv u into
For instance, employing the correlator (2.8), let us explicitly compute the correlator
tr cKc
1
c
1
To evaluate the above double integral, we perform the change of variables t1 ! uv, t2 !
Therefore, we have just shown that
tr B
1
1 + K
cKc
c
cKc =
cKc =
1
1 + K
c =
cKc =
cKc
2
6
3
2
2
15
4
;
;
;
15
4
4
2
;
2
2
Employing these results (2.16){(2.23) into equation (2.7) and adding up all terms, we
obtain the value for the cubic term
tr[
] = 0, it is guaranteed that the energy
associated with the solution (2.1) is directly proportional to the kinetic term
term by means of the curly L0 level expansion of the solution.
As we are going to show, when we insert the curly L0 level expansion of the solution into
the kinetic term, we are required to evaluate two point vertices for string elds containing
the operators L^
, B^ and c~p. These two point vertices can be evaluated by means of the
socalled conservation laws which will be studied in the next section.
3
Conservation laws and the two point vertex in the sliver frame
The operators employed in the basis of curly L0 eigenstates are given in terms of the basic
operators L^
, B^ and c~p. These operators are related to the worldsheet energy
momentum tensor T (z), the b(z) and c(z) ghosts elds respectively. We are going to derive the
conservation law for the L^ operator
L^ =
I dz
2 i
(1 + z2)(arctan z + arccotz) T (z) :
Using the conformal map z~ = 2 arctan z, we can write the expression of the L^ operator
in the sliver frame
L^ =
I dz~
2 i
"(Rez~) T~(z~) ;
where "(x) is the step function equal to
1 for positive or negative values of its argument
respectively.
{ 5 {
We need conservation laws such that the operator L^ acting on the two point vertex,
which we denote as V2 , can be expressed in terms of nonnegative Virasoro modes de ned
HJEP01(28)6
on the sliver frame1
where an and bn are coe cients that will be determined below.
v(2)(z~2)
puncture, v(1)(z~1)
To derive a conservation law of the form (3.5), we need a vector eld which behaves as
"(Rez~2) + O(z~2) around puncture 2, and has the following behavior in the other
O(z~1). A vector eld which does this job is given by
The expression of the conservation law for Virasoro modes de ned on the sliver frame
v(z) = (1 + z2)arccotz:
V2
X2 I
j=1 Cj 2 i
1
v(j)(z~j )T~(z~j )dz~j = 0 ;
For vertex operators i de ned on the sliver frame, the two functions f1 and f2 which
appear in the de nition of the two point vertex f1
1(0)f2
2(0) are given by
f1(z~1) = tan
(1 + z~1) ;
f2(z~2) = tan
2
2 z~2 :
(3.3)
(3.4)
(3.5)
(3.6)
(3.7)
(3.8)
(3.9)
(3.10)
is given by2
jpuncture.
where v(j)(z~j ) = (@z~j fj (z~j )) 1v(fj (z~j )), and Cj is a closed contour which encircles the
Using
equations
(3.3), (3.4)
and
(3.6) into
the
de nition
(@z~j fj (z~j )) 1v(fj (z~j )) of the vector elds v(1)(z~1) and v(2)(z~2), we nd that
v(j)(z~j )
=
v(1)(z~1) =
z~1
v(2)(z~2) = "(Rez~2)
z~2:
Due to the presence of the step function we see that the vector eld v(2)(z~2) is discontinuous
around puncture 2, since we are interested in the conservation law of the operator de ned
in equation (3.2), this kind of discontinuity is what we want. Using (3.7) and noting that
integration amounts to the replacement vn(i)z~in ! vn(i)L(ni) 1, we can immediately write the
conservation law
V2
L0
(1) + L^(2)
(2)
L0
= 0 :
1We are going to use the following notation O(i) to refer an operator O de ned around the ith puncture.
2This formula can be derived using the general prescription for conservation laws shown in
references [21, 22].
{ 6 {
We can write this conservation law (3.10) in the standard form as given in equation (3.5)
By the symmetry property of the two vertex, the same identity (3.11) holds after replacing
Regarding the conservation law for the B^ operator, since the b ghost is a conformal
eld of dimension two, the conservation laws for operators involving this eld are identical
HJEP01(28)6
to those for the Virasoro operators
Employing these conservation laws for the operators L^ and B^, together with the
commutator and anticommutator relations
[L(0i); L^(j)] = ij ^(j);
L
[B0(i); L^(j)] = ij ^(j);
B
[L(0i); B^(j)] = ij ^(j);
B
fB0(i); B^(j)g = 0;
[L(0i); c~(pj)] =
ij p c(pj) ;
fB0(i); c~(pj)g = ij 0;p ;
we can show that all two point correlation functions involving string elds constructed
out of the operators L^, B^ and c~p can be reduced to the evaluation of the following basic
correlators
To evaluate explicitly the above correlators (3.17) and (3.18), the following formulas
where the correlator c(x)c(y)c(z)
in general is given by
will be very useful
Sa;b
Ca;b
2 i
2 i
I dz za sin(bz) =
I dz za cos(bz) =
b a 1 cos
b a 1 sin
( a)
( a)
{ 7 {
sin
a
2 ;
a
For instance, let us compute correlator (3.17). Using (3.19) into equation (3.17), we have
It is clear that the above equation (3.22) can be written in terms of the functions (3.20)
and (3.21), so that we arrive to an explicit expression for the correlator (3.17)
In the same way, we can also derive the explicit expression for the correlator (3.18)
Note that in addition to the conservation laws, we will be required to know the action of
the BRST charge QB on the operators L^, B^ and c~p
[QB; L^(j)] = 0;
fQB; B^(j)g = L^(j);
fQB; c~(pj)g =
k)c~(pj)kc~(kj):
As an illustration of the use of conservation laws, we are going to compute a particular
correlator involving the operators B^ and L^
string elds
We choose, as an example, the following
1
X
k= 1
(1
= B^L^c~0c~1j0i;
= c~1j0i:
{ 8 {
i
i
i
(3.25)
(3.26)
(3.27)
Using these string elds, let us evaluate the correlator
Inserting equation (3.27) into equation (3.28) and using (3.26), we obtain
Using the conservation law (3.14) and the anticommutator relations (3.16), from
equation (3.29) we get
Employing the conservation laws (3.12), (3.14) and the commutator and anticommutator
relations (3.15), (3.16), from equation (3.30) we arrive to
tr[ QB ] = V2 c~0
t1 c t2 Bc t3 = X
(x1 p + y1 p)L^nc~pj0i
1
1
X
where we have used equation (3.24). These kind of computations can be automated in a
computer. Next, we are going to apply the results shown in this section to evaluate the
kinetic term by means of the curly L0 level expansion of the real solution (2.1).
4
Curly L0 level expansion analysis of the real solution
Since the kinetic term does not depend on the QBexact terms, we are going to consider
only the rst term of
given in equation (2.1). Let us de ne this term as
^ =
1
1
{ 9 {
Using the integral representation of 1=(1 + K)
we can write (4.1) as
^ =
tc + c t + c tBc
+
dsdt e s t sc t :
By writing the basic string elds K, B in terms of the operators L^, B^, and using the
modes c~p of the ghost eld c(z) de ned in the z~conformal frame z~ = 2 arctan z, we can
show that
4
2
=
8
2
;
(3.31)
(3.28)
(3.29)
(4.1)
(4.2)
(4.3)
where
=
1
2
1
2
where in this case
double integral
dsdt e s t
x
1 p =
(t1 + t2 + t3); x =
(t3
t1
t2); y =
Employing equation (4.4), it is possible to derive the curly L0 level expansion of the
string eld de ned in equation (4.3). As a pedagogical illustration, let us explicitly
compute the curly L0 level expansion of the last term appearing on the right hand side of
equation (4.3)
dsdt e s t sc t =
dsdt e s t
(x1 p + y1 p)L^nc~pj0i;
As we can see from equations (4.6) and (4.7), we are required to evaluate the following
dsdt e s t
x
1 p =
dsdt e s t
2 n+p 1( s
t + 1)n(t
s)1 p
:
(4.8)
Performing the change of variables s ! uv, t ! u
above integral (4.8), we obtain
uv, R01 dsdt ! R01 du R01 dv u into the
e u2 n+p 1(1
u)nu2 p(1
2v)1 p
1) 2 n+p 2 Z 1
du e u(1
u)nu2 p
1
2
1
X
1
X
n!
n
2n!
1
2
0
F (n; 2
p);
M
k
1
2
n!
n!
(4.6)
(4.7)
(4.9)
(4.11)
(4.12)
(4.13)
where we have de ned
F (M; N ) =
u)M uN =
X( 1)M k
(M + N
k)!
(4.10)
Proceeding in the same way, we can also calculate the curly L0 level expansion of the
rst terms appearing on the right hand side of equation (4.3). Adding up all the results,
we show that the string eld (4.1) has the following curly L0 level expansion
^ =
1
X
1
X
where the coe cients fn;p and fn;p;q are given by
fn;p =
fn;p;q =
(1
( 1)p) 2 n+p 4 3F (n; 1
p) + 2 1 p F (n; 2
(( 1)q
( 1)p) 2 n+p+q 6F (n; 2
p
q)
:
p)
;
To compute the kinetic term, we start by replacing the string eld ^ with zL0 ^ , so that
states in the curly L0 level expansion will acquire di erent integer powers of z at di erent
levels. As we are going to see, the parameter z is needed because we need to express the
kinetic term as a formal power series expansion if we want to use Pade approximants. After
doing our calculations, we will simply set z = 1.
Let us start with the evaluation of the kinetic term as a formal power series expansion
in z. By inserting the expansion (4.11) of the string eld ^ into the kinetic term, and using
the conservation laws studied in section 3 to evaluate the corresponding two point vertices,
we obtain
tr[zL0 ^ QB zL0 ^ ] =
+
+
4
2z2 +
41
4
293
2
1
4
Considering terms up to order z6, and setting z = 1, from equation (4.14) we get 3328%
of the expected result (2.4). In principle, we can compute the curly L0 level expansion of
the kinetic term up to any desired order, however as we increase the order, the involved
tasks demand a lot of computing time. We have determined the series (4.14) up to order
z18, and setting z = 1, we obtain about 1:5036
1015% of the expected result. As we can
see, if we naively set z = 1 and sum the series, we are left with a nonconvergent result.
Recall that in numerical curly L0 level truncation computations, a regularization
technique based on Pade approximants provides desired results for gauge invariant quantities
like the energy [6, 20, 23, 24]. Let us see if after applying Pade approximants, we can
recover the expected result.
term as follows
To start with Pade approximants, rst let us de ne the normalized value of the kinetic
Since the series for the kinetic term (4.14) is known up to order z18, we can write the series
for E^(z) up to order z20, and after considering a numerical value for , we obtain
(4.15)
(4.16)
In general, to construct a Pade approximant of order Pnn(z) for the normalized value of the
kinetic term (4.15), we need to truncate the series (4.16) up to order z2n.
2:84231
3:28987z3
Expanding the right hand side of (4.17) around z = 0 up to order z4 and equating the
coe cients of z0, z1, z2, z3, z4 with the expansion (4.16), we get a system of algebraic
equations for the unknown coe cients a0, a1, a2, b1, and b2. Solving those equations
we get
HJEP01(28)6
a0 =
Replacing the value of these coe cients inside the de nition of P22(z) (4.17), and evaluating
this at z = 1, we get the following value
(4.17)
(4.19)
(4.20)
The results of our calculations are summarized in table 1. As we can see, the value
of E^(z) at z = 1 by means of Pade approximants con rms the expected analytical result
1. Although the convergence to the expected answer gets
irregE^(1) = 32 tr[ ^ QB ^ ] !
right value.
ular at n = 4, by considering higher level contributions, we will eventually reach to the
Using an alternative resummation technique, we would like to con rm the expected
answer for the normalized value of the kinetic term. We have used a second method which
is based on a combination of Pade and Borel resummation. We replace the Borel transform
of E^(z), which is de ned as E^(z)Borel = P Ekzk=k!, by its Pade approximant Pnn(z)Borel
and then evaluate the integral
Penn(z) =
dt e t Pnn(zt)Borel
at z = 1. In the third column of table 1, we list the results obtained for E^(1) by means
of PadeBorel approximation. Note that starting at the value of n = 4, PadeBorel does a
little better than Pade.
5
L0 level expansion analysis of the real solution
To expand the string eld (4.3) in the Virasoro basis of L0 eigenstates, we are going to use
the following formulas
e t1K ce t2K Bce t3K
P22(z = 1) =
0:935125008:
=
r cos2
x
r
+
r cos2
y
r
(r
2y)
Pnn
1:3333333333
0:9351250080
0:7462344772
0:9803952323
0:9800827399
0:9997340118
Penn
1:3333333333
0:6792579899
0:9160629680
0:9938587065
1:0020031889
1:0017620332
1:3333333333
9:2413341787
3327:8214730
2:56791
9:62763
4:94676
107
109
1015
2 2
3
z tr[zL0 ^ QB zL0 ^ ] evaluated at z = 1. The second column shows the Pnn Pade
last column, P02n represents a trivial approximation, a naively summed series.
approximation. The third column shows the corresponding Penn PadeBorel approximation. In the
HJEP01(28)6
(5.1)
(5.2)
(5.3)
(5.4)
(5.5)
where the operator Uer is de ned as
To nd the coe cients un;r appearing in the exponentials, we use
r
2
2
r
tan
arctan z
f2;u2;r
f4;u4;r
f6;u6;r
f8;u8;r
f10;u10;r
fN;uN;r (z)
f2;u2;r (f4;u4;r (f6;u6;r (f8;u8;r (f10;u10;r (
(fN;uN;r (z)) : : : ))))) ;
where the function fn;un;r (z) is given by
fn;un;r (z) =
(1
z
un;rnzn)1=n :
Employing the set of equations (5.1) (5.3) for the string eld (4.3), we obtain
0
+
+
dt
^ =
Z 1
e tr sin2
2r
where r = 1 + t.
e s t(1 + s + t)2 cos2
(t s)
2(1+s+t)
8
0 2 tan
(t s)
2(1+s+t)
1 + s + t
A ; (5.6)
Uer c
2r Uerb 2kc
t
2 tan 2r
r
+ c
t
2 tan 2r
r
t
2 tan 2r
r
c
t
2 tan 2r
r
1
k=1
Uer
= lim
N!1
= lim
N!1
By writing the c ghost in terms of its modes c(z) = Pm cm=zm 1 and employing
equations (5.3) and (5.6), the string
eld ^ can be readily expanded and the individual
coe cients can be numerically integrated. For instance, let us write the expansion of ^ up
to level four states
^ = 0:45457753c1j0i + 0:17214438c 1j0i
the resulting string eld zL0 ^ , we de ne, the analogue of equation (4.15)
As in the case of the curly L0 level expansion analysis, to evaluate the normalized
value of the vacuum energy, rst we perform the replacement ^ ! zL0 ^ and then using
The normalized value of the vacuum energy is obtained just by setting z = 1. Since the
kinetic term is diagonal in L0 eigenstates, the coe cients of the energy (5.8) at order z2L
are exactly the contributions from
elds at level L. We have expanded the string eld
^ given in equation (5.6) up to level twelve states, and hence the series of E~(z) can be
determined up to the order z24
If we naively evaluate the truncated vacuum energy (5.9), i.e., setting z = 1 in the
series before using Pade or PadeBorel approximations, we obtain a nonconvergent result.
Note that the series (5.9) is less divergent than the series (4.16) that has been obtained in
the case of the curly L0 level expansion analysis of the energy.
Let us resum the divergent series (5.9). To obtain the Pade or PadeBorel
approximation of order Pnn for the energy, we will need to know the series expansion of E~(z) up
to the order z2n. The results of these numerical calculations are summarized in table 2.
6
Summary and discussion
We have analyzed the validity of the recently proposed real tachyon vacuum solution [11],
in open bosonic string eld theory. We have found that the solution solves in a non trivial
way the equation of motion when contracted with itself. Let us point out that a similar
test of consistency was performed by Okawa [18], Fuchs, Kroyter [19] and Arroyo [20] for
the case of the original Schnabl's solution [3].
As a second test of consistency, we have analyzed the solution from a numerical point of
view. Using either the curly L0, or the Virasoro L0 level expansion of the solution, we have
found that the expression representing the energy is given in terms of a divergent series,
(5.7)
HJEP01(28)6
(5.8)
(5.9)
z tr[zL0 ^ QB zL0 ^ ] evaluated at z = 1. The second column shows the Pnn Pade
last column, P02n represents a trivial approximation, a naively summed series.
approximation. The third column shows the corresponding Penn PadeBorel approximation. In the
which nevertheless can be resummed, either by means of Pade technique or a combination
of PadeBorel resummation to bring the expected result in agreement with Sen's conjecture.
It would be interesting to analyze other real solutions. For instance, the tachyon
vacuum solution corresponding to the regularized identity based solution [8]. The real
version of this solution, obtained by means of a similarity transformation, contains square
roots and consequently the analytical and numerical computations of the energy become
cumbersome [9, 23]. Employing the prescription studied in reference [11], it should be
possible to nd an alternative real version for this regularized identity based solution.
Finally, regarding to the modi ed cubic superstring eld theory [
25
] and Berkovits
nonpolynomial open superstring eld theory [26], since these theories are based on Witten's
associative star product, their mathematical setup shares the same algebraic structure of
the open bosonic string eld theory, and thus the prescription developed in reference [11]
and the results shown in this paper should be extended to construct and study new real
solutions in the superstring context like the ones discussed in references [24, 27{32].
Acknowledgments
I would like to thank Ted Erler and Max Jokel for useful discussions.
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.
[INSPIRE].
[1] E. Witten, Noncommutative geometry and string eld theory, Nucl. Phys. B 268 (1986) 253
[2] M.R. Gaberdiel and B. Zwiebach, Tensor constructions of open string theories. 1:
foundations, Nucl. Phys. B 505 (1997) 569 [hepth/9705038] [INSPIRE].
[3] M. Schnabl, Analytic solution for tachyon condensation in open string eld theory, Adv.
Theor. Math. Phys. 10 (2006) 433 [hepth/0511286] [INSPIRE].
[4] M. Schnabl, Algebraic solutions in open string eld theory  A lightning review, Acta
Polytechn. 50 (2010) 102 [arXiv:1004.4858] [INSPIRE].
[5] T. Erler and C. Maccaferri, The phantom term in open string eld theory, JHEP 06 (2012)
[6] T. Erler and M. Schnabl, A simple analytic solution for tachyon condensation, JHEP 10
[7] E.A. Arroyo, Generating ErlerSchnabltype solution for tachyon vacuum in cubic superstring
eld theory, J. Phys. A 43 (2010) 445403 [arXiv:1004.3030] [INSPIRE].
[8] S. Zeze, Regularization of identity based solution in string eld theory, JHEP 10 (2010) 070
[9] E.A. Arroyo, Comments on regularization of identity based solutions in string eld theory,
[10] T. Erler and C. Maccaferri, Connecting solutions in open string eld theory with singular
gauge transformations, JHEP 04 (2012) 107 [arXiv:1201.5119] [INSPIRE].
[11] M. Jokel, Real tachyon vacuum solution without square roots, arXiv:1704.02391 [INSPIRE].
[12] I. Ellwood, B. Feng, Y.H. He and N. Moeller, The identity string eld and the tachyon
vacuum, JHEP 07 (2001) 016 [hepth/0105024] [INSPIRE].
[13] I. Ellwood and M. Schnabl, Proof of vanishing cohomology at the tachyon vacuum, JHEP 02
(2007) 096 [hepth/0606142] [INSPIRE].
[14] S. Inatomi, I. Kishimoto and T. Takahashi, Homotopy operators and oneloop vacuum energy
at the tachyon vacuum, Prog. Theor. Phys. 126 (2011) 1077 [arXiv:1106.5314] [INSPIRE].
[15] A. Sen, Descent relations among bosonic Dbranes, Int. J. Mod. Phys. A 14 (1999) 4061
[hepth/9902105] [INSPIRE].
[INSPIRE].
[hepth/0611200] [INSPIRE].
[16] A. Sen, Universality of the tachyon potential, JHEP 12 (1999) 027 [hepth/9911116]
[17] T. Erler, Split string formalism and the closed string vacuum, JHEP 05 (2007) 083
[18] Y. Okawa, Comments on Schnabl's analytic solution for tachyon condensation in Witten's
open string eld theory, JHEP 04 (2006) 055 [hepth/0603159] [INSPIRE].
[19] E. Fuchs and M. Kroyter, On the validity of the solution of string eld theory, JHEP 05
(2006) 006 [hepth/0603195] [INSPIRE].
Phys. A 42 (2009) 375402 [arXiv:0905.2014] [INSPIRE].
[20] E.A. Arroyo, Cubic interaction term for Schnabl's solution using Pade approximants, J.
[21] L. Rastelli and B. Zwiebach, Tachyon potentials, star products and universality, JHEP 09
(2001) 038 [hepth/0006240] [INSPIRE].
033 [arXiv:1103.4830] [INSPIRE].
(2011) 079 [arXiv:1109.5354] [INSPIRE].
[22] E.A. Arroyo, Conservation laws and tachyon potentials in the sliver frame, JHEP 06 (2011)
[23] E. Aldo Arroyo, Level truncation analysis of regularized identity based solutions, JHEP 11
[24] E. Aldo Arroyo, Level truncation analysis of a simple tachyon vacuum solution in cubic
superstring eld theory, JHEP 12 (2014) 069 [arXiv:1409.1890] [INSPIRE].
theory, JHEP 11 (2013) 007 [arXiv:1308.4400] [INSPIRE].
2014 (2014) 063B03 [arXiv:1306.1865] [INSPIRE].
[arXiv:1204.0213] [INSPIRE].
[arXiv:1009.1865] [INSPIRE].
[25] I. Ya . Arefeva, P.B. Medvedev and A.P. Zubarev , New representation for string eld solves the consistency problem for open superstring eld theory, Nucl . Phys. B 341 ( 1990 ) 464 [26] N. Berkovits , SuperPoincare invariant superstring eld theory, Nucl . Phys. B 450 ( 1995 ) 90 [Erratum ibid . B 459 ( 1996 ) 439] [ hepth/9503099] [INSPIRE].
[27] E. Aldo Arroyo , Comments on multibrane solutions in cubic superstring eld theory , PTEP [28] E. Aldo Arroyo , Multibrane solutions in cubic superstring eld theory , JHEP 06 ( 2012 ) 157 [29] E.A. Arroyo , A singular oneparameter family of solutions in cubic superstring eld theory,