Marginal deformation for the photon in superstring field theory
aa r X i v : . [ h e p - t h ] N ov Preprint typeset in JHEP style - HYPER VERSION
AEI-2007-042
Marginal deformation for the photon in superstringfield theory
Ehud Fuchs, Michael Kroyter
Max-Planck-Institut f¨ur GravitationsphysikAlbert-Einstein-Institut14476 Golm, Germany [email protected], [email protected]
Abstract:
We find solutions of supersymmetric string field theory that correspond tothe photon marginal deformation in the boundary conformal field theory. We revisit thebosonic string marginal deformation and generate a real solution for it. We find a mapbetween the solutions of bosonic and supersymmetric string field theories and suggest auniversal solution to superstring field theory.
Keywords:
String Field Theory. ontents
1. Introduction 1
2. Revisiting the bosonic string 5
3. A map between bosonic and supersymmetric solutions 104. Supersymmetric marginal deformations 11
5. The universal superstring solution 156. Conclusions 16A. Split string formalism 16B. Integrated strip formalism 16
1. Introduction
Our understanding of open bosonic string field theory [1] has deepened following Schnabl’sanalytic solution [2], as can be seen from the papers that followed [3–14]. Specifically,marginal deformations [15–17] were found in [18, 19] and used to study the rolling tachyonin [20] (for earlier related works see [21–28]). A different approach to generate such solutionswas given in [29]. The added value of the new approach is that the handling of marginaldeformations that correspond to operators with a singular OPE is much easier.For open superstring field theory [30–32] marginal deformation solutions were foundin [33–35]. The purpose of this paper is to generalize the methods of [29] to the supersym-metric theory. This gives the marginal deformation solution corresponding to the photonwhose OPE is singular.There is a similarity between pure gauge solutions to the bosonic Chern-Simon-liketheory and the equation of motion of the supersymmetric WZW-like theory. This similaritywas recently used to generate superstring marginal deformations [33]. To adapt the method– 1 –f [29] from the bosonic string to the superstring, we exploit a different similarity, one whichrelates the pure gauge solutions of the two theories.The results in [29] rely on the fact that solutions of bosonic string field theory can beformally written as pure gauge solutionsΨ = Γ(Φ) − Q Γ(Φ) , (1.1)where Q is the BRST charge. In the relation above, Φ is a string field from which thesolution is generated and Γ(Φ) is a function of the formΓ(Φ) = 1 + Φ + O (Φ ) , (1.2)where the product used is the star product and the “1” represents the identity state, whichguarantees that the function Γ(Φ) is invertible. The first order term generates the lineargauge transformation Q Φ.The reason that Ψ is a physical solution has to do with the singular nature of Φ. In [29],the singularity for the photon marginal deformation was due to the linear dependence ofΦ on x , which is the zero mode of the scalar field X (space-time indices are not explicitlymentioned throughout the paper as we always work with a single space-like scalar field).In order for the solution to make sense we require that Ψ is x -independent ∂ x Ψ = ∂ x (cid:0) Γ(Φ) − Q Γ(Φ) (cid:1) = 0 . (1.3)This resembles the equation of motion of the superstring field [30, 31] η (cid:0) G − QG ) = 0 , (1.4)where η is the superstring ghost field zero mode, which behaves as a second BRST charge(more details on the superstring field theory we are using are given in section 1.1). However,this similarity is not the one we wish to exploit. Instead, we wish to compare the puregauge solutions of both theories. In the supersymmetric theory the infinitesimal gaugetransformation depends on two gauge fields Λ , ˜Λ δG = − ( Q ˜Λ) G + G ( η Λ) . (1.5)The integrated form of this infinitesimal gauge transformation is G λ = e − λQ ˜Λ G e λη o Λ . (1.6)The two exponents above may be replaced by any functions of the form (1.2), since thiscorresponds to a field redefinition. Because we are interested in pure gauge solutions, G is the identity state and G is G = ˜Γ( ˜Φ) − Γ(Φ) , ˜Φ ≡ Q ˜Λ , Φ ≡ η Λ . (1.7)The gauge field Φ of the bosonic string obviously does not have insertions of thesuperstring ghost field ξ , which is the conjugate of the η ghost. Therefore, it is η -exactand can be used to define Λ such that Φ in both theories are the sameΛ = ξ ( z )Φ bosonic ⇒ Φ SUSY = Φ bosonic ≡ Φ , (1.8)– 2 –or any z . ˜Φ can also be based on Φ in a similar way. Here, we get a new state since Φ isnot Q -exact ˜Λ = P ( z )Φ ⇒ ˜Φ = Φ − P ( z ) Q Φ , (1.9)where P ( z ) is the inverse operator of Q described in section 1.1. This gives a mapping ofbosonic solutions to supersymmetric ones. One gets G = 1 + P Ψ + ... , (1.10)where the ellipsis stand for corrections, such as higher order λ corrections in the marginaldeformation case. This mapping seems to be a natural one, since up to higher ordercorrections the ghost number zero superstring field is obtained from the ghost number onebosonic string field by the action of the P operator. For the marginal deformation thiscanonical choice does not work because of the additional constrain of x -independence.A short calculation shows that the bosonic solution can be written asΨ( G ) = G − QG . (1.11)Therefore it is clear that if the supersymmetric solution is x -independent then the bosonicsolution follows suit ∂ x G = 0 ⇒ ∂ x Ψ = 0 . (1.12)The relation (1.11) between solutions of the bosonic and supersymmetric theories, is notone-to-one, different G ’s can result in the same Ψ. Thus, (1.12) does not hold in theother direction. The x -independence of Ψ does not impose any condition on how ˜Φenters G . However, as we will see, for any Ψ one can find a corresponding G that is x -independent. We find a map G (Ψ) satisfying ∂ x G (Ψ) = 0, such that (1.11) is obeyed,that is Ψ( G (Ψ)) = Ψ.The rest of the paper is organized as follows. We end this introduction by presenting thesuperstring field theory that we use. Then, in section 2 we summaries the bosonic marginaldeformation solutions [29] and extend the formalism to generate solutions satisfying thereality condition in 2.2. Next, in section 3, we find the condition for x -independence forsolutions of the supersymmetric theory. In section 4 we present the marginal deformationof the superstring. First, we show in 4.1 that the photon of the supersymmetric theorycan also be written as an exact state. Then, we generate a solution to all orders in 4.2 anda real solution in 4.3. In section 5 we suggest that our method can be used to generatethe universal superstring solution corresponding to Schnabl’s solution [2]. In this case itis possible to use (1.9). We wrap things up with conclusions in 6. In the appendices werelate our solutions to those of [33, 34]. There are two main versions of supersymmetric open string field theory. It was shown thatthe one introduced in [36] suffers from singularities due to collision of picture-changingoperators [37]. While it is plausible that a modification of this theory of the form presentedin [38–40] can save the day, we prefer to use the more established superstring field theory,due to Berkovits [30–32]. – 3 –he action in the NS sector is a generalization of WZW theory, where the two differ-entials ∂, ¯ ∂ are replaced by Q, η . The action can be extended to include also the Ramondsector, albeit not in a Lorentz-covariant way. Only fields in the NS sector get a vev. Theequation of motion for the NS sector is (1.4). The string field G in this equation depends onthe variables X µ , ψ µ , as well as on the b, c ghosts and on the “bosonized” superghosts [41].The bosonization is given by β = ∂ξe − φ , γ = ηe φ , (1.13)where ξ, η are a conjugate pair of fermions and φ is a scalar field, such that the solitons e ± φ are fermions. It is clear that the superghosts do not depend on the zero mode of ξ . Thus,the physical space is in the “small Hilbert space”, not including the zero mode, while the“large Hilbert space” contains one more copy of the small one with ξ acting on it.An important peculiarity of the RNS string is the existence of an infinity of vertexoperators for any given physical state. The various “pictures” of the vertex operators areeasy to understand using the ξ, η, φ variables. Each operator is assigned a picture-numberas in table 1.Now, given a vertex operator V , another ver- operator h n g n p b c -1 1 0 η ξ e qφ − q ( q +2)2 qβ = ∂ξe − φ -1 0 γ = ηe φ − J B P Table 1:
The conformal weight h , ghostnumber n g and picture number n p ofthe superstring field theory operators wework with. tex operator describing the same state, but in adifferent picture, is obtained by [41]˜ V = [ Q, ξV ] . (1.14)While this state is exact in the large Hilbert space,it is only closed in the small Hilbert space, due tothe appearance of ξ in its definition. Scatteringamplitudes can be calculated with any set of rep-resentatives from the equivalence classes of vertexoperators, as long as the picture number is exactlysaturated, as shown in [41].In Berkovits’ string field theory, the states livein the large hilbert space. However, since η actsas a generator of gauge transformation (in the lin-earized theory), the whole small hilbert space is composed of gauge degrees of freedom.Since the other part of the space is just ξ times the small hilbert space, it has exactly thecorrect amount of degrees of freedom to represent the string with Q as the BRST opera-tor. In this way the theory is described without the need to use explicit picture changingoperators. Thus avoiding the potential problems of other formulations.An important property of the large Hilbert space is that the BRST charge Q = I dzJ B ( z ) = I dz (cid:16) c ( T m + T ξη + T φ ) + c∂cb + ηe φ G m − η∂ηe φ b (cid:17) , (1.15)has an inverse in this space { Q, P ( z ) } = 1 , P ( z ) ≡ − ξ∂ξe − φ c ( z ) . (1.16)– 4 –o verify this we use the following identities T ξη = − η∂ξ , η ( z ) ξ (0) ∼ z , (1.17) T φ = − ∂φ∂φ − ∂ φ , φ ( z ) φ (0) ∼ − log z ,φ ( z ) e qφ (0) ∼ − q log ze qφ (0) , e q φ ( z ) e q φ (0) = z − q q e q φ ( z )+ q φ (0) . (1.18)Some other useful identities include Q = P ( z ) = η = ξ ( z ) = { Q, η } = 0 , { η , ξ ( z ) } = 1 . (1.19)These relations reveal a duality under exchange of Q with η and P ( z ) with ξ ( z ).
2. Revisiting the bosonic string
The bosonic string marginal solution of [29] was based on the fact that the physical photonstate can be written as an exact stateΨ = c∂X (0) | i = QX (0) | i . (2.1)This means that any pure gauge string field (1.1), which automatically satisfies the equationof motion, is a candidate photon marginal solution provided that it generates the first orderstate. This only requires Γ(Φ) to be of the form (1.2) andΦ = λX (0) | i + O ( λ ) . (2.2)We refer to different choices of Γ(Φ) as “different schemes” [29]. The solution to linearorder is scheme independent, and we can generate identical solutions to all orders usingdifferent schemes by modifying the higher order terms of Φ.For a solution to be meaningful, it also has to be x -independent. This can be achievedby an appropriate choice of the non-linear terms of Φ. We refer to such terms as counterterms. For the “left” and “right” schemesΓ L (Φ) = 11 − Φ ⇒ Ψ L = (1 − Φ) Q − Φ , (2.3)Γ R (Φ) = 1 + Φ ⇒ Ψ R = 11 + Φ Q (1 + Φ) , (2.4)we have an explicit expression for the counter terms that generates such a solution. Concen-trating on Ψ L , it would be x -independent, provided that Φ satisfies the linear differentialequation ∂ x Φ = λ (1 − Φ)Ω , (2.5)where Ω is the vacuum state. – 5 –t is the specific form of the function Γ L (Φ), which allows us to easily calculate deriva-tives despite the fact that we are working with a non-commutative algebra ∂ − Φ = 11 − Φ ∂ Φ 11 − Φ , (2.6)where ∂ can stand for any derivation. This gives ∂ x Ψ L = − ∂ x Φ Q − Φ + (1 − Φ) Q (cid:16) − Φ ∂ x Φ 11 − Φ (cid:17) = − λ (1 − Φ)Ω Q − Φ + λ (1 − Φ) Q (cid:16) Ω 11 − Φ (cid:17) = 0 . (2.7)To solve (2.5) we expand Φ = ∞ X n =1 λ n Φ n . (2.8)This reveals that the differential equation (2.5) is actually an infinite set of differentialequations ∂ x Φ = Ω , ∂ x Φ n = − Φ n − Ω . (2.9)Solving these linear equations order by order is straightforward, but although there aremany possible solutions, we only have one solution in closed formΦ n = − ( − n n ! ( X n , , . . . , | {z } n − ) . (2.10)Here we are using the n -vector notation to represent the wedge state | n + 1 i [42–44], wherethe vector elements represent the operator insertions at the n canonical sites of the wedgestate. Normal ordering at each site is implicit and 1 stands for the identity insertion, i.e.no insertion. This is illustrated in figure 1.Actually, at each order the number of degrees of freedom for generating a solution isdim(Φ n ) = (cid:18) n − n (cid:19) . (2.11)These degrees of freedom are in general complex. They correspond to the number of gaugedegrees of freedom within our ansatz. Next we would like to find a solution that satisfies the string field reality condition. Thereality condition states that hermitian conjugation and BPZ conjugation agree [1]. In ourvector notation the reality condition translates to the following statement. Write the statein the opposite orientation, with a factor of ( −
1) for every ∂X or ∂c insertion and nofactors for X and c and complex conjugate the coefficients. If this procedure returns theoriginal state then the state is real. For simplicity, we consider only real coefficients, as itturns out that this is sufficient for constructing a real string field. In particular we choosethe function Γ(Φ) of (1.1) to be a real function. As we will see the reality of Ψ implies– 6 – n − ( n +1) π ( n +1) π π Figure 1:
Graphical representation of the state Φ n (2.10). The worldsheet is a semi-infinitecylinder (the double-arrowed lines are identified with each other) of circumference ( n +1) π , wherethe coordinate patch is marked in gray. The canonical upper-half-plane coordinate ξ is mapped tothis cylinder using the transformation z = n +12 arctan( n +1 ξ ). The operator X n is a product of n scalar fields X ( z ) in the cylinder coordinates, where normal ordering is implicit. The 1’s stand forno operator insertion and in this sense they are redundant. They are only presented to clarify therelation to the n -vector notation in (2.10). that Φ should be imaginary and this implies that the deformation parameter λ should beimaginary .Since Φ is built only from X insertions the reality of Φ is simply related to its symmetry.The component λ n Φ n is imaginary provided it is symmetric under inversion when n is oddand antisymmetric when n is even. To evaluate the number of degrees of freedom weconsider the space of solutions of the homogeneous equation ∂ x Φ n = 0 . (2.12)The space of solutions of this equation is given by the quotient of the space of homogeneouspolynomials of degree n in n variables by the space of homogeneous polynomials of degree n − n variables. The dimension of this space is given by (2.11). We now divideboth spaces into symmetric and antisymmetric parts. The derivative ∂ x does not changethe symmetry property and, as we shall soon demonstrate, it is also possible to defineintegration in a way that respects the symmetry. Hence, the number of (anti-)symmetricdegrees of freedom is just the dimension of the quotient space of the two (anti-)symmetricspaces. The combinatorics is different for the cases of n odd/even. The result can besummarized by dim(Φ S,An ) = 12 (cid:18) n − n (cid:19) ± − n (cid:18) n − n (cid:19)! , (2.13)where the plus sign stands for the symmetric case.We have only two solutions in a closed form, the one described above in the left-schemeand a corresponding solution in the right-scheme. It is easy to see that these solutions are– 7 –ot real. We can, however, generate different solutions. Let us work in the left-scheme.At level two, imposing the reality condition, and using only real coefficients the uniquesolution is Φ = − (cid:0) ( X ,
1) + 2(
X, X ) − (1 , X ) (cid:1) . (2.14)At level three there are two degrees of freedom for choosing a real solution.Already in the expression for the counter terms at level two we have the term ( X, X ),which we interpret as ‘changing the scheme’ [29]. Thus, it may seem beneficial to startwith a scheme where the symmetry is more transparent. We want a systematic procedurefor generating real solutions. We define Φ ∗ to be the string field obtained from Φ by acombination of hermitian and BPZ conjugations. FromΨ ∗ = Γ(Φ ∗ ) Q Γ(Φ ∗ ) − , (2.15)we see that the reality condition can be written as Γ(Φ) = Γ(Φ ∗ ) − . (2.16)This condition is generically non-linear in Φ , Φ ∗ . However for schemes of the formΓ(Φ) = Γ( − Φ) − , (2.17)we get that the reality condition on Ψ gives the linear conditionΦ ∗ = − Φ . (2.18)It is indeed natural to require that Φ is imaginary since λ Φ is imaginary. The three otherschemes that were specifically considered in [29], i.e., the symmetric scheme, the exponentscheme and the square root scheme, are given byΓ S (Φ) = 1 + Φ2 − Φ2 , Γ E (Φ) = e Φ , Γ R (Φ) = r − Φ . (2.19)They all obey (2.18).For the symmetric scheme we can use the algebraic relation between the two Φ’s toobtain a differential equation analogous to (2.5), ∂ x Φ = λ (1 − Φ2 )Ω(1 + Φ2 ) . (2.20)Note that this equation is invariant under conjugation (2.18), since the conjugate of ∂ x is − ∂ x . This yields a recursion relation for Φ k , ∂ x Φ k = 12 ΩΦ k − −
12 Φ k − Ω − k − X j =1 Φ j ΩΦ k − − j . (2.21) Expanding Γ(Φ) in λ , this condition fixes the real part of the n th order in term of the lower orders. – 8 –e prove that real solutions to the above equation exist within our ansatz by providingan explicit integration recipe that is manifestly imaginary. This not only proves that a realsolution exists, but also gives an easy algorithm to find it order by order. In fact, one candefine explicitly infinitely many different recursion relations leading to real solutions. Weprovide explicit results for the first few coefficients one gets using some of these algorithms.We also give a closed form expression for one of the possible recursion relations.Given a site k , one can define the integration “localized at this site” of a length- n vector by Z k ( X j , .., X j k , .., X j n ) ≡ j k + 1 ( X j , .., X j k +1 , .., X j n ) − j k + 1)( j k + 2) X m = k ∂ X m ( X j , .., X j k +2 , .., X j n ) (2.22)+ 1( j k + 1)( j k + 2)( j k + 3) X m = k ∂ X m X m = k ∂ X m ( X j , .., X j k +2 , .., X j n ) − . . . The number of terms is finite, since the total power is finite. The result of applying morethan P m = k j m derivatives is zero. We are performing an integration by parts, such thatthe power at the k th site is raised, while the power at other sites is reduced.Since the power of X is always raised by one in the integration, the combination λ ( R k + R n − k ) is imaginary (recall that the number of X ’s equals the number of the λ ’s inΦ and that λ is imaginary). The integration operations are linear and so any combinationof the form Z ~α n ≡ n X k =1 α kn Z k , n X k =1 α kn = 1 , a kn = a n +1 − kn , (2.23)yields an imaginary integration prescription, which leads to a well defined recursion relation.Integrating (2.21) using such a recursion relation gives a solution that is imaginary byconstruction.For example, the choice α n = α nn = , α k / ∈{ ,n } n = 0, gives at the first few ordersΦ =( X ) , (2.24)Φ = 14 (cid:16) (1 , X ) − ( X , (cid:17) , (2.25)Φ = 148 (cid:16) ( X , ,
1) + 6( X , X, − X , , X ) − X, X , − X, , X ) − , X , X ) + 6(1 , X, X ) + (1 , , X ) (cid:17) . (2.26)Another simple choice is α kn = n . This gives another real solution that differs starting fromthe third orderΦ = 172 (cid:16) ( X , ,
1) + 9( X , X, − X , , X ) − X, X , − X, X, X ) − X, , X ) − , X , − , X , X ) + 9(1 , X, X ) + (1 , , X ) (cid:17) . (2.27)– 9 –he first choice seems more natural since it does not involve the scheme changing state( X, X, X ).Yet another possible integration scheme is to integrate each term of (2.21) at the Ω-site.This is not given by a choice of ~α n ’s, but it is easy to see that it also leads to a symmetricintegration prescription and therefore to a real solution. The recursion relation can bewritten explicitly in this case asΦ k = k X n =1 ( − n n ! (cid:16) ( ∂ n − x Φ k − , X n ) − ( X n , ∂ n − x Φ k − ) (cid:17) + k X n =1 n X l =1 k − X j =1 ( − n l − n − l )! n ( ∂ l − x Φ j , X n , ∂ n − lx Φ k − − j ) . (2.28)The third order term in this case isΦ = 124 (cid:0) , X, X ) + 3( X , X, − , X , − X, X, X ) (cid:1) , (2.29)while the fourth order term isΦ = 196 (cid:0) ( X , , , − (1 , , , X ) (cid:1) + 124 (cid:0) (1 , X, , X ) + (1 , X , X, − ( X , , X, − (1 , X, X , (cid:1) + 132 (cid:0) (1 , , X , X ) + ( X , , X , − ( X , X , , − (1 , X , , X ) (cid:1) + 116 (cid:0) ( X, X, X ,
1) + (
X, X , , X ) + ( X , , X, X ) (2.30) − (1 , X , X, X ) − ( X, , X , X ) − ( X, X, , X ) (cid:1) . The closed form expression for the recursion relations (2.28) allows us to calculate higherorder terms. The number of terms in Φ n seems to grow exponentially fast. For n = 1 .. , , , , , , , ,
3. A map between bosonic and supersymmetric solutions
In this section we show how an x -independent solution for the superstring can be builtfrom an x -independent solution of the bosonic string, such that (1.11) holds. To that endit is useful to define Ξ Γ ( x ) ≡ ( ∂ x Γ)Γ − . (3.1)It is interesting to observe that Γ is the path-ordered exponential of Ξ Γ Γ = P exp Z x Ξ Γ ( x ) dx ≡ ∞ X n =1 Z x Ξ Γ ( x ) dx Z x Ξ Γ ( x ) dx · · · Z x n − Ξ Γ ( x n ) dx n . (3.2)The freedom in defining the above integration goes beyond setting lower limits to theintegrals, as x can be related to X ( z ) insertions at any point on the boundary, givinga continuum of degrees of freedom. Restricting the resulting expressions to the form ofour ansatz, leaves us with the same expressions for Γ and the same ambiguity of defining– 10 –he integration scheme discussed in the previous section. A similar construction was usedin [35] to generate a real solutions from a real Ξ. The difference is that in [35], one integratesover the gauge parameter λ rather than over x .The condition for x -independence of Ψ is equivalent to the condition that Ξ Γ is Q -closed, since the definition of Ψ (1.1), implies Q Ξ Γ = Γ( ∂ x Ψ)Γ − = 0 . (3.3)For a supersymmetric solution of the form (1.7), x -independence impliesΞ Γ − Ξ ˜Γ = ˜Γ( ∂ x G )Γ − = 0 . (3.4)Thus, the condition we were after can be written as ∂ x ΓΓ − = ∂ x ˜Γ˜Γ − = Ξ , (3.5)where Ξ is arbitrary.Now, since ˜Φ is exact Q ˜Φ = 0 ⇒ Q ˜Γ = 0 ⇒ Q Ξ ˜Γ = Q Ξ Γ = 0 . (3.6)Thus, the x -independence of G results in a closed Ξ Γ . Meaning that the bosonic condi-tion (3.3) follows from the supersymmetric condition (3.4), in accordance with (1.12).The solutions representing the photon marginal deformation in the bosonic theory inthe left, right and symmetric schemes, all result in the expressionΞ Γ = λ Ω . (3.7)This relation is not modified if we replace Φ by ˜Φ. Therefore, all schemes can be usedinterchangeably to create supersymmetric solutions.
4. Supersymmetric marginal deformations
It is not a priori clear to which superstring state the bosonic photon marginal deformationwould be mapped. First, in 4.1 we show that the photon state of the supersymmetrictheory can be written as a pure gauge state. This proves that the full superstring photonmarginal deformation can be generated using our methods. Then, we demonstrate how toget explicit solutions to all orders in 4.2 and real solutions in 4.3.
Expanding the superstring field G = 1 + λG + O ( λ ) , (4.1)yields the linear order of the superstring field equation of motion (1.4) η QG = 0 . (4.2)– 11 –he photon state G = cξe − φ ψ (0) | i , (4.3)solves this equation. Here, ψ has an implicit µ index and is of conformal weight . Likein the bosonic case, we would like to write this state as a pure gauge state generated bya singular gauge transformation. This will allow us to generate the higher order terms forthis solution. In superstring field theory there are two gauge fields from which pure gaugestates can be built. Expanding the infinitesimal gauge transformation (1.5) to linear orderin λ gives G = − Q ˜Λ + η Λ . (4.4)Notice that G has the ξ operator in it, implying that it lies in the large Hilbert space.Like in the bosonic case we have to enlarge the Hilbert space using x . The ψ operatorwill be generated thanks to the γG m factor in the BRST charge G m = i √ ψ∂ z X ⇒ [ Q, X ] = c∂X − i √ ηe φ ψ . (4.5)Then it is natural to guess˜Λ = P X (0) | i ⇒ ˜Φ = Q ˜Λ = X (0) | i − P QX (0) | i . (4.6)The first term is redundant and can be canceled by the other gauge fieldΛ = ξX (0) | i ⇒ Φ = η Λ = X (0) | i . (4.7)In total we get G = − ˜Φ + Φ = P QX (0) | i = P ( c∂X − i √ ηe φ ψ )(0) | i = cξe − φ ψ (0) | i , (4.8)which is exactly what we want. To get a solution to the non-linear equation of motion we need to use the integrated gaugetransformation (1.7). Plugging the first order gauge parameters (4.6), (4.7) into (1.7)produces x -dependence at higher orders, no matter what functions Γ(Φ) , ˜Γ( ˜Φ) are used.We therefore need to add counter terms.One could try to use the bosonic Φ, where the counter terms are known (for exam-ple the left scheme solution (2.10)) together with the ˜Φ defined by (1.9). This gives an x -dependent solution, as can be seen by a direct calculation. Alternatively, we can findand solve a set of differential equations analogous to the ones of the bosonic case. This ispresented below. Note that we use the conventions of [29] for ∂X on the boundary (eq. 2.5 there), where ∂ denotesderivation with respect to the boundary coordinate z +¯ z . This convention results in simple expressions, sowe continue to follow it. This is what we mean by ∂X everywhere, except in the definition of G m , wherewe write explicitly ∂ z X . ∂ z X and ∂X differ by a factor of 2. For the other operators there is no such issue,because they are holomorphic. – 12 –e choose to work in the left scheme, for which we have a closed form solution in thebosonic theory. Relying on the relation between the bosonic and supersymmetric solutionswe write G L in a form similar to Ψ L G L = Γ L ( ˜Φ) − Γ L (Φ) = (1 − ˜Φ) 11 − Φ . (4.9)In accordance with section 3, we require that both Φ and ˜Φ satisfy equations similar tothe bosonic case ∂ x Φ = λ (1 − Φ)Ω , ∂ x ˜Φ = λ (1 − ˜Φ)Ω . (4.10)Not surprisingly, this gives an x -independent solution ∂ x G L = 0 . (4.11)It is insufficient to solve (4.10), since these equations do not by themselves imply theequation of motion (1.4). We want to find gauge fields Λ , ˜Λ generating these Φ , ˜Φ and asolution as in (1.7). We can write the following equations for the gauge fields ∂ x Λ = λ (cid:0) ξ (0)Ω − ΛΩ (cid:1) , ∂ x ˜Λ = λ (cid:0) P (0)Ω − ˜ΛΩ (cid:1) , (4.12)from which (4.10) directly follow. The position of the P and ξ operators was explicitlyshown to emphasize that they operate on the vacuum state and not star-multiply it. Thesolution for the gauge fields isΛ = ∞ X n =1 λ n Λ n , Λ n = − ( − n n ! ( ξX n , . . . , , (4.13)˜Λ = ∞ X n =1 λ n ˜Λ n , ˜Λ n = − ( − n n ! ( P X n , . . . , , (4.14)which yields the string fieldsΦ = η Λ = − ∞ X n =1 ( − λ ) n n ! ( X n , . . . , , (4.15)˜Φ = Q ˜Λ = − ∞ X n =1 ( − λ ) n n ! ( X n − nY X n − − n ( n − ZX n − , . . . , . (4.16)For this calculation we have used the commutation relation[ Q, X n ] = − i √ nηe φ ψX n − + nc∂XX n − − n ( n − ∂cX n − , (4.17)and defined Y ≡ − i √ P ηe φ ψ = − i √ cξe − φ ψ , Z ≡ c∂cξ∂ξe − φ . (4.18)The operator Z has the unique property that all its quantum numbers are zero.The field Φ of the superstring looks exactly the same as the field Φ of the bosonicstring. The fact that Φ is η closed means that the related gauge transformation can besimply written as in (1.8), Λ = ξ Φ . (4.19)This state does not obey (4.12), but it only differs from (4.13) by an η -closed term. Thus,both gauge fields result in exactly the same solution.– 13 – .3 Real solutions We now want to identify a real solution. The reality condition for the superstring field is G ∗ = G − . (4.20)The fields X , Y and Z are all real. We assume that Φ and ˜Φ are chosen such that theykeep the imaginary nature of their lowest order. Then for G to be real, the functions thatgenerate it need to satisfy (2.17) just as in the bosonic case. The next step is to imitatethe bosonic symmetric solution G = Γ S ( ˜Φ) − Γ S (Φ) = 1 − ˜Φ1 + ˜Φ 1 + Φ1 − Φ , (4.21)and require ∂ x Φ = λ (1 −
12 Φ)Ω(1 + 12 Φ) , ∂ x ˜Φ = λ (1 −
12 ˜Φ)Ω(1 + 12 ˜Φ) , (4.22)to get an x -independent solution. Just like in the left scheme solution, the expression forthe supersymmetric Φ is the same as that of Φ of the bosonic string. For ˜Φ we need tosolve the equation ∂ x ˜Λ = λ (cid:16) P (0)Ω −
12 ˜ΛΩ + 12 Ω ˜Λ −
18 ˜ΛΩ( Q ˜Λ) −
18 ( Q ˜Λ)Ω ˜Λ (cid:17) , (4.23)where we have chosen the symmetric form of the equation.We can use an integration choice analogous to that of the bosonic case (2.28) of inte-grating at the Ω site. This can be explicitly written as˜Λ k = k X n =1 ( − n n ! (cid:16) ( ∂ n − x ˜Λ k − , X n ) − ( X n , ∂ n − x ˜Λ k − ) (cid:17) + k X n =1 n X l =1 k − X j =1 ( − n l − n − l )! n ·· (cid:16) ( ∂ l − x Q ˜Λ j , X n , ∂ n − lx ˜Λ k − − j ) + ( ∂ l − x ˜Λ j , X n , ∂ n − lx Q ˜Λ k − − j ) (cid:17) . (4.24)This results in ˜Λ = 14 (cid:16) ( P, X ) − ( X , P ) (cid:17) + 12 (cid:16) ( X, P X ) − ( P X, X ) (cid:17) . (4.25)Note that unlike for the bosonic case, there is a freedom in choosing a real solution alreadyat the second order since the location of the P insertion should be specified.It is possible to simplify the expressions by choosing a different integration prescription,namely to integrate in the location of the P insertion. It should be understood that the P ’sappearing in expressions that result from Q ˜Λ in (4.23) are not the ones where integrationshould be performed, since a Q is acting on them. With this understanding, every Φ k hasexactly one site with a P insertion and our algorithm is well-defined. The second orderresult is then˜Λ = 14 (cid:16) (1 , P X ) − ( P X , (cid:17) , (4.26)˜Φ = 14 (cid:16) (1 , X − XY − Z ) − ( X − XY − Z, (cid:17) , (4.27) G = 12 (cid:16) (1 , XY ) − ( XY,
1) + (
Y, X ) − ( X, Y ) + (
Y, Y ) − ( Z,
1) + (1 , Z ) (cid:17) , (4.28)– 14 –here for the evaluation of G we have taken the expression for Φ from the bosonic string.Calculating higher order terms is straightforward, but not very illuminating.
5. The universal superstring solution
Schnabl’s original solution for the bosonic string [2] can also be written as a gauge trans-formation [3] Ψ λ = (1 − Φ) Q − Φ , Φ = λπ B † c (0) | i . (5.1)Φ may also be viewed as a singular gauge transformation since it generates a field whichis both exact and satisfies the Schnabl gauge B Q Φ = 0 . (5.2)The Siegel gauge does not seem to permit such states.Let us define a similarity transformation like the one we used for regularizing thethree-vertex [9] B s ≡ s − L B s L , Φ s ≡ s − L Φ . (5.3)For any finite s , states in the Schnabl gauge transform into states in the B s gauge. Inthe limit s → B † (and L † ).A conceptual difference between Schnabl’s universal solution and our marginal defor-mation is that for small λ his solution is indeed a pure gauge solution. Only at the criticalvalue λ = 1 does it become a physical solution. Still, we can speculate that the relationbetween bosonic and superstring solutions also holds for this case. The state G λ = (1 − ˜Φ) 11 − Φ , Φ = λπ B † c (0) | i , ˜Φ = QP (0)Φ . (5.4)is clearly a solution to the superstring field equation of motion. Φ was copied from thebosonic string and since it is built upon the vacuum state, there seems to be no ambiguityabout the location of the P insertion in ˜Φ. One can check that G λ satisfies Schnabl’s gauge B ( G λ −
1) = 0 . (5.5)We suggest that at the critical value of λ this could be the universal solution for thesuperstring. Generically, there is no tachyon in superstring field theory, so we should notthink of this state as being the tachyon vacuum. We believe that this solution representsa state with no D -branes and therefore has an empty cohomology.Like in the bosonic case, the study of this state should require some kind of regular-ization. We leave this study for future work.– 15 – . Conclusions It seems that all known solutions to bosonic and supersymmetric string field theories canbe written as pure gauge solutions. The difference between different solutions and differentapproaches is in the choice of the gauge field. The approach of this paper and [29] giveselegant results that generalize automatically to singular currents, but works only for thephoton operator. The approach of [18, 19, 33–35] works for all non-singular currents, butrequires complicated counter terms for handling singular currents. The generalization ofour approach to other operators was discussed in [29]. It would be interesting to completethis program.
Acknowledgments
We would like to thank Sudarshan Ananth, Rob Potting and Stefan Theisen for usefuldiscussions. The work of M. K. is supported by a Minerva fellowship. The work of E. F.is supported by the German-Israeli Project cooperation (DIP H.52).
A. Split string formalism
In order to compare our solution to that obtained by other authors it may be useful towrite it using the formalism of [33]. Insertion of X n will be described by an insertion overthe identity string field, X n ≡ X n | i = X ⋆ . . . ⋆ X | {z } n times . (A.1)Normal ordering in this expression is implicit, therefore the r.h.s. cannot be strictly viewedas a chain of matrix multiplications.Then, between any two insertion sites there is a strip of string that can be representedby F = Ω. For example, the bosonic left solution is given by1 − Φ = 1 + F ∞ X k =1 ( − λ ) k k ! X k F k − ≡ F ∞ X k =0 ( − λ ) k k ! X k F k − . (A.2)This can be written in short as1 − Φ =
F e ∂ α ∂ β e − αλX e β Ω (cid:12)(cid:12)(cid:12) α = β =0 F − . (A.3)Using the bosonic part of (4.17) we can write the solution asΨ = λF c∂XF − (1 − Φ)Ω(1 − Φ) − + λ F ∂cF − (1 − Φ)Ω (1 − Φ) − . (A.4) B. Integrated strip formalism
The marginal deformations in [18,19,34,35] were all based on the fact that the inverse of L can be written as an integration over the width of a strip of string. Here, we demonstratethat these solution can also be viewed as pure gauge solutions.– 16 –or the bosonic string, our solution is based on the fact that Ψ , which is closed Q Ψ = 0 , (B.1)can be written as an exact state Ψ = Q Φ . (B.2)Using the integrated strip one can define the state J , which satisfies QJ = 1 . (B.3)This state is defined as an integral of a wedge state with length varying between zero and π . Since π is the length of the local coordinate patch, the integral is over states whichremove string strips and as such, are not generally defined. The expression we get for thephysical state is, however, well defined. We can use this state to writeΦ = J Ψ ⇒ Q Φ = ( QJ )Ψ − J ( Q Ψ ) = Ψ , (B.4)which is exactly what we need. It is a formal gauge field, not strictly existing due tothe appearance of J in its definition and its gauge variation gives the correct first ordersolution. These are also the properties of our gauge field x | i .The full solution, ignoring the issue of singular OPE’s isΨ n = ( Q Φ )Φ n − = Ψ ( J Ψ ) n − . (B.5)This is exactly the form of the solution of [34]. The structure of this solution is like ours,yet the states involved are different. Specifically, X (0) | i 6 = J c∂X (0) | i , (B.6)and the solutions differ, but are presumably gauge equivalent.The supersymmetric theory requires a different J state. This time J satisfies therelation Qη J = 1 . (B.7)We can use this state to writeΛ = λ ( QG ) J ⇒ Φ = η Λ = − λ ( QG )( η J ) , (B.8)˜Λ = − λG ( η J ) ⇒ ˜Φ = Q ˜Λ = − λG − λ ( QG )( η J ) , (B.9)where we used (4.2). This means that every state of the form G = ˜Γ( ˜Φ) − Γ(Φ) , (B.10)solves the equation of motion, with the right linear term G , if the functions Γ , ˜Γ are ofthe form (1.2). The superstring marginal solution of [34], G − = 1 − λ − λ ( QG )( η J ) G , (B.11)is reproduced by choosing Γ(Φ) = 1 + Φ , ˜Γ( ˜Φ) = 1 + ˜Φ . (B.12)– 17 – eferences [1] E. Witten, Noncommutative geometry and string field theory , Nucl. Phys.
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