String field theory solution corresponding to constant background magnetic field

Following the method recently proposed by Erler and Maccaferri, we construct solutions to the equation of motion of Witten's cubic string field theory, which describe constant magnetic field background. We study the boundary condition changing operators relevant to such background and calculate the operator product expansions of them. We obtain solutions whose classical action coincide with the Born-Infeld action.


Introduction
: σ(z),σ(z). The boundary on the left of σ on the real axis corresponds to the boundary condition of BCFT 0 and the boundary on the right corresponds to that of BCFT * , and vice versa forσ. The order of the star product and that in the figure coincide if we take the direction of the real axis as above [4].
where I is the identity string field and σ(s),σ(s) are the BCC operators such that σ(s) changes the open string boundary condition from the one corresponding to BCFT * to the one corresponding to BCFT 0 , andσ(s) changes in reverse, as indicated in figure 1. We assume that σ(s),σ(s) are matter primary fields of weight 0 and satisfy the operator product expansion (OPE) for s > 0, where g * is the disk partition function of BCFT * and g 0 is that of BCFT 0 . We have the following algebraic relations involving σ,σ : With all these string fields, it is possible to construct a solution Ψ given by where Ψ tv is the Erler-Schnabl solution [4] for the tachyon vacuum and Σ,Σ are the string fields Using the formulaΣ which can be derived from (2.2), (2.4), and (2.6), it is possible to calculate the energy and the gauge-invariant observable [29][30][31][32][33] as These results imply that the solution corresponds to the boundary conformal field theory BCFT * . Therefore, in order to construct a solution corresponding to BCFT * , we need to construct primary Here σ * (s),σ * (s) are the BCC operators of weight h which satisfȳ for s > 0, and they act as the identity operator in the time direction.

BCC operators for a constant background magnetic field
We would like to construct a solution to the equation of motion (2.1) which corresponds to a constant magnetic field background. Here we consider the bosonic string theory in 26-dimensional Minkowski space-time and take BCFT 0 to be the usual one for a Dp-brane, i.e. Neumann boundary conditions for X µ (µ = 0, · · · , p) and Dirichlet boundary conditions for X I (I = p + 1, · · · , 25). We take BCFT * to be the one with a constant magnetic field, namely with F 0µ = 0. Our goal is to study the BCC operators in this setup and calculate the OPEs, which are necessary for the construction of the Erler-Maccaferri solution.

Canonical quantization
The BCC operators correspond to states of open strings with one end satisfying the BCFT 0 boundary condition and the other end satisfying the BCFT * boundary condition. Therefore it is possible to deduce properties of the BCC operators by studying the open string states in such sectors. Let us consider the worldsheet theory of the open string which is given by the conformal field theory (CFT) on the strip where the Neumann boundary conditions are imposed at σ = 0 and the other boundary at σ = π is coupled to the electromagnetic fields asˆA In the case at hand, we have a constant magnetic field (3.1) and we take Since F µν is a real antisymmetric tensor with F 0µ = 0, we can make a rotation to put it into block diagonal form We concentrate on one of the blocks, 0 and introduce complex coordinates, in these two dimensions. The boundary conditions for these variables turn out to be The open string theory with such boundary conditions is studied in [23], the results of which are reviewed in appendix A. The mode expansions of X andX are given by where z = e τ +iσ and λ is related to the magnetic field as 2πα ′ F 12 = tan πλ (0 ≤ λ < 1). In order for X 1 , X 2 to be real, (α k+λ ) † =α −k−λ and (x) † =x. α k+λ ,α k−λ , x, andx satisfy the commutation Equation (3.10) implies that a noncommutative space arises due to the background gauge field.
The energy-momentum tensor is given by and we get the Virasoro generators where : · : denotes the normal ordering. We construct a basis of Fock space using the oscillators To every ket state, there corresponds a BCC operator which may be used to construct the Erler-Maccaferri solution. In this paper, for simplicity, we restrict ourselves to the BCC operators corresponding to the ground states. From (3.12), we can see that these operators are primary fields with conformal A ground state can be expressed as a linear combination of |y that satisfies We can define the BCC operators σ y * ,σ y * such that Another choice of basis of the ground states is given by the eigenstates of the operator F 12 xx which is included in the angular momentum J associated with the rotational symmetry X → e iφ X,X → e −iφX : where a normal ordering constant is determined by the parity along the x 2 direction 5 . Since the zero modes satisfy x † =x and (3.10), the operator F 12x x is a kind of number counting operator and its expectation value must be positive or negative definite, depending on the signature of F 12 . So, the operator x is regarded as the annihilation operator andx as the creation operator if −F 12 > 0, and the roles of x andx are inverted if −F 12 < 0. Hence, the vacuum is defined as x |Ω = 0 for −F 12 > 0 or x |Ω = 0 for −F 12 < 0. We can define the BCC operators σ Ω * ,σ Ω * such that The eigenstate of the angular momentum is given asx n |Ω (n = 0, 1, 2, · · · ) for −F 12 > 0 and its eigenvalues are J = −(n + 1/2). For −F 12 < 0, x n |Ω is the eigenstate of J corresponding to the eigenstate n + 1/2. Classically, this is interpreted as circular motion in a rotational direction for a fixed direction of the magnetic field and, quantum mechanically, its angular momentum is discretized.

Toroidally compactified theory
We can follow the same procedure as above and deal with the case where the directions X 1 , X 2 are toroidally compactified as We obtain the same mode expansions (3.7), (3.8) and the commutation relations (3.9), (3.10). In this case, we need to introduce two unitary operators to get the representation of the zero-mode algebra (3.10) consistent with the periodicity (3.15). Since the Dirac quantization implies for some integer N , we can see that the U, V satisfy the relation It is well known that the algebra (3.18) has a |N |-dimensional representation. For |N | = 1, we can take if we diagonalize the operator V , the representation is explicitly given in matrix form as Let |k be the eigenstate of V corresponding to the eigenvalue ω k (k ∈ Z, |k| ≤ |N | 2 ). We normalize the eigenstates as k|l = δ k, l . (3.20) We can define the BCC operators σ k * ,σ k * so that The theory discussed in the previous subsection corresponds to the limit R 1 , R 2 , |N | → ∞ with fixed. The states |y defined in (3.13) can be given by the limit

Correlation functions
For the construction of Erler-Maccaferri solutions, it is necessary to obtain the OPEs of the BCC operators. We calculate the three-point and four-point correlation functions of the BCC operators and derive the OPE from these. Let us first consider the theory in the compactified space (3.15) and study the three-point function of which can be expressed as l| e ip·X (z,z) |k in the operator language. As is demonstrated in appendix B, this correlation function can be evaluated using (3.8), (3.9), and (3.10), and we obtain With the three-point function (3.22), the OPEs of the operators σ k * ,σ l * are obtained as for s > 0. g 0 is equal to the volume (2π) 2 R 1 R 2 of the two-dimensional space, and all we have to do is to obtain g * , which can be derived from the four-point function. The four-point functions of the BCC operators are calculated in appendix C, and we have (3.24) In order to evaluate g * , we examine the limit x → 1 − 0, which can be studied by rewriting it into the N 2 a n 2 .
From (3.25), we find that, for x ∼ 1, Comparing this with (3.23), we get 26) and the OPEs of

28)
g * can be expressed as which coincides with the contribution to the Born-Infeld action from the two dimensions we are dealing with.
The noncompact case can be dealt with in the same way. The three-point function and the four-point function of σ Ω ,σ Ω can be calculated to be For z ∈ R, the two-point functions of the BCFT 0 on the upper half plane are normalized as and those for the BCFT * should be Equations (3.31) and (3.32) imply although both g 0 and g * are infinite. We can derive the OPE for s > 0. They are given as an integral of the exponential operator over the continuous momenta and it is difficult to construct BCC operators satisfying (2.3) from these. OPEs of σ y * ,σ y * can be obtained either by calculating the correlation functions or simply by taking the limit (3.21) of (3.27). We get

Toroidally compactified theory
In the case of the theory in the toroidally compactified space, it is straightforward to construct σ * ,σ * .
Since (3.27) imply for small positive s, one can take From these, one can construct the Erler-Maccaferri solution (2.5) which describes the D-branes with gauge field strength F 12 . From (3.28), we can see that the action of the background is given by the difference between the Born-Infeld action and the D-brane tension.
It is also possible to construct solutions corresponding to multiple brane solutions with the F 12 back-ground by considering from which we can construct string fields Σ k ,Σ l similarly to (2.6) satisfyinḡ so that

Noncompact case
Unlike the compactified case, (3.34) and (3.33) include extra ln s dependence for small positive s, and therefore we cannot choose σ y * ,σ y * or σ Ω * ,σ Ω * as the σ * ,σ * satisfying the OPE (2.7). One somewhat artificial way to construct such BCC operators is to make the following linear combinations of σ y * ,σ y * : for a > 0. It is straightforward to derive the following OPEs for s > 0 from (3.34): From σ * ,σ * , we can construct solutions corresponding to the D-brane with the F µν background in the way explained in the previous subsection. Similarly, by taking different linear combinations of σ y * ,σ y * , 6 More general solutions can be given by where A ij denotes a Hermitian |N | × |N | matrix satisfying A can be considered as a projection operator.
such as for s > 0 and we can construct solutions corresponding to multiple D-branes with F µν from them.

Concluding remarks
In this paper, we have constructed the Erler-Maccaferri solutions corresponding to constant magnetic field configurations on D-branes. In order to do so, we have calculated the correlation functions of the BCC operators and obtained their OPEs of them in the cases of the theories in toroidally compactified and noncompact space. We have shown that the Born-Infeld action appears as the action for such backgrounds.
There are several important things to be studied about the solutions obtained in this paper. In the case of toroidally compactified space, the Chern number of the U (1) gauge field is nonvanishing.
Therefore the configuration should be a topologically nontrivial one, from the low energy point of view. One question is how such configurations are realized as those of the string field. Such a question may be studied by examining the position space profile of the gauge field, as was done in [22]. From the OPE of the operators σ * ,σ * , one can calculate the profile, which is easily seen to be a periodic function of the coordinates x 1 , x 2 . Therefore one expects that the gauge field profile has jump discontinuities as a function of x 1 , x 2 in order for the configuration to be topologically nontrivial 7 . We leave this for future work.
With the solution constructed in this paper, we should be able to deduce all the interesting features of the open string theory around the background. In particular, the relation to the noncommutative geometry should be seen by analyzing the solution. One thing that can be done would be to get the relation between the worldsheet variables of BCFT 0 and BCFT * by using the string field theory technique.
In [22], it is pointed out that the correspondence between the states in BCFT 0 and BCFT * can be given by the string fields Σ,Σ. It will be interesting to explore such a correspondence and compare with the one given in [34,35].

A First quantization of open strings in background gauge field
In this appendix, we discuss the first quantization of open strings in background gauge fields. It has been discussed in [23] and we present their results here in order to fix our notation.
Let us consider the worldsheet theory of the variables X,X defined in subsection 3.1. Here, we study the general case where the charges at the ends of the strings are q 1 , q 2 . The action is where θ = −2πα ′ F 12 . The system discussed in subsection 3.1 corresponds to the case q 1 = 0, q 2 = 1.
The canonical momentum is defined as and they satisfy the canonical commutation relations with all other commutators vanishing. The boundary conditions are given by First we consider the case q 1 + q 2 = 0. The variables X,X can be expanded as where q i θ = − tan πλ i (i = 1, 2) . ψ k (t, σ) are the mode functions which are solutions to the wave equation with boundary conditions (A.5): Their complex conjugates ψ * k (t, σ) satisfy the boundary conditions (A.6). SinceX is the Hermitian conjugate of X, the operators x,x, α k+λ1+λ2 , andα k+λ1+λ2 obey (A.10) The orthogonality relations are given bŷ π 0 dσψ * k (t, σ)Dψ l (t, σ) = π(k + λ 1 + λ 2 )δ k, l , Since ψ k (t, σ) and the constant mode form a complete set, the operators x,x, α k+λ1+λ2 , andα k+λ1+λ2 are expressed as From these expressions and the canonical commutation relations, it follows that The energy-momentum tensor is defined as and we get the Virasoro generator Let us turn to the case q 1 + q 2 = 0, which corresponds to a neutral string. An easy way to deal with this case is to take the limit λ 1 = −λ + ǫ, λ 2 = λ + ǫ, ǫ → 0 of the above results. Although the limit of the zero modes requires some care, we finally obtain the mode expansions 1 n α n e −int cos(nσ + πλ) , (A.18) 1 nα n e −int cos(nσ − πλ) (A. 19) and the commutation relations with all other commutators vanishing. The Virasoro generators are given by The Fock vacuum | p can be defined to satisfy | p corresponds to the primary field e i p· X (z,z) = e i(pX+pX) (z,z) with weight α ′ p 2 cos 2 πλ | p = e i p· X (0, 0) |0 .
The operator e i p· X (0, 0) here is normal ordered and can be expressed more precisely as Here, for regularization, we replace the local operator X(z,z) by an integral along the small contour around z = 0. Taking the limit ǫ → 0 with the factor on the second line of (A.21), we get the operator e i p· X (0, 0) normal ordered. This expression is useful in the calculation of three-point functions.

B Three-point functions
In this appendix, we show how to calculate the correlation functions of the form for z ∈ R, which play crucial roles in deriving the OPE of the BCC operators in subsection 3.3. Here, | ′ , | are states in the Fock space defined in subsections 3.1 and 3.2, and satisfy for k ≥ 0. Such correlation functions can be evaluated by essentially following the method in [36]. For z > 0, we take the primary field e i p· X in (B.1) to be the one corresponding to the delta function normalized ground state in the BCFT 0 . Therefore it should coincide with (A.21) in the λ = 0 case, and we get where R [· · · ] denotes the radial ordering. Substituting the mode expansions (3.7), (3.8) into (B.3), it is straightforward to calculate the expectation value on the right-hand side of (B.3) and we obtain where F (α, β, γ; z) is the hypergeometric function 8 and Using the formula [37] F (α, β, α + β; z) = Γ(α + β) for |arg(1 − z)| < π, |1 − z| < 1, the right-hand side of (B.3) is evaluated to be and ψ(x) is the digamma function. For z < 0, the three-point function (B.1) can be calculated in the same way using 8 We note the relation: and (B.4) with z = e πi |z|. We eventually obtain
Therefore we find that the four-point function can be expressed as where the constant C p is determined by taking the limit x → 0 in (C.20). Taking the limit of the right-hand side of (C.20), we get where δ is the one which appears in (B.5). On the other hand, using the OPE (3.23), we obtain 2 ω n 1 n 2 2 +n2j δ i−j, n1 (modN ) σ k * e −ip·X(1,1) σ l * , (C. 22) where p i = n i /R i (i = 1, 2). Substituting (3.22) into (C.22) and comparing it with (C.21), we can derive (C.24) Here, the sums over n, m correspond to those over the momenta in the