BRST Ghost-Vertex Operator in Witten's Cubic Open String Field Theory on Multiple $Dp$-branes

The Becchi-Rouet-Stora-Tyutin (BRST) ghost field is a key element in constructing Witten's cubic open string field theory. However, to date, the ghost sector of the string field theory has not received a great deal of attention. In this study, we address the BRST ghost on multiple $Dp$-branes, which carries non-Abelian indices and couples to a non-Ablelian gauge field. We found that the massless components of the BRST ghost field can play the role of the Faddeev-Popov ghost in the non-Alelian gauge field, such that the string field theory maintains the local non-Abelian gauge invariance.

In a recent study [1], we extended the Witten open string field theory [2,3] on a single D25brane to a cubic open string field theory on multiple Dp-branes, p = −1, 0, · · · , 25. On multiple Dp-branes, both the string field and gauge parameters carry non-Abelian group indices. We expect that the Faddeev-Popov ghost [4] structure originates from the low-energy sector of the BRST ghost field. Siegel [5,6] pointed out that the massless component of the BRST ghost field could be the Faddeev-Popov ghost of gauge theory. However, his discussion was limited to the U (1) gauge field [7], which is the low-energy sector of open string field theory on a single D25-brane.
In non-Abelian gauge theory, which describes the low-energy sector of an open string on multiple Dp-branes, the Faddeev-Popov ghost field interacts with the non-Abelian gauge field. Therefore, it is necessary to examine cubic string coupling to confirm that the Faddeev-Popov ghost structure is consistent with the non-Abelian gauge symmetry of the low-energy sector of strings on multiple Dp-branes. For this purpose, we construct the BRST ghost-vertex operator for Witten's cubic open string field theory on multiple Dp-branes. Because the BRST ghost fields transform nontrivially under conformal transformation, it is not easy to find a propagator on the string world sheet to evaluate the Polyakov string path integral, which leads us to the cubic string vertex in the ghost sector.
In this study, we construct the three-string vertex operator in the ghost sector explicitly for Witten's cubic open string field theory for multiple Dp-branes. In the next section, we define Witten's cubic open string field theory on multiple Dp-branes with the overlapping functions of the string and BRST ghost coordinates. We could directly convert the overlapping functions into their Fock space representations [8][9][10][11][12][13] to obtain the vertex operators. However, this procedure requires inverting infinite-dimensional matrices that could possibly not be uniquely defined. Moreover, the obtained vertex operators could possibly not correctly reproduce the scattering amplitudes represented by the Polyakov string path integrals. These problems were identified earlier by Cremmer and Gervais [14]. To avoid these problems, we apply a different strategy, called the Mandelstam procedure [15][16][17][18], advocated for the light-cone string field theory by Mandelstam where Q is the BRST operator. Here, the star product * with the BRST ghost coordinates is defined as This action is invariant under the BRST gauge transformation On multiple Dp-branes, both string field Ψ and gauge parameter field can carry U (N ) group indices.

A. BRST ghost coordinates of open string field theory
The ghost part of the Polyakov string path integral and action are given by The ghost coordinates b and c are Grassmann-odd fields on the two-dimensional space of conformal dimensions (2, 0) and (−1, 0). To evaluate the Polyakov string path integral and construct the vertex operators, we must know the propagators of the ghost fields on the string world sheet.
However, it is difficult to construct the propagators of the ghost fields of open strings directly on the string world sheet. We know that on the complex plane (closed string), the holomorphic and anti-holomorphic parts of the propagator are given as Thus, we must identify how the BRST ghost fields of an open string transform under conformal transformation.
To understand the conformal transformation of open string BRST ghost fields, it is convenient to consider an open string as a folded closed string [19]. The folding condition is as follows (on a cylindrical surface): which are equivalent to b n =b n , c n =c n .
Considering these folding conditions, we can define the ghost fields of open strings as follows: If the folding condition is imposed, the string world sheet of the closed string, which is a cylindrical surface for the free string, is folded into a strip, which corresponds to the string world sheet of an open string. The conformal mapping from the complex plane onto the cylindrical surface (the world sheet of a free closed string) is given by Under this conformal mapping, the b-c ghost fields transform as It follows from the conformal transformation of the b-c ghost coordinates and folding construction that the free propagator of the b-c ghost fields of an open string on a strip must be given as Details of the construction of Green's function and the calculations of the Neumann functions of the b-c ghost fields are given in the Appendix.

III. CONSTRUCTION OF VERTEX OPERATORS FOR BRST GHOST
Assuming that we found Green's function of the ghost fields on the string world sheet, in this section, we construct the vertex operator for the BRST ghost fields. The world sheet of the three-open-string interaction is displayed in Fig. 1. [20] 0 We can use Green's function for the ghost field on the string world sheet as G(η r , ξ r ; η s , ξ s ) = δ rs n≥−1 cos nη r cos nη s + n,m≥−1Ḡ rs nm e |n|ξr+|m|ξs cos nη r cos mη s .
With the boundary values of the ghost fields fixed by {b (r) , c (r) , r = 1, 2, 3}, we can evaluate the Polyakov string path integral, which defines the scattering amplitude as where ∂M (r) and r = 1, 2, 3 are the temporal boundaries of the string world sheet. Using Green's function, Eq. (15), we can rewrite F ghost as Now, we rewrite this expression of F ghost into an operatorial form. It is useful to consider a set of anticommuting operators We can construct a coherent state for these anticommuting operators with a set of eigenvalues θ and χ as follows: By comparing the coherent states in Eq. (19) with F ghost , the first line of Eq. (17) must correspond to the normalization factor of the corresponding coherent state.
Using the coherent state r |b (r) , c (r) , we can rewrite the Polyakov string path integral in the ghost sector in an operatorial form: Here,b (r) 0ĉ

IV. FADDEEV-POPOV GHOST OF NON-ABLEIAN GAUGE FIELD THEORY AND BRST GHOSTS IN OPEN STRING THEORY
We expand the string state in the ghost sector to identify the Faddeev-Popov ghost in the asymptotic region (in cylindrical space) as follows: Note that the component fieldsη 1 (x) and χ 1 (x) are massless. From the kinetic term of the string field Ψ gh |Q|Ψ gh , the component fields obtain their kinetic terms χ 1 ∂η 1 +η 1 ∂χ 1 .
If we collect the massless component field terms in the vertex, Eq. (23), we obtain As we demonstrate in the Appendix, Thus, the BRST ghost field terms in Eq. (25) reduce to the following term: To calculate the gauge-ghost field coupling, we choose the string state as follows: Now the gauge-ghost field coupling terms can be written as It can be further rewritten as Here, we make use of the previous results on open string Neumann functions [1] From calculations ofḠ rs 11 given in the Appendix, we havē ,Ḡ 21 00 = 0, where g = 2 6 3 5 5. This is the cubic interction part of the Faddeev-Popov ghost action

V. CONCLUSIONS AND DISCUSSIONS
In this study, we investigated the BRST ghost coordinate fields for open strings on multiple Dpbranes. On multiple Dp-branes, the string fields carry non-Abelian group indices and the massless components become non-Abelian gauge fields. In string field theory, the local gauge symmetry appears to be fixed covariantly; hence, the role of the ghost field, the massless component of the Abelian gauge symmetry is maintained in the low-energy region. However, the coupling constants of the ghost gauge fields and those of the cubic gauge fields do not agree. This could be due to the conic singularity of Witten's string field theory. It has also been noted before that the coupling of cubic gauge fields and that of quartic gauge fields are not in agreement with each other [22,23]; this was in a study of Witten's cubic string field theory using the level truncation method. This motivates us to study the ghost sector of cubic string field theory in a proper-time gauge [20,[24][25][26][27][28][29]. In a proper-time gauge, the coupling of cubic gauge fields and that of quartic gauge fields agree with each other [28].
The second issue this paper brings is the extension of this work to closed string field theory. It would be interesting to determine if the Kawai-Lewellen-Tye (KLT) relation [30] holds for the ghost sector, such that the relationship between the general and gauge covariances could be understood at a deeper level. This extension would complete our recent work on closed cubic string field theory [21].
I implicitly assume the Siegel gauge, which fixes the BRST invariance in a way compitable with the Lorentz gauge for gauge field. It may be interesting to examine the effect of choosing different gauge such as the Schnabl gauge [31,32], which simplifies the star product drastically.
This work could also shed light on the double copy theory [33][34][35][36], which is based on the proposal "gravity = gauge × gauge". A classical solution to Einstein gravity could possibly be obtained as a product of two copies of the non-Abelian gauge theory. In this approach, the ghost sector could help us understand the relationship between the general covariance of gravity theory and local gauge invariance of non-Abelian gauge theory. These issues will be discussed in subsequent papers.

(A3)
For Witten's BRST ghost field, the conformal mapping from the string world sheet, described by ζ r , r = 1, 2, 3, onto the upper half of the complex plane, denoted by z r , is defined by the following two consecutive mappings: where the local coordinates of the three patches are given as ζ r = ξ r + iη r , r = 1, 2, 3. At the interaction point, B is mapped to the origin of the disk and the external strings are located at e − 2πi 3 , 1, and e 2πi 3 , respectively. In a compact form, Then, each local coordinate patch on the unit disk is mapped onto the upper half plane by the following conformal transformation: The external strings are mapped to three points on the real line: Z n = tan n−2 3 π , n = 1, 2, 3, or explicitly, The Schwarz-Christoffel mapping from the local coordinate patch on the string work sheet to the upper (lower) half complex plane is expressed by series expansions [21] , c Here, we note that as z r → Z r and z s → Z s , Thus, a comparison of Eq. (B1) with Eq. (A2) yields To be explicit, •Ḡ rs n0 , n ≥ 0: Differentiating Eq. (A1) and Eq. (A2) with respect to ζ r , Taking the limit where z s → Z s (ω s → 0) (the leading term is proportional to 1/ω s ), Performing a contour integral around ω r = 0 (z r = Z r ), for n ≥ 0, Performing a contour integral dω r dω s (ω r ) −n−3 (ω s ) −m around ω r = 0 (z r = Z r ) and (B12) Note that this equation does not determine when m = 1,Ḡ rs n1 .