Supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2017R1D1A1A02017805)

^{3}and published under licence by Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd

We study covariant open bosonic string field theories on multiple

Article funded by SCOAP^{3}

String theories are defined only in critical dimensions: 10 dimensions for the super-string theories and 26 dimensions for the bosonic string theories. However, the quantum field theories, which describe open strings in the low energy region, can be defined in any dimension less than or equal to the critical dimension _{critical} if we construct the string field theories on _{critical}−1. Thus, string field theory provides a unique framework to explore low dimensional quantum field theories in a unified manner. The purpose of this work is twofold. First, we shall construct covariant string field theories on

The core strategy we shall adopt in the present work is deformed cubic open string field theory [

The deformation procedure transforms the non-planar world sheet diagrams of Witten’s cubic open string field theory [_{critical}–

On a ^{μ}^{i}_{critical}−1, are normal to the ^{μ}^{i}^{I}^{i}

The string propagator is obtained by evaluating the path integral on a strip with the Polyakov string action,^{1} = ^{2} = _{αn}_{α}_{n}_{αn}^{inσ}, with ^{I}

Because the end point of the open string is attached to one of ^{2} different quantum states and consequently, the string field Ψ carries the group indices of ^{a}^{2} − 1 are generators of the

It is not difficult to extend Witten’s cubic open string field theory [^{I}^{I}

Now we shall deform the cubic open string field theory on multiple ^{I}

Figure ^{I}_{A}^{I}

The world sheet diagram of the three-string scattering.

Auxiliary patch to be effectively removed by deformation.

Comparison of two string momentum bases.

If we choose the Neumann condition as the boundary conditions for the end points of the string on the patch, we may think of the patch as a world sheet of an open string propagating freely from the initial state on _{A}_{A}

It follows from consideration of this deformation that the initial states of the first string and the second string should be given as_{1}⟩ ⊗ |Ψ1⟩, Eq. (

It is important to note that we deform the cubic open string field theory only by choosing the external string states given as Eq. (

Effectively removing the auxiliary patch from the world sheet diagram of the three-string scattering by choosing the external string states appropriately, we find that the deformed world sheet diagram is the same as the planar diagram of the string field theory in the proper-time gauge [_{1} = 1, _{2} = 0, _{3} = ∞. The three temporal boundaries labelled _{r}_{0} = −2ln2. To obtain the Fock space representation of the three-string vertex, we need to solve Green’s equation on the world sheet of the three-string scattering. However, it is not a simple task to solve Green’s equation directly on the world sheet. Green’s functions on the world sheet may be obtained by using a conformal transformation (inverse Schwarz-Christoffel transformation) of the well-known Green’s functions on the upper half plane, which are given by

Three-string scattering diagram of string field theory in the proper-time gauge.

Construction of the Fock space representations of multi-string vertices in the case of the Neumann Green’s function _{N}_{D}_{r}_{s}_{r}_{r}

To be explicit, we may write the Fock space representation of the three-string vertex in terms of the Neumann function as^{(2)} − ^{(1)}. The three-string interaction may be written as

In the zero-slope limit, the external string states correpond to massless gauge fields ^{μ}^{i}^{μ}^{i}_{AAA}_{Aφφ}_{AAA}_{YM}

Here we only need to evaluate the term _{Aφφ}

The four-string scattering amplitude may be written at the tree level as

If we choose the external string states appropriately to encode physical information only on the halves of the external strings, as in the case of the three-string scattering, the non-planar diagram of the cubic string field theory may reduce to the planar diagram of string field theory in the proper-time gauge, as depicted in Fig.

Deformation of the four-string scattering diagram.

We may fix the _{r}_{1} and _{2} are two interaction times on the world sheet,

To evaluate the effective action in the zero-slope limit, we choose the external string states as_{AAAA}

The effective four-scalar field action can be also calculated in a similar way. The four-scalar vertex is obtained by choosing the external string states as

Effective four-scalar field interactions.

In the zero-slope limit, we have shown that there is an interaction term for scalar fields and the gauge fields _{Aφφ}_{φφφ}

Now we shall calculate the effective interaction term for the scalar field and the gauge field

From _{AAA}_{Aφφ}_{μ}_{i}_{AAAA}_{φφφφ}

Effective gauge-scalar field interactions.

If we collect the effective actions which are represented by field theoretical actions for the _{0} is the free field actions for the gauge fields and the scalar fields, which may be derived easily from the kinetic term of the string free field action trΨ * _{μ} A^{μ}

If ^{i}

What is more interesting is the case where ^{I}^{I}^{I}

The four-string scattering amplitude for the open strings on multiple ^{I}^{I}^{I}

In this present work, we have discussed cubic open string field theories on multiple

We defined the open string field theory on multiple

If we choose

We have shown that the deformed cubic open string field theory, if properly defined on multiple

Recently, the cubic open string field theories on