String Field Theory, Gauge Theory and the Landau-Stueckelberg action

By George Shiber Monday, June 5, 2017Friday, October 26, 2018 In Grand Unification Physics, M Theory, Mathematical Physics, Mathematics, Physics, Super String Theory, Supersymmetry, Witten Theory Gauge Field Theory, String Field Theory

Continuing from my last post where I discussed the triangular interplay between string-string duality, string field theory, and the action of Dp/M5-branes, here I shall discuss Stueckelberg string fields and derive the BRST invariance of the Landau-Stueckelberg action. Recalling that the action of M-theory in the Witten gauge is:

$\begin{array}{l}{S_M} = \frac{1}{{{k^9}}}\int\limits_{{\rm{world - volumes}}} {{d^{11}}} \sqrt {\frac{{ - {g_{\mu \nu }}}}{{ - \gamma }}} {T_p}^{10}d\Omega {\left( {{\phi _{INST}}} \right)^{26}}\left( {{R_{icci}} - A_\mu ^H\frac{1}{{48}}G_4^2} \right) + \\\sum\limits_{Dp} {D_\mu ^{SuSy}} {e^{ - H_3^b}}/S_{Dp}^{WV} + \sum\limits_{Dp} {D_\nu ^{SuSy}} {e^{H_3^b}}/S_{Dp}^{SV}\end{array}$

with $k$ the kappa symmetry term, ${g_{mn}}$ the metric on ${M^{11}}$ , and ${x^m}$ the corresponding coordinates with ${B_{mnp}}$ an antisymmetric 3-tensor. Hence, the worldvolume ${M^3}$ is:

$R \times {S^1} \times {S^1}/{Z_2}$

and the worldsheet action:

${S_{het}} = {S_{st}} + {S_{KK}} + {S_{\bmod }}$

being the sum of three terms:

${S_{st}} = \int {{d^2}} \sigma \frac{1}{2}\left( {{g_{mn}}{\eta ^{ij}} + {b_{mn}}{\varepsilon ^{ij}}} \right){\partial _i}{x^m}{\partial _j}{x^n}$

${S_{KK}}\int {{d^2}} \sigma {\varepsilon ^{ij}}{\partial _i}{x^I}{\partial _j}{x^m}A_m^I$

${S_{\,\bmod \,}} = \int {{d^2}} \sigma \frac{1}{2}\left( {{g_{IJ}}{\eta ^{ij}} + {b_{IJ}}{\varepsilon ^{ij}}} \right){\partial _i}{x^J}{\partial _j}{x^I}$

and the index I = 1, … , 22 labels 22 gauge fields: 16 coming from the internal dimensions of the heterotic string, and the other 6 gauge fields are the KK modes of the metric and antisymmetric tensor. The action ${S_{\bmod }}$ has a massless spectrum given by moduli fields corresponding to deformations of the Narain lattice and thus take values in the group manifold:

$\frac{{SO\left( {19,3} \right)}}{{SO\left( {19} \right) \times SO\left( 3 \right)}}$

Something deep has occurred: all the gauge fields of the action ${S_{het}}$ have appeared within a two-dimensional theory, and not a three-dimensional theory

which is precisely the long wavelength limit behavior of the open membrane:

the gauge fields are defined in terms of fields that live on 10-dimensional boundaries of M-theory

In the closed membrane case:

the gauge fields are defined in terms of 11-dimensional fields

which brought us to the connection between string field theory and Dp-branes. Recall that one derives the string propagator by an evaluation of the Witten super-symmetric quantum path integral on a fiber-strip with the Polyakov string action:

$G\left[ {{X_1};{X_2}} \right] = \int {D\left[ h \right]} D\left[ X \right]\exp \left( {iS} \right)$

with:

$S = - \frac{1}{{4\pi \alpha '}}\int_M {d\tau d\sigma } \sqrt { - h} {h^{\alpha \beta }}\frac{{\partial {X^I}}}{{\partial {\sigma ^\alpha }}}\frac{{\partial {X^J}}}{{\partial {\sigma ^\beta }}}{\eta _{IJ}}$

for $I,J = 0,...,d$ and the Regge parameter clear from context. In the proper-time gauge and the normal modes of the lapse and shift function in 2-D, the Polyakov metric has the following property:

$\sqrt { - h} {h^{\alpha \beta }} = \frac{1}{{{N_1}}}\left( {\begin{array}{*{20}{c}}{ - 1}&{{N_2}}\\{{N_2}}&{{{\left( {{N_1}} \right)}^2} - {{\left( {{N_2}} \right)}^2}}\end{array}} \right)$

allowing us to derive the open string field Polyakov propagator on the Dp-branes:

$\begin{array}{c}G\left[ {{X_1};{X_2}} \right) = \int_0^\infty {ds} \left\langle {{X_1}\left| {\exp \left[ { - is\left( {{L_0} - i\tilde \varepsilon } \right)} \right]} \right|{X_2}} \right\rangle \\ = \left\langle {{X_1}\left| {\frac{1}{{{L_0} - i\tilde \varepsilon }}} \right|{X_2}} \right\rangle \end{array}$

with:

${L_0} = \frac{{{p^\mu }{p_\mu }}}{2} + \sum\limits_{n = 1} {\frac{1}{2}} \left( {p_n^Ip_n^J + {n^2}x_n^Ix_n^J} \right){\eta _{IJ}} - 1$

and the momentum operators are given by:

${P^\mu }\left( \sigma \right) = \frac{1}{\pi }{\left( {{p^\mu } + \sqrt 2 \sum\limits_{n = 1} {p_n^\mu \cos \left( {n\sigma } \right)} } \right)_{,\mu = 0,1,...,d}}$

${P^i}\left( \sigma \right) = \frac{{\sqrt 2 }}{\pi }{\sum\limits_{n = 1} {p_n^i\sin \left( {n\sigma } \right)} _{,i = 0,1,...,d}}$

Since open string end-points are topologically glued to $N$ Dp-branes, open strings must have ${N^2}$ inequivalent quantum states and thus, the string field $\Psi$ has to carry the gauge group indices of $U\left( N \right)$ :

$\Psi \left[ X \right] = \frac{1}{{\sqrt 2 }}{\Psi ^0}\left[ X \right] + {\Psi ^a}\left[ X \right]{T^a}$

where ${T^a}$ are the generators of the SU(N) group, with $a = 1,...,{N^2} - 1$ . Hence, the string propagator on multi-Dp-branes takes the following form, with contraction and indices ordering:

$\begin{array}{l}{G^{ab}}\left[ {{X_1};{X_2}} \right] = i\left\langle {T{\Psi ^a}\left[ {{X_1}} \right]{\Psi ^b}\left[ {{X_2}} \right]} \right\rangle \\ = i\int D \left[ X \right]{\Psi ^a}\left[ {{X_1}} \right]{\Psi ^b}\left[ {{X_2}} \right]\exp \left\{ { - i\int {D\left[ X \right]{\rm{tr}}\Psi \left( {{L_0} + i\tilde \varepsilon } \right)\Psi } } \right\}\end{array}$

which yields the field theory action:

${S_0} = \int {D\left[ X \right]} {\rm{tr}}\Psi \left( {{L_0} - i\tilde \varepsilon } \right)\Psi$

BRST-invariantly as:

${S_0} = \int {{\rm{tr}}\Psi } * Q_{BRST}^{generators}\Psi$

Hence, the above field theory action implies that the string-string duality associates to every Dp–Brane a solution corresponding to the d–dimensional string–frame Lagrangian:

$\begin{array}{c}{\mathcal{L}_{S,d}} = \sqrt {\left| g \right|} \left\{ {{e^{ - 2\phi }}} \right.\left[ {R - 4{{\left( {\partial \phi } \right)}^2}} \right] + \\\frac{{{{( - )}^{p + 1}}}}{{2\left( {p + 2} \right)!}}{e^{a\phi }}\left. {F_{\left( {p + 2} \right)}^2} \right\}\end{array}$

with $\phi$ the dilaton, ${F_{\left( {p + 2} \right)}}$ the curvature of a (p + 1)–form gauge field:

${F_{\left( {p + 2} \right)}} = d{A_{\left( {p + 1} \right)}}$

where the two–index NS/NS tensor ${B^{(1)}}$ and the dual six-index heterotic five–brane tensor $\tilde B_{het}^{(1)}$ are given by:

$S_{WZ}^{(1)} = \int {{d^2}} \xi {B^{(1)}}$

and

$S_{WZ}^{(5)} = \int {{d^6}} \xi \tilde B_{het}^{(1)}$

Now we have the general form of a 10-D p-brane solution:

$\left\{ {\begin{array}{*{20}{c}}{ds_{S,d}^2 = {H^\alpha }dx_{\left( {p + 1} \right)}^2 - {H^\beta }dx_{\left( {D - p - 1} \right)}^2}\\{{e^{2\phi }} = {H^\gamma }}\\{{F_{0...pi}} = \delta {\partial _i}{H^{\tilde \varepsilon }}}\end{array}} \right.$

with:

$\left\{ {\begin{array}{*{20}{c}}{\alpha = \frac{1}{N}\left( {2 - a} \right)}\\{\beta = - \frac{1}{N}\left( {2 + a} \right)}\end{array}} \right.$

and:

$\left\{ {\begin{array}{*{20}{c}}{\gamma = \frac{1}{N}\left[ {2\left( {p + 1} \right) + \left( {2 + a} \right)\left( {1 - \frac{1}{2}d} \right)} \right]}\\{{\delta ^2} = - \frac{4}{N},\quad \,\tilde \varepsilon = - 1}\end{array}} \right.$

with

$N = \left( {p + 1} \right)a + \left( {1 - \frac{1}{2}d} \right){\left( {1 + \frac{1}{2}a} \right)^2}$

The general form of 11-D Mp–branes solutions, noting the absence of the dilaton field, with the following Lagrangian:

${\mathcal{L}_{Ein,d}} = \sqrt {\left| g \right|} \left[ {R + \frac{1}{2}{{\left( {\partial \phi } \right)}^2} + \frac{{{{( - )}^{p + 1}}}}{{2\left( {p + 2} \right)!}}{e^{\alpha \phi }}F_{\left( {p + 2} \right)}^2} \right]$

is:

$\begin{array}{*{20}{c}}{\alpha = - \frac{4}{N}\left( {d - p - 3} \right),}&{\beta = \frac{4}{N}\left( {p + 1} \right)}\\{\gamma = \frac{{4a}}{N}\left( {d - 2} \right),}&\begin{array}{l}{\delta ^2} = \frac{4}{N}\left( {d - 2} \right)\\\tilde \varepsilon = - 1\end{array}\end{array}$

Hence, the M2-brane solution is:

$ds_{Ein,11}^2 = {H^{ - 2/3}}dx_{\left( 3 \right)}^2 - {H^{1/3}}dx_{\left( 8 \right)}^2$

${F_{012i}} = {\partial _i}{H^{ - 1}}$

squaring the field strength gives the following M5-brane solution:

$ds_{Ein,11}^2 = {H^{ - 1/3}}dx_{\left( 6 \right)}^2 - {H^{2/3}}dx_{\left( 5 \right)}^2$

${F_{012345i}} = {\partial _i}{H^{ - 1}}$

In the string-frame Ramond-Ramond gauge field Lagrangian:

Dp-brane solutions have the following form:

$ds_{S,10}^2 = {H^{ - 1/2}}dx_{\left( {p + 1} \right)}^2 - {H^{1/2}}dx_{\left( {9 - p} \right)}^2$

${e^{2\phi }} = {H^{ - \frac{1}{2}\left( {p - 3} \right)}}$

${F_{0...pi}} = {\partial _i}{H^{ - 1}}$

From the string-string duality above and ${\mathcal{L}_{Ein,d}}$ , we can derive the kinetic term of Dp–branes in terms of the Born–Infeld action with the following form:

${S^{Dp}} = \int {{d^{p + 1}}} \xi {e^{ - \phi }}\sqrt {\left| {\det \left( {{g_{ij}} + {{\tilde F}_{ij}}} \right)} \right|}$

with the embedding metric and the gauge field world-volume curvature manifest, entailing the existence of a WZ/RR term that couples to Dp-branes:

$S_{WZ}^{Dp} = \int {{d^{p + 1}}} \xi \tilde {\rm A}{e^{\tilde F}}$

$\tilde {\rm A} = \sum\nolimits_{q = 0}^9 {{A_{\left( {q + 1} \right)}}}$

and where the heterotic 5–brane, the IIA five–brane and the D5–brane dual potentials are given by:

$^ * d{B^{(1)}} = d\tilde B_{het}^{(1)}$

$^ * d{B^{(1)}} = d\tilde B_{{\rm{IIA}}}^{(1)} - \frac{{105}}{4}CdC - 7{A^{(1)}}G\left( {\tilde C} \right)$

$^ * d{B^{(1)}} = d\tilde B_{{\rm{IIB}}}^{(1)} + Dd{B^{(2)}} - \frac{1}{4}{{\tilde \varepsilon }^{kl}}{B^{(2)}}{B^{(k)}}d{B^{(1)}}$

Parallels for the M5-brane are formally similar. We have the quadratic kinetic term:

${S^{M5}} = \int {{d^6}} \xi \sqrt {\left| g \right|} \left[ {1 + \frac{1}{2}{\mathcal{H}^2} + \wp \left( {{\mathcal{H}^4}} \right)} \right]$

with the WZ term:

$S_{WZ}^{M5} = \int {{d^6}} \xi \left[ {\frac{1}{{70}}\tilde C + \frac{3}{4}\mathcal{H}C} \right]$

and the dual 6–form potential:

$d\tilde C - \frac{{105}}{4}CdC = {\,^ * }dC$

By the field-property of the Polyakov propagator on the Dp-branes:

combined with the string-string duality, we can prove that all Dp-and-Mn–brane solutions preserve half of the SUSY. With the SUSY rules for the gravitino and dilatino in the string-frame given by:

$\delta {\psi _\mu } = {\partial _\mu }\tilde \varepsilon - \frac{1}{4}{\omega _\mu }^{ab}{\gamma _{ab}}\tilde \varepsilon + \frac{{{{( - )}^p}}}{{8\left( {p + 2} \right)!}}{e^\phi }F \cdot \gamma {\gamma _\mu }{{\tilde \varepsilon '}_{(p)}}$

$\delta \lambda = {\gamma ^\mu }\left( {{\partial _\mu }\phi } \right)\tilde \varepsilon + \frac{{3 - p}}{{4\left( {p + 2} \right)!}}{e^\phi }F \cdot {\gamma _\mu }{{\tilde \varepsilon '}_{(p)}}$

$F \cdot \gamma \equiv {F_{{\mu _1},,,{\mu _{p + 2}}}}{\gamma ^{{\mu _1},,,{\mu _{p + 2}}}}$

Let us consider the gauge covariantization of the proper-time gauge and the Ramond-Ramond gauge discussed above. The action for the covariant bosonic open string field theory is implicitly defined by the BRST operator $Q$ :

${Q^c} = - \frac{1}{2}\left\langle {{\Phi _1},Q{\Phi _1}} \right\rangle$

with respect to the BPZ conjugation-derived inner product, where the string field has the following Fock space expansion:

${\Phi _1} = {\phi ^{\left( 0 \right)}} + {c_0}{\omega ^{\left( { - 1} \right)}}$

where the following holds:

${\phi ^{\left( 0 \right)}} = \int {\frac{{{d^{26}}p}}{{{{\left( {2\pi } \right)}^{26}}}}} \left[ {\sum\limits_{\left| f \right\rangle } {\left| {{f^{\left( 0 \right)}}} \right\rangle } {\mkern 1mu} {\psi _{\left| f \right\rangle }}\left( p \right)} \right]$

${\omega ^{\left( { - 1} \right)}} = \int {\frac{{{d^{26}}p}}{{{{\left( {2\pi } \right)}^{26}}}}} \left[ {\sum\limits_{\left| g \right\rangle } {\left| {{g^{\left( { - 1} \right)}}} \right\rangle } \,{\psi _{\left| g \right\rangle }}\left( p \right)} \right]$

for the bosonic case, and:

$\left\{ {\begin{array}{*{20}{c}}{{\psi _{\left| f \right\rangle }}\left( p \right)}\\{{\psi _{\left| g \right\rangle }}\left( p \right)}\end{array}} \right.$

are the associated space-time fields. We can now write the action as:

${S^c} = - \frac{1}{2}\left\langle {\left( {{\phi ^{\left( 0 \right)}} - \frac{1}{{{L_0}}}{{\tilde Q}_\omega }^{\left( { - 1} \right)}} \right),{c_0}L\left( {{\phi ^{\left( 0 \right)}} - \frac{1}{{{L_0}}}{{\tilde Q}_\omega }^{\left( { - 1} \right)}} \right)} \right\rangle$

and is invariant under the gauge transformation:

$\delta {\Phi _1} = Q{\Lambda _0}$

with the gauge parameter being a Grassmann string field of

${N^g} = 0$

given as:

${\Lambda _0} = {\lambda ^{\left( { - 1} \right)}} + {c_0}{\rho ^{\left( { - 2} \right)}}$

In terms of ${\phi ^{\left( 0 \right)}}$ and ${\omega ^{\left( { - 1} \right)}}$ , the gauge transformation is expressible as:

$\delta {\phi ^{\left( 0 \right)}} = \tilde Q{\lambda ^{\left( { - 1} \right)}} + M{\rho ^{\left( { - 2} \right)}}$

$\delta {\omega ^{\left( { - 1} \right)}} = {L_0}{\lambda ^{\left( { - 1} \right)}} - \tilde Q{\rho ^{\left( { - 2} \right)}}$

It follows then that:

${\zeta ^{\left( { - 1} \right)}} = \tilde Q{\phi ^{\left( 0 \right)}} + M{\omega ^{\left( { - 1} \right)}}$

is gauge invariant. Hence, in terms of ${\zeta ^{\left( { - 1} \right)}}$ , the action:

becomes:

$\begin{array}{c}{S^c} = - \frac{1}{2}\left( {\left\langle {{\phi ^{\left( 0 \right)}},{c_0}{L_0}{\phi ^{\left( 0 \right)}}} \right\rangle - \left\langle {\tilde Q{\phi ^{\left( 0 \right)}},{c_0}{W_1}\left( {\tilde Q{\phi ^{\left( 0 \right)}}} \right)} \right\rangle } \right.\\ + \left. {\left\langle {{\zeta ^{\left( 1 \right)}},{c_0}{W_1}{\zeta ^{\left( 1 \right)}}} \right\rangle } \right)\end{array}$

Note that the gauge invariance of each of:

${\left\langle {{\zeta ^{\left( 1 \right)}},{c_0}{W_1}{\zeta ^{\left( 1 \right)}}} \right\rangle }$

and

${\left\langle {{\phi ^{\left( 0 \right)}},{c_0}{L_0}{\phi ^{\left( 0 \right)}}} \right\rangle - \left\langle {\tilde Q{\phi ^{\left( 0 \right)}},{c_0}{W_1}\left( {\tilde Q{\phi ^{\left( 0 \right)}}} \right)} \right\rangle }$

entails that the string field, by conformal gauge theory, has the following form:

$\phi _{N \le 1}^{\left( 0 \right)} = \int {\frac{{{d^{26}}p}}{{{{\left( {2\pi } \right)}^{26}}}}} \frac{1}{{\sqrt {\alpha '} }}\left( {\phi \left( p \right)\left| {0,p; \downarrow } \right\rangle + {A_\mu }\left( p \right)\alpha _{ - 1}^\mu \left| {0,p, \downarrow } \right\rangle } \right)$

$\omega _{N \le 1}^{\left( 0 \right)} = \int {\frac{{{d^{26}}p}}{{{{\left( {2\pi } \right)}^{26}}}}} \frac{i}{{\sqrt 2 }}\chi \left( p \right){b_{ - 1}}\left| {0,p; \downarrow } \right\rangle$

and we also have:

$\zeta _{N \le 1}^{\left( 1 \right)} = \int {\frac{{{d^{26}}p}}{{{{\left( {2\pi } \right)}^{26}}}}} \sqrt 2 \left( { - i\chi \left( p \right) + {A_\mu }\left( p \right){p^\mu }} \right){c_{ - 1}}\left| {0,p; \downarrow } \right\rangle$

Thus, our action becomes a sum of two gauge invariant terms:

$- \frac{1}{2}\left( {\left\langle {{\phi ^{\left( 0 \right)}},{c_0}{L_0}{\phi ^{\left( 0 \right)}}} \right\rangle - \left\langle {\tilde Q{\phi ^{\left( 0 \right)}},{c_0}{W_1}\left( {\tilde Q{\phi ^{\left( 0 \right)}}} \right)} \right\rangle } \right)\left| {_{N \le 1}} \right.$

and

$- \frac{1}{2}\left\langle {{\zeta ^{\left( 1 \right)}},{c_0}{W_1}{\zeta ^{\left( 1 \right)}}} \right\rangle \left| {_{N \le 1}} \right.$

Now, crucially:

${\left\langle {\tilde Q{\phi ^{\left( 0 \right)}},{c_0}{W_1}\left( {\tilde Q{\phi ^{\left( 0 \right)}}} \right)} \right\rangle }$

is equivalent to a gauge invariant action of massless vector field $- \frac{1}{4}{F_{\mu \nu }}{F^{\mu \nu }}$

and by the metaplecticity of:

$\left\langle {{\zeta ^{\left( 1 \right)}},{c_0}{W_1}{\zeta ^{\left( 1 \right)}}} \right\rangle$

it follows that gauge transformation up to level N = 1 is expandible in terms of a gauge parameter $\lambda$ as:

$\delta {A_\mu }\left( p \right) = i{p_\mu }\lambda$

$\delta \chi \left( p \right) = {p^2}\lambda$

We perform now a Virasoro reparametrization of the evolving string surface as a transformation:

$\delta \left| \Phi \right\rangle = i\sum\limits_{n = - \infty }^\infty {{b_n}{L_{ - n}}} \left| \Phi \right\rangle$

with $\left| \Phi \right\rangle$

the wave-function of the string, which in string field theory, must be interpreted as a functional $\Phi \left[ {x\left( \sigma \right)} \right]$ , giving us the functional action:

${S_R} = - \frac{1}{2}\left\langle {\Phi \left| {\left. {{K_R}\Phi } \right\rangle } \right.} \right.$

where the inner product is defined in terms of integrals over the whole string configuration space and $K$ the string field kinetic energy operator. By reparametrization invariance, we can derive the following:

$\delta {S_R} = - \frac{i}{2}\sum\limits_n {{b_n}} \left( {\Phi \left[ {{K_R},{L_{ - n}}} \right]\Phi } \right)$

and the following relations can be easily checked:

${L_n}\Phi \left[ {x\left( \sigma \right)} \right] = {0_{,\quad n\, > 0}}$

$\left\{ {\begin{array}{*{20}{c}}{\left[ {K,{L_0}} \right] = 0}\\{K{L_{ - n}} = {0_{,\quad n\, > 0}}}\end{array}} \right.$

Now, what makes $K$ unique contrastively to ${K_R}$ is its invariance under an large group of extra symmetries in addition to reparametrization implicitly expressed by:

$\delta \left| \Phi \right\rangle = i\sum\limits_{n = - \infty }^\infty {{b_n}{L_{ - n}}} \left| \Phi \right\rangle$

specifically, given by shifts:

$\delta \Phi \left[ {x\left( \sigma \right)} \right] = \mathcal{L}_{ - i}^{\left( n \right)}{\Psi _{ni}}\left[ {x\left( \sigma \right)} \right]$

and is a type of meta-gauge symmetry acting directly on the metaplectic phase space. Let us study some properties of this metaplectic gauge group as well as $K$ .

We expand $\Phi$ in terms of the eigenstates of the mass operator:

$2\left( {{L_0} - 1} \right) = {p^2} + 2\left\{ {\sum\limits_{n > 0} {{\alpha _{ - n}}} \cdot {\alpha _n} - 1} \right\} = {p^2} + {M^2}$

where the state ${\Phi ^{\left( 0 \right)}}$ is annihilated by all ${\alpha _n}$ . Then the string field functional can be written as:

$\begin{array}{l}\Phi \left[ {x\left( \sigma \right)} \right] = \left\{ {\phi \left( x \right)} \right. - i{A^\mu }\left( x \right)\alpha _{ - 1}^\mu - \\\frac{1}{2}{h^{\mu \nu }}\left( x \right)\alpha _{ - 1}^\mu \alpha _{ - 1}^\nu i{v^\mu }\alpha _{ - 2}^\mu \left. { + ...} \right\}{\Phi ^{\left( 0 \right)}}\end{array}$

The kinetic gauge of $\Phi$ is given by the action of ${L_{ - n}}$ on new string functionals. At first order we get the following equation of motion:

$\begin{array}{c}{L_{ - 1}}\Psi \left[ {x\left( \sigma \right)} \right] = \left( {p \cdot {\alpha _{ - 1}} + {\alpha _{ - 2}} \cdot + ...} \right)\left\{ {{\phi _\Psi }\left( x \right) - iA_\Psi ^\mu \left( x \right)\alpha _{ - 1}^\mu + ...} \right\}{\Phi ^{\left( 0 \right)}}\\ = \left\{ { - i{\partial ^\mu }{\phi _\Psi }\left( x \right) \cdot \alpha _{ - 1}^\mu + ...} \right\}{\Phi ^{\left( 0 \right)}}\end{array}$

and since ${A^\mu }\left( x \right)$ satisfies:

$\delta {A^\mu } = {\partial ^\mu }{\phi _\Psi }$

it follows that:

has the property of linearized Yang-Mills gauge invariance and the following transformation laws can be derived:

$\delta {h^{\mu \nu }} = \left( {{\partial ^\mu }A_\Psi ^\nu + {\partial ^\nu }A_\Psi ^\mu } \right) - {\eta ^{\mu \nu }}{\phi _\Xi }$

$\delta {v^\mu } = A_\Psi ^\mu + {\partial ^\mu }{\phi _\Xi }$

where $\Xi$ is the Chan-Paton field term.

With $\left| h \right\rangle$ a 0-level state and an eigenstate of ${L_0}$ with eigenvalue $h$ , we have the following definition for the contravariant form $M_{ij}^{\left( n \right)}$

$M_{ij}^{\left( n \right)}\left( h \right) = \left\langle h \right|\mathcal{L}_i^{\left( n \right)}\mathcal{L}_{ - j}^{\left( n \right)}\left| h \right\rangle$

Now we must define:

${\Pi ^{\left( n \right)}} = 1 - \mathcal{L}_{ - i}^{\left( n \right)}M_{ij}^{\left( n \right) - 1}\left( {{L_0}} \right)\mathcal{L}_j^{\left( n \right)}$

satisfying:

${\Pi ^{\left( n \right)}} = {\Phi _m} = {\Phi _m}\,\,,\quad m < n$

${\Pi ^{\left( n \right)}} = {\Phi _n} = {\Pi ^{\left( n \right)}}\mathcal{L}_{ - k}^{\left( n \right)}{\Phi _0} = 0$

with the 0-th projection operator:

$P = {\Pi ^{\left( 1 \right)}}{\Pi ^{\left( 2 \right)}}...{\Pi ^{\left( n \right)}}...$

Combining, we have at n-th-mass level a Klein-Gordon equation:

$- \int {{d^d}} x\frac{1}{{2f\left( {\lambda \left( n \right)} \right)}}{A_\mu }\left( {{\eta ^{\mu \nu }}{p^n} - {p^\mu }{p^\nu }} \right){A_\nu } = \int {{d^d}} x\left( { - \frac{1}{{4f\left( {\lambda '\left( n \right)} \right)}}F_{\mu \nu }^n} \right)$

which is gauge-invariant, thus the existence of ${A_\mu }$ .

Adding the Stueckelberg string fields to the fundamental string field $\Phi$ , we have the local Stueckelberg action:

$S = - \frac{1}{2}\left( {\Phi - \sum\limits_n {{L_{ - n}}S_n^{Stk}\left| {2\left( {{L_0} - 1} \right)} \right|\Phi - \sum\limits_n {{L_{ - n}}S_n^{Stk}} } } \right)$

We now define N-th-projection operators:

${P_N} = {\Pi ^{\left( {N + 1} \right)}}{\Pi ^{\left( {N + 2} \right)}}...$

along with:

${S_N} = - \frac{1}{2}\left( {{\Phi _N}\left| {2\left( {{L_0} - 1} \right){P_N}} \right|{\Phi _N}} \right)$

with

${\Phi _N} = \Phi - {P_N}\sum\limits_n {{L_{ - n}}S_n^{Stk}}$

Now, since we have:

${S_N} \equiv {S_{N - 1}}$

it follows that the Stueckelberg action:

$S = - \frac{1}{2}\left( {\Phi - \sum\limits_n {{L_{ - n}}S_n^{Stk}\left| {2\left( {{L_0} - 1} \right)} \right|\Phi - \sum\limits_n {{L_{ - n}}S_n^{Stk}} } } \right)$

is equivalent to the kinetic metaplectic gauge field action:

$S = - \frac{1}{2}\left( {\Phi \left| {K\Phi } \right.} \right)$

Now we introduce a new gauge:

$\left\{ {\begin{array}{*{20}{c}}{{b_0}{c_0}\tilde Q{\Phi _1} = 0}\\{\tilde Q{\phi ^{\left( 0 \right)}} = 0}\end{array}} \right.$

At N=1 gauge field level, we thus have:

${p^\mu }{A_\mu }\left( p \right) = 0$

which is a hybrid Landau-Stueckelberg gauge. So for

${L_0} \ne 0$

this fixes the gauge invariance of the Stueckelberg action ${S_{Stk}}$ above.

Note that under this condition, since the following holds:

$\tilde Q\left( {{\phi ^{\left( 0 \right)}} + \delta {\phi ^{\left( 0 \right)}}} \right) = \tilde Q\left( {{\phi ^{\left( 0 \right)}} + \tilde Q\frac{1}{{{L_0}}}{W_1}\left( {\tilde Q{\phi ^{\left( 0 \right)}}} \right)} \right) = 0$

any string field:

${\Phi _1} = {\phi ^{\left( 0 \right)}} + {c_0}{\omega ^{\left( { - 1} \right)}}\quad ,\;{L_0} \ne 0$

satisfies the Landau-Stueckelberg gauge condition given:

$\delta {\Phi _1} = Q{\Lambda _0}$

with:

${\Lambda _0} = \frac{1}{{{L_0}}}{W_1}\left( {\tilde Q{\phi ^{\left( 0 \right)}}} \right)$

${\Lambda _0} = {\lambda ^{\left( { - 1} \right)}} + {c_0}{\rho ^{\left( { - 2} \right)}}$

This in turn entails that:

$\tilde Q\left( {\tilde Q{\lambda ^{\left( { - 1} \right)}} + M{\rho ^{\left( { - 2} \right)}}} \right) = M\left( { - {L_0}{\lambda ^{\left( { - 1} \right)}} + \tilde Q{\rho ^{\left( { - 2} \right)}}} \right) = 0$

holds.

The action for this gauge condition is then:

$\begin{array}{c}{S_{LS}} = - \frac{1}{2}\sum\limits_{n = - \infty }^\infty {\left\langle {{\Phi _n},Q{\Phi _{ - n + 2}}} \right\rangle } - \frac{g}{3}\sum\limits_{l + m + n = 3} {\left\langle {{\Phi _l},{\Phi _m} * {\Phi _n}} \right\rangle } \\ + \sum\limits_{n = 2}^\infty {\left( {\left\langle {{{\left( {{\vartheta _{LS}}B} \right)}_{ - n + 3}},{\Phi _n}} \right\rangle + \left\langle {{{\left( {{\vartheta _{LS}}B} \right)}_n},{\Phi _{ - n + 3}}} \right\rangle } \right)} \end{array}$

with:

${\left( {{\vartheta _{LS}}B} \right)_n} = {c_0}{b_0}{M^{n - 2}}\tilde Q{B_{3 - n}}$

${\left( {{\vartheta _{LS}}B} \right)_{ - n + 3}}{c_0}{b_0}{W_{n - 1}}\tilde Q{B_n} + {b_0}{\rm{bpz}}\left( {1 - {{\bar P}_{\tilde Q{M^{n - 2}}}}} \right){{B'}_{4 - n}}$