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

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:

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

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:

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

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:

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

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:

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

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:

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

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

with the odd/even Grassmann string fields and the projection operator defined implicitly by:

    \[{\bar P_{\tilde Q{M^{n - 2}}}}\left| {{f^{\left( {n - 2} \right)}}} \right\rangle \in \tilde Q{M^{n - 2}}{\hat F^{ - n + 1}}\]

Putting all together, it follows that the action {S_{LS}} is BRST invariant.

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