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:

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