• 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:

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


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

    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:

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


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

    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:

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


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

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

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


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


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


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

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

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


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

    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:

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