String-String Duality, String Field Theory and M/D-p-Branes

The D=6 string-string duality, crucial for allowing the interchanging of the roles of 4-D spacetime and string-world-sheet loop expansion, entails that there is an equivalence between the K-3 membrane action and the {T^3} \times {S^1}/{Z^2} orbifold action. Here are some thoughts and reflections.

In the bosonic sector, the membrane action is:

    \[\begin{array}{l}S = {S_M} + \int_{\partial {M^3}} {\left\{ {\frac{1}{2}} \right.} \left( {{g_{mn}}{\eta ^{ij}} + {b_{mn}}{\varepsilon ^{ij}}} \right)\\{{\not \partial }_i}{x^m}{{\not \partial }_j}{x^n} + \frac{1}{2}\left( {{g_{IJ}}{\eta ^{ij}} + {b_{IJ}}{\varepsilon ^{ij}}} \right)\\{{\not \partial }_i}{x^I}{{\not \partial }_j}{x^J} + {\varepsilon ^{ij}}{{\not \partial }_i}{x^J}{{\not \partial }_j}{x^m}\left. {A_m^J(x)} \right\}\end{array}\]

where:

    \[\begin{array}{l}{S_M} = \int_{{M^3}} {\left( {\sqrt { - {g_{mn}}{{\not \partial }_i}{x^m}{{\not \partial }_j}{x^n}} } \right.} + \\\frac{1}{6}{\varepsilon ^{ijk}}{{\not \partial }_i}{x^m}{{\not \partial }_j}{x^n}{{\not \partial }_k}{x^p}\left. {{B_{mnp}}} \right)\end{array}\]

Recall I derived the total action:

    \[\begin{array}{l}{S^{Total}} = \frac{1}{{2\pi {\alpha ^\dagger }12}}\int\limits_{{\rm{world - volumes}}} {{d^{26}}} x\,d\,\Omega {\left( {{\phi _{INST}}} \right)^2}\sqrt {\frac{{ - {g_{\mu \nu }}}}{{ - \gamma }}} \,{e^{ - {c_{2n}}/{\Upsilon _\kappa }(\cos \varphi )}} \cdot \\\left( {{R_{icci}} - 4{{\left( {{{\not D}^{SuSy}}\left( {{\phi _{INST}}} \right)} \right)}^2}} \right) + \frac{1}{{12}}H_{3,\mu \nu \lambda }^bH_3^{b,\mu \nu \lambda }/A_\mu ^H + \sum\limits_{D - p - branes} {S_{Dp}^{WV}} \end{array}\]

which is highly non-trivial since Clifford algebras are a quantization of exterior algebras. Applying to the Einstein-Minkowski fibre-bundle, we get via Gaussian matrix elimination, an expansion of {\not D^{SuSy}} via Green’s-functions, yielding the on-shell action of M-theory in the Witten gauge:

    \[{S_M} = \frac{1}{{{k^9}}}\int\limits_{{\rm{world - volumes}}} {{d^{11}}} \sqrt {\frac{{ - {g_{\mu \nu }}}}{{ - \gamma }}} {T_p}^{10}{\mkern 1mu} d{\mkern 1mu} \Omega {\left( {{\phi _{INST}}} \right)^{26}}\left( {{R_{icci}} - A_\mu ^H\frac{1}{{48}}G_4^2} \right) + \sum\limits_{Dp} {\not D_\mu ^{SuSy}} {e^{ - H_3^b}}/S_{Dp}^{WV} + \sum\limits_{Dp} {\not D_\nu ^{SuSy}} {e^{H_3^b}}/S_{Dp}^{SV}\]

with k the kappa symmetry term. With {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:

    \[R \times {S^1} \times {S^1}/{Z_2}\]

The bosonic sector lives on the boundary of the open membrane: two copies of R \times {S^1}, which naturally couple to the U(1) connections {A^J}.

Now, double dimensional reduction of the twisted supermembrane on:

    \[{M^{10}} \times {S^1}/{Z_2}\]

of

    \[\begin{array}{l}S = {S_M} + \int_{\partial {M^3}} {\left\{ {\frac{1}{2}} \right.} \left( {{g_{mn}}{\eta ^{ij}} + {b_{mn}}{\varepsilon ^{ij}}} \right)\\{{\not \partial }_i}{x^m}{{\not \partial }_j}{x^n} + \frac{1}{2}\left( {{g_{IJ}}{\eta ^{ij}} + {b_{IJ}}{\varepsilon ^{ij}}} \right)\\{{\not \partial }_i}{x^I}{{\not \partial }_j}{x^J} + {\varepsilon ^{ij}}{{\not \partial }_i}{x^J}{{\not \partial }_j}{x^m}\left. {A_m^J(x)} \right\}\end{array}\]

entails that the bosonic sector is that of the heterotic string:

    \[\begin{array}{l}{S_h}\int {{d^2}} \sigma \left\{ {\frac{1}{2}} \right.\left( {{g_{mn}}{\eta ^{ij}} + {b_{mn}}{\varepsilon ^{ij}}} \right){{\not \partial }_i}{x^m}{{\not \partial }_j}{x^n}\\ + \frac{1}{2}\left( {{g_{IJ}}{\eta ^{ij}} + {b_{IJ}}{\varepsilon ^{ij}}} \right){{\not \partial }_i}{x^I}{{\not \partial }_j}{x^I} + \\{\varepsilon ^{ij}}{{\not \partial }_i}{x^I}{{\not \partial }_n}{x^m}\left. {A_m^{(I)}(x)} \right\}\end{array}\]

with gauge group indices I = 1, … , 16.

It gets interesting when we consider:

    \[{M^{10}} = {T^3}{\rm{ }} \times {\rm{ }}{M^7}\]

with dimension:

    \[dim{\rm{ }}{H^1}\left( {{M^7}} \right) = 0\]

since the worldsheet action:

    \[{S_{het}} = {S_{st}} + {S_{KK}} + {S_{\bmod }}\]

is now just a sum of three terms:

    \[{S_{st}} = \int {{d^2}} \sigma \frac{1}{2}\left( {{g_{mn}}{\eta ^{ij}} + {b_{mn}}{\varepsilon ^{ij}}} \right){\not \partial _i}{x^m}{\not \partial _j}{x^n}\]

    \[{S_{KK}}\int {{d^2}} \sigma {\varepsilon ^{ij}}{\not \partial _i}{x^I}{\not \partial _j}{x^m}A_m^I\]

    \[{S_{\bmod }} = \int {{d^2}} \sigma \frac{1}{2}\left( {{g_{IJ}}{\eta ^{ij}} + {b_{IJ}}{\varepsilon ^{ij}}} \right){\not \partial _i}{x^J}{\not \partial _j}{x^I}\]

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:

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

Now, something fundamentally 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

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

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

In the closed membrane case:

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

Hence, the gauge fields of the closed membrane must be defined over M3 and not over its boundary, unlike the closed membrane, whose action on K3 \times {M^7} is:

    \[\begin{array}{l}{{S'}_M} = \int_{{M^3}} {{d^3}} \zeta \left( {\sqrt { - {g_{mn}}{{\not \partial }_i}{x^m}{{\not \partial }_j}{x^n}} } \right.\\ + \frac{1}{6}{\varepsilon ^{ijk}}{{\not \partial }_i}{x^m}{{\not \partial }_j}{x^n}\left. {{{\not \partial }_k}{x^p}{B_{mnp}}} \right)\end{array}\]

where {M^3} is {T^2} \times R with the spacetime being {M^7} \times K3.

Hence, the closed membrane action {S'_M} on {M^7} \times K3 reduces to:

    \[{S'_M} = {S'_{st}} + {S'_{KK}} + {S'_{\bmod }}\]

with:

    \[{S'_{st}} = \int {{d^3}} \sigma \sqrt { - {g_{mn}}{{\not \partial }_i}{x^m}{{\not \partial }_j}{x^n}} + \frac{1}{6}{B_{mnp}}{\varepsilon ^{ijk}}{\not \partial _i}{x^m}{\not \partial _j}{x^n}{\not \partial _k}{x^p}\]

    \[{S'_{KK}} = \frac{1}{6}\int {{d^3}} \sigma {\varepsilon ^{ijk}}{\not \partial _i}{x^a}{\not \partial _j}{x^b}{\not \partial _k}{x^m}{B_{abm}}\]

and

    \[{S'_{\bmod }} = \int {{d^3}} \sigma \sqrt { - {g_{ab}}{{\not \partial }_i}{x^a}{{\not \partial }_j}{x^b}} + \frac{1}{6}{\varepsilon ^{ijk}}{\not \partial _i}{x^a}{\not \partial _j}{x^b}{\not \partial _k}{x^c}{B_{abc}}\]

and since K3 surfaces have no one-cycles, it follows that the three-form potential that appears in {S'_{KK}} of the action:

    \[{S'_{\bmod }} = \int {{d^3}} \sigma \sqrt { - {g_{ab}}{{\not \partial }_i}{x^a}{{\not \partial }_j}{x^b}} + \frac{1}{6}{\varepsilon ^{ijk}}{\not \partial _i}{x^a}{\not \partial _j}{x^b}{\not \partial _k}{x^c}{B_{abc}}\]

can be expanded in terms of the cocycles of K3.

For the 22 2-cocycles of K3, one can decompose B in a similar way for the two-form potential:

    \[{B_{abm}} = b_{ab}^I\left( {{x^a}} \right)C_m^I\left( {{x^r}} \right)\]

with I = 1, …, 22 labeling the two-cycles of K3. So after insertion into {S'_{KK}}, we can derive:

    \[\int_{{M^3}} {{\varepsilon ^{ijk}}} {\not \partial _i}{x^m}{\not \partial _j}{x^b}{\not \partial _k}{x^a}b_{ab}^I\left( {{x^c}} \right)C_m^I\left( {{x^r}} \right)\]

Applying reparametrization invariance, one can set:

    \[\rho = {x^{11}}\]

where \rho is a worldvolume coordinate, and now one performs a dimensional reduction of:

    \[\int_{{M^3}} {{\varepsilon ^{ijk}}} {\not \partial _i}{x^m}{\not \partial _j}{x^b}{\not \partial _k}{x^a}b_{ab}^I\left( {{x^c}} \right)C_m^I\left( {{x^r}} \right)\]

Here are the key propositions relevant to the membrane/string duality of the low energy theory in D=7.

  • the kinetic terms for the gauge fields in D=7 supergravity are:

    \[\int_{{M^7}} {\sqrt { - {g^{\left( 7 \right)}}} } {a_{IJ}}F_{mn}^I{F^{Jmn}}\]

derived by a split of the 4-4 field strength H = dB, of the 11-dimensional supergravity action:

    \[{H_{abmn}} = b_{ab}^IF_{mn}^I\]

from the following term:

    \[\begin{array}{l}\int_{{M^{11}}} {\sqrt { - {g^{\left( {11} \right)}}} } {H^2} = \int_{{M^7}} {\sqrt { - {g^{\left( 7 \right)}}} } F_{mn}^I{F^{Jmn}}\\\int_{K3} {\sqrt { - {g^{\left( {K3} \right)}}} } b_{ab}^I{b^{Jab}}\end{array}\]

  • Membrane/string duality in D=7 requires the existence of a point in the moduli space of K3 where all the 22 gauge fields are enhanced via U(1) gauging: this is key to preserving kappa symmetry. Thus, at the point in the moduli space when the 22 two-cycles vanish the following holds:

    \[\left\{ {\begin{array}{*{20}{c}}{{{\not \partial }_{{x^{11}}}}b_{ab}^I = 0}\\{{{\not \partial }_{{x^{11}}}}g_{ab}^I = 0}\end{array}} \right.\]

  • Hence, dimensional reduction yields:

    \[\int_{{M^2}} {{\varepsilon ^{ij}}} {\not \partial _i}{x^m}{\not \partial _j}{x^b}b_{11b}^IC_m^I\]

So, the S-duality map:

    \[\left\{ {\begin{array}{*{20}{c}}{b_{a11}^I{{\not \partial }_j}{x^a} \to {{\not \partial }_j}{x^I}}\\{C_m^I \to A_m^I}\end{array}} \right.\]

takes:

    \[\int_{{M^2}} {{\varepsilon ^{ij}}} {\not \partial _i}{x^m}{\not \partial _j}{x^b}b_{11b}^IC_m^I\]

to:

    \[\int_{{M^2}} {{\varepsilon ^{ij}}} {\not \partial _i}{x^I}A_m^I\]

and is equivalent to the term {S_{KK}} in:

    \[{S_{\bmod }} = \int {{d^2}} \sigma \frac{1}{2}\left( {{g_{IJ}}{\eta ^{ij}} + {b_{IJ}}{\varepsilon ^{ij}}} \right){\not \partial _i}{x^J}{\not \partial _j}{x^I}\]

So, the above map acts on the induced metric on the worldvolume. It follows then that the term in {S'_{\bmod }} in:

    \[{S'_{\bmod }} = \int {{d^3}} \sigma \sqrt { - {g_{ab}}{{\not \partial }_i}{x^a}{{\not \partial }_j}{x^b}} + \frac{1}{6}{\varepsilon ^{ijk}}{\not \partial _i}{x^a}{\not \partial _j}{x^b}{\not \partial _k}{x^c}{B_{abc}}\]

yields, after a double dimensional reduction of {x^{11}}, the following:

    \[\int {{d^2}} \sigma \frac{1}{2}\left( {{g_{IJ}}{\eta ^{ij}} + {b_{IJ}}{\varepsilon ^{ij}}} \right){\not \partial _i}{x^J}{\not \partial _j}{x^I}\]

with:

    \[\left\{ {\begin{array}{*{20}{c}}{{g_{IJ}} = {g_{ab}}b_I^{11a}b_J^{11b}}\\{{b_{IJ}} = {B_{ab11}}b_I^{11a}b_J^{11b}}\end{array}} \right.\]

which yields an equivalence between:

    \[\int {{d^2}} \sigma \frac{1}{2}\left( {{g_{IJ}}{\eta ^{ij}} + {b_{IJ}}{\varepsilon ^{ij}}} \right){\not \partial _i}{x^J}{\not \partial _j}{x^I}\]

and

    \[{S_{\bmod }} = \int {{d^2}} \sigma \frac{1}{2}\left( {{g_{IJ}}{\eta ^{ij}} + {b_{IJ}}{\varepsilon ^{ij}}} \right){\not \partial _i}{x^J}{\not \partial _j}{x^I}\]

Thus, the S-duality map that takes {S_{KK}} to {S'_{KK}} also takes {S_{\bmod }} to the dimensionally reduced {S'_{\bmod }}.

To achieve the matching of gauge sectors of the closed and open membrane, we must generate the gauge fields of the closed membrane before dimensionally reducing the theory, as opposed to the gauge fields of the open membrane, which are always generated within the two-dimensional theory. This explains the origin of strong-weak duality in string theory. The strong coupling limit of type IIA string is 11-dimensional supergravity which is believed to arise as the long wavelength limit of supermembrane theory. So, gauge fields present in the 3-dimensional theory will be strongly interacting, and will continue to be strongly interacting after dimensional reduction to a two-dimensional theory. However, the open membrane has its gauge fields appearing in two dimensional theories, which are therefore weakly interacting.

So, we must consider the spacetime part of the action for the closed membrane:

    \[\begin{array}{l}{S_M} = \int_{{M^3}} {\left( {\sqrt { - {g_{mn}}{{\not \partial }_i}{x^m}{{\not \partial }_j}{x^n}} } \right.} + \\\frac{1}{6}{\varepsilon ^{ijk}}{{\not \partial }_i}{x^m}{{\not \partial }_j}{x^n}{{\not \partial }_k}{x^p}\left. {{B_{mnp}}} \right)\end{array}\]

The term:

    \[\int_{{M^3}} {\sqrt { - {g_{mn}}{{\not \partial }_i}{x^m}{{\not \partial }_j}{x^n}} } \]

can be dimensionally reduced to:

    \[\int_{{M^2}} {\sqrt { - {g_{mn}}{{\not \partial }_i}{x^m}{{\not \partial }_j}{x^n}} } \]

which is equivalent to the first term in:

    \[{S_{st}} = \int {{d^2}} \sigma \frac{1}{2}\left( {{g_{mn}}{\eta ^{ij}} + {b_{mn}}{\varepsilon ^{ij}}} \right){\not \partial _i}{x^m}{\not \partial _j}{x^n}\]

and the term:

    \[\int_{{M^3}} {{\varepsilon ^{ijk}}} {\not \partial _i}{x^m}{\not \partial _j}{x^n}{\not \partial _k}{x^p}{B_{pmn}}\]

maps to:

    \[\int_W {d{\Sigma ^{mnpq}}} {H_{mnpq}}\]

with H = dB and W members of {H_4}\left( {{M^7}} \right)

Now, since the term is topological, and S-duality of the seven dimensional space entails:

    \[{H^3}\left( {{M^7}} \right) = {H^4}\left( {{M^7}} \right)\]

then one can reduce:

    \[\int_W {d{\Sigma ^{mnpq}}} {H_{mnpq}}\]

to:

    \[\int_{ * W} {d{\Sigma ^{mnp}}} {H_{mnp}}\]

with * the Hodge dual and in turn, allows us to further reduce to:

    \[\int_{{M^2}} {{\varepsilon ^{ij}}} {\not \partial _i}{x^m}{\not \partial _j}{x^n}{b_{nm}}\]

Therefore the b-term in the spacetime string action is a direct consequence of the duality of the seven dimensional duality between 3- and 4-forms, and so the dimensional reduction of {S'_{st}} yields the term {S_{st}}, and this is tantamount to mapping the closed membrane action on K3 to the open membrane action on {T^3} \times {S^1}/{Z^2}, thus D=6 string-string duality follows and both theories will have the same spacetime supersymmetry since they have the same massless spectra

This naturally brings 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:

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

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

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

for IIA:

    \[{{\tilde \varepsilon '}_{(p)}} = \left\{ {\begin{array}{*{20}{c}}{\tilde \varepsilon \quad \quad \quad p = 0}\\{{\gamma _{11}}\tilde \varepsilon \quad \quad \quad p = 2}\\{\tilde \varepsilon \quad \quad \quad p = 4}\\{{\gamma _{11}}\tilde \varepsilon \quad \quad \quad p = 6}\\{\tilde \varepsilon \quad \quad \quad p = 8}\end{array}} \right.\]

and for IIB:

    \[{{\tilde \varepsilon '}_{(p)}} = \left\{ {\begin{array}{*{20}{c}}{{\rm{i}}\tilde \varepsilon \quad \quad \quad p = - 1}\\{{\rm{i}}{{\tilde \varepsilon }^ * }\quad \quad \quad p = 1}\\{{\rm{i}}\tilde \varepsilon \quad \quad \quad p = 3}\\{{\rm{i}}{{\tilde \varepsilon }^ * }\quad \quad \quad p = 5}\\{{\rm{i}}\tilde \varepsilon \quad \quad \quad p = 7}\end{array}} \right.\]

Since the Killing spinor is given by:

    \[\left\{ {\begin{array}{*{20}{c}}{\tilde \varepsilon = {H^{ - 1/8}}{{\tilde \varepsilon }_0}}\\{\tilde \varepsilon + {\gamma _{01...p}}{{\tilde \varepsilon '}_{(p)}} = 0}\end{array}} \right.\]

where {\tilde \varepsilon _0} is a constant spinor.

End of proof.

Hence, the triangular interplay between string-string duality, string-field theory, and the action of Dp/M5-branes establishes a duality between 4-D spacetime and string-world-sheet loop expansion, entailing the equivalence between the K-3 membrane action and the {T^3} \times {S^1}/{Z^2} orbifold action. Here is a classic by Edward Witten et al. on why that is important.

 

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