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:

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


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

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:

    \[{M^{10}} \times {S^1}/{Z_2}\]