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:

    \[{M^{10}} \times {S^1}/{Z_2}\]

String Field Theory, 4-D Space-Time and the Superstring

Let us see how deeply interconnected string field theory, 4-D space-time and the topological superstring are. Aside: I highly recommend the book on the cover picture. In the context of topological sigma models with Calabi-Yau target space, the BRST action is:

    \[S = it\int_\Sigma {{d^2}} z\left\{ {\tilde Q',V} \right\} + t\int_\Sigma {{\Phi ^ * }} \left( {\hat K} \right)\]

with:

    \[V = {g_{i\bar j}}\left( {\psi _z^{\bar i}{{\not \partial }_{\bar z}}{\phi ^j} + {{\not \partial }_z}{\phi ^{\bar i}}\psi _{\bar z}^j} \right)\]

and

    \[\int_\Sigma {{\Phi ^*}} \left( {\hat K} \right)\]

is the integral of the pullback of the Kähler form \hat K, and is:

    \[\int_\Sigma {{\Phi ^*}} \left( {\hat K} \right) = \int_\Sigma {{d^2}} z\left( {{{\not \partial }_z}{\phi ^i}{{\not \partial }_{\bar z}}{\phi ^{\bar j}}{g_{i\bar j}} - {{\not \partial }_{\bar z}}{\phi ^i}{{\not \partial }_z}{\phi ^{\bar j}}{g_{i\bar j}}} \right)\]

In the A model, given that:

    \[\begin{array}{l}W = \int_\Sigma {\left( { - {\theta _i}\not D{\rho ^i} - \frac{i}{2}{R_{i\overline i j\overline j }}{\rho ^i}} \right.} \\\left. { \wedge {\rho ^j}{\eta ^{\overline i }}{\theta _k}{g^{k\overline j }}} \right)\end{array}\]

and:

    \[V = {g_{i\overline j }}\left( {\rho _z^i{{\not \partial }_z}{\phi ^{\bar j}} + \rho _{\bar z}^i{{\not \partial }_z}{\phi ^{\bar j}}} \right)\]

the action above can be expressed as:

    \[S = it\int {\left\{ {Q,V} \right\}} + tW\]

where Q is the BRST holomorphic operator. Since Edward Witten showed in ‘Chern-Simons Gauge Theory as a String Theory’