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:

    \[{M^{10}} \times {S^1}/{Z_2}\]

How the String Theory Worldsheet ‘Knows’ All About Spacetime Physics

This is how the worldsheet quantum theory knows all about spacetime physics. Setting the stage first. Since branes are ‘generalizations’, and are BPS, the supergravity solution in the multi-brane harmonic function form is:

    \[H_p^{{\rm{array}}} = 1 + \sum\limits_{n = - \infty }^{ + \infty } {\frac{{r_p^{7 - p}}}{{{{\left| {{{\widetilde r}^2} + {{\left( {{X^{p + 1}} - 2\pi nR} \right)}^2}} \right|}^{(7 - p)/2}}}}} \]

with:

    \[{r^2} = {\left( {{X^{p + 1}}} \right)^2} + {\left( {{X^{p + 2}}} \right)^2} + ... + {\left( {{X^{p + 9}}} \right)^2} = {\widetilde r^2} + {\left( {{X^{p + 1}}} \right)^2}\]

Thus, I can now derive:

    \[H_p^{{\rm{array}}} \sim 1 + \frac{{r_p^{7 - p}}}{{2\pi R}}\frac{1}{{{{\widetilde r}^{6 - p}}}}\int\limits_{ - \infty }^\infty {\frac{{du}}{{{{\left( {1 + {u^2}} \right)}^{\left( {7 - p} \right)/2}}}}} \]

Hence, the integral is:

    \[\int\limits_{n = - \infty }^{ + \infty } {\frac{{du}}{{{{\left( {1 + {u^2}} \right)}^{\left( {7 - {p^n}} \right)/2}}}}} = \frac{{\sqrt {2\pi n{R^n}} {\mkern 1mu} \Gamma \left[ {\frac{1}{2}\left( {6 - {p^n}} \right)} \right]}}{{\Gamma \left[ {\frac{1}{2}\left( {7 - {p^n}} \right)} \right]}}\]

After checking renormalization, one gets:

    \[H_p^{{\rm{array}}} \sim H_{p + 1}^{{\rm{array}}} = 1 + \frac{{\sqrt {\alpha '} r_{p + 1}^{7 - \,\left( {p + 1} \right)}}}{R}\frac{1}{{{{\widetilde r}^{7 - \,\left( {p + 1} \right)}}}}\]

which is the correct harmonic function for a D(p+1)-brane. The relevance of H_{p + 1}^{{\rm{array}}} is that via Green’s functional analysis, it yields the string coupling of the dual 25-D theory:

{e^{{\Phi _{bos}}}} = {e^{\Phi _{bos}^{{e^{{\phi _{si}}}}}}}\frac{{{{\alpha '}^{1/2}}}}{{2\pi nR}}

which is key to the T-duality transformation properties of propagating background matter fields in 4-dimensional space-time, with {\Phi _{bos}} the bosonic field configuration corresponding to the string world-sheet, whose variable is {\phi _{si}}, yielding the two following key relations:

    \[\begin{array}{c}({T_p}\left( {2\pi \sqrt {\alpha '} } \right){e^{ - {\Phi _{bos}}}}\prod\limits_{i = 1}^{p - 1} {\left( {2\pi {R_p}} \right)} = \\{T_{p - 1}}{e^{ - \Phi _{bos}^{1/2}}}\prod\limits_{i = 1}^{p - 1} {\left( {2\pi {R_{p - 1}}} \right)} \end{array}\]

and

    \[\frac{d}{{{d_{{\sigma _p}}}}}\int\limits_{{\rm{worldvolumes}}}^p {{e^{H_{p + 1}^{{\rm{array}}}}}} + \underbrace {\sum\limits_{{\sigma _p}}^D {{{\left( {S_p^D} \right)}^{ - H_{p + 1}^{{\rm{array}}}}}} }_{{\rm{topologies}}}\]