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:

    \[\begin{array}{*{20}{l}}{S = {S_M} + \int_{\partial {M^3}} {\left\{ {\frac{1}{2}} \right.} \left( {{g_{mn}}{\eta ^{ij}} + {b_{mn}}{\varepsilon ^{ij}}} \right)}\\{{\partial _i}{x^m}{\partial _j}{x^n} + \frac{1}{2}\left( {{g_{IJ}}{\eta ^{ij}} + {b_{IJ}}{\varepsilon ^{ij}}} \right)}\\{{\partial _i}{x^I}{\partial _j}{x^J} + {\varepsilon ^{ij}}{\partial _i}{x^J}{\partial _j}{x^m}\left. {A_m^J(x)} \right\}}\end{array}\]

where:

    \[\begin{array}{*{20}{l}}{{S_M} = \int_{{M^3}} {\left( {\sqrt { - {g_{mn}}{\partial _i}{x^m}{\partial _j}{x^n}} } \right.} + }\\{\frac{1}{6}{\varepsilon ^{ijk}}{\partial _i}{x^m}{\partial _j}{x^n}{\partial _k}{x^p}\left. {{B_{mnp}}} \right)}\end{array}\]

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 {D^{SuSy}} via Green’s-functions, yielding the on-shell action of M-theory in the Witten gauge:

    \[\begin{array}{l}{S_M} = \frac{1}{{{k^9}}}\int\limits_{{\rm{world - volumes}}} {{d^{11}}} \sqrt {\frac{{ - {g_{\mu \nu }}}}{{ - \gamma }}} {T_p}^{10}d\Omega {\left( {{\phi _{INST}}} \right)^{26}}\left( {{R_{icci}} - A_\mu ^H\frac{1}{{48}}G_4^2} \right) + \\\sum\limits_{Dp} {D_\mu ^{SuSy}} {e^{ - H_3^b}}/S_{Dp}^{WV} + \sum\limits_{Dp} {D_\nu ^{SuSy}} {e^{H_3^b}}/S_{Dp}^{SV}\end{array}\]

with k the kappa symmetry term. 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 Theory, the Witten Index and the Seiberg-Lebesgue Problem

String/M-[F]-theory remains by far the most promising – only? – theoretical paradigm for both, grand unification and quantization of general relativity. With the Dp-action given by:

    \[S_p^D = - {T_p}\int\limits_{{\rm{worldvolumes}}} {{d^{p + 1}}} \xi \frac{{{D_{\mu \nu }}L}}{{{\partial _{{v_a}}}}}{e^{ - {\Phi _{bos}}}}{\rm{de}}{{\rm{t}}^{1/2}}G_{ab}^{\exp \left( {H_{p + 1}^{{\rm{array}}}} \right)}\]

for contextualization, note that a necessary condition for the world-sheet Dirac propagator {\delta ^{\left( 2 \right)}}\left( {{\sigma _i} - {\sigma _j}} \right):

    \[S = i\int {{d^2}} {\sigma _1}{d^2}{\sigma _2}\sum\limits_{i,j = + , - } {{\psi _i}\left( {{\sigma _1}} \right)} {A_{ij}}\left( {{\sigma _1},{\sigma _2}} \right)\psi \left( {{\sigma _2}} \right)\]

to be integrable, is that the Seiberg vacuum fluctuation of the string world-sheet:

    \[{S_\eta } = \frac{1}{\beta }\sum\limits_{\frac{{i2\pi }}{\beta }} {{{\left( {i\frac{{2n + 1}}{\beta }} \right)}^\pi }} W + \alpha '{R_{\left( 2 \right)}}\Phi \]

with

    \[W \equiv {h^{mn}}{\partial _m}{X^a}{\partial _n}{X^b}{g_{ab}}\left( X \right)\]

and \beta the bosonic frequency, be analytically summable. The string world-sheet is given by:

    \[{S_{ws}} = \frac{1}{{4\pi \alpha '}}\int\limits_{c + o} {d\tilde \sigma } d\tau '\sqrt h \left( {W + \alpha '{R_{\left( 2 \right)}}\Phi } \right)\]

A major problem is that by the Heisenberg’s uncertainty principle:

    \[\left( {\Delta A/} \right)\left( {\left| {\frac{{d\left\langle A \right\rangle }}{{dt}}} \right|} \right)\left( {\Delta H} \right) \ge \hbar /2\]

the string time-parameter on the world sheet {\sigma _t} with Feynman propagator in Euclidean signature being:

    \[\begin{array}{c}G\left( {x,y} \right) = \int_0^\infty {d{\sigma _t}} G\left( {x,y;{\sigma _t}} \right)\\ = \int {\frac{{{d^D}p}}{{{{\left( {2\pi } \right)}^D}}}} \exp \left[ {ip \cdot \left( {y - x} \right)} \right]\frac{2}{{{p^2} + {m^2}}}\end{array}\]

violates