The String-String Duality, K3 Geometry, and Dimensional Reduction

This post is on the String-String Duality. In particular, the D=6 string-string duality, which is crucial since it allows interchanging the roles of 4-D spacetime and string-world-sheet loop expansion, and this is mathematically essential for phenomenologically adequate string-compactifications. Here I will prove an equivalence between K3 membrane action and {T^3} \times {S^1}/{Z^2} orbifold action and show how it entails D=6 string-string duality. Working 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, via Clifford algebraic symmetry, 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 deep since Clifford algebras are a quantization of exterior algebras, and applying to the ‘Einstein-Minkowski’ tangent bundle, we get via Gaussian matrix elimination, an expansion of {\not D^{SuSy}} via Green-functions, that yields M-Theory’s action:

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

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. The worldvolume {M^3} is:

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

Schrödinger’s Equation, Geometrodynamics, and Quantum Mechanics

Without getting into the mathematical aspects of the ‘quantum mechanics = Bayes theory in the complex-number-system program’, I will show that a geometrodynamic derivation of quantum mechanics is possible based on a modified Schrödinger equation whose action-propagator yields an interpretation of ‘particle’ as curvature in spacetime, and relate it to the conflict I derived between the Lindblad quantum-jump equation:

    \[\begin{array}{l}{\rm{d}}\hat \rho _S^C = - i\left[ {{{\hat H}_S},\hat \rho _S^C} \right]{\rm{d}}t - \\\frac{1}{2}\sum\limits_\mu {{\kappa _\mu }} \left[ {{{\hat L}_\mu },\left[ {{{\hat L}_\mu },\hat \rho _S^C} \right]} \right]{\rm{d}}t + \\\sum\limits_\mu {\sqrt {{\kappa _\mu }} } W\left[ {{{\hat L}_\mu }} \right]\hat \rho _S^C{\rm{d}}{W_\mu }\end{array}\]

where:

    \[\begin{array}{l}W\left[ {\hat L} \right]\hat \rho \equiv \hat L\hat \rho + \hat \rho {{\hat L}^\dagger } - \\\hat \rho {\rm{Tr}}\left\{ {\hat L\hat \rho + \hat \rho {{\hat L}^\dagger }} \right\}\end{array}\]

with {\rm{d}}{W_\mu } the Weiner-quantum-increments, and the collapse-action-functional corresponding to the collapse equation, given by:

    \[\begin{array}{*{20}{l}}{d\left| {{\psi _t}} \right\rangle = \left[ { - \frac{i}{\hbar }} \right.Hdt + \sqrt \lambda \int {{d^3}} x\left( {N(\bar x) - {{\left\langle {N(\bar x)} \right\rangle }_t}} \right)}\\{ \cdot d{W_t}(\bar x) - \frac{\lambda }{2}\int {{d^3}} x\left. {{{\left( {N(\bar x) - {{\left\langle {N(\bar x)} \right\rangle }_t}} \right)}^2}dt} \right]\left| {{\psi _t}} \right\rangle }\end{array}\]

where the collapse-action-functional is:

    \[\int {\exp \frac{i}{\hbar }} {S_{1,2}}\left[ \Gamma \right]\left[ {d\Gamma } \right] = {K_{{t_2},{t_1}}}\left( {{x_2},{x_1}} \right)\]

and {S_{1,2}}\left[ \Gamma \right] is given by:

    \[{S_{1,2}}\left[ \Gamma \right] = \int\limits_{{t_1}}^{{t_2}} {\left( {{T^\Gamma }\left( {\dot X,\dot x} \right) - {U_q}\left( {x,X} \right)h\left( {t - {t_3}} \right) - U\left( {x,t} \right)} \right)} dt\]