A Weyl-Plebanski 7-D Embedding of General Relativity

I will derive an isomorphism between a Cartan-3-form Weyl-theory in 7-D and general relativity in Plebanski formulation and discuss the depth and impact. Basically, that the 4D Lagrangian:

\begin{array}{l}{L_{4D}}/{\phi ^4} = {B^i}{F^i} - \frac{1}{2}{M^{ij}}{B^j}\\ + 2\mu \left( {\det \left( {\tilde {\rm I} + {\phi ^2}M} \right) - {\phi ^3}} \right)vo{l_\Sigma }\end{array}

is an exact description of general relativity, where {L_{4D}} is given by:

\begin{array}{l}{L_{4D}} = {\phi ^4}{B^i}{F^i} + \frac{{{\phi ^2}}}{2}{B^i}{B^i} + \\3{\phi ^3}{\left( {\det (X)} \right)^{1/3}}vo{l_\Sigma }\end{array}

and the metric-volume form implicitly defining vo{l_\Sigma }, namely, vo{l_C}, is:

\begin{array}{l}vo{l_C} = \frac{{{\phi ^3}}}{6}{\varepsilon ^{ijk}}{W^i}{W^j}{W^k}\\{\left( {\det (X)} \right)^{1/3}}vo{l_\Sigma }\end{array}

with the 3-form C on a 7-D manifold M is explicitly defined by the {G_2}-holomorphic metric:

{g_C}\left( {\xi ,\eta } \right)vo{l_C} = \frac{1}{6}{i_\xi }C \wedge {i_\eta }C \wedge C

with the action:

S\left[ C \right] = \frac{1}{2}\int_M C \wedge dC + 6\lambda vo{l_C}

Such holomorphic 3-forms in 7-D are analytically related to the exceptional group {G_2}, which can be interpreted as the subgroup of {\rm{GL}}\left( 7 \right) that metaplectically stabilizes generic 3-forms.

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}


\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}