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:

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

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

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

with the action:

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.

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:

where:

Recall I derived, via Clifford algebraic symmetry, the total action:

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:

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: