The Equations we know of

First is General Relativity. The action can be put in this form:

S_{E H^{'}} = \frac{1}{8 κ} \int \sqrt{- g} (g^{a b} g^{d e} g^{c f} + 2 g^{a c} g^{b f} g^{d e} + 3 g^{a d} g^{b e} g^{c f} - 6 g^{a d} g^{b f} g^{c e}) \partial_{c} g_{a b} \partial_{f} g_{d e} d x^{4}

Where

g^{a b} = \frac{1}{4!} ε^{a p q r} ε^{a u v w} g_{p u} g_{q v} g_{r w} / det (g)

We can get Maxwell's equations from the above by a Kaluza-Klein compactification on (D+n) dimensions. They are:

S_{Y M} = \int \sqrt{- g} \frac{1}{2} (g^{a b} g^{c d} - g^{a c} g^{b d}) (\partial_{a} A_{b} + i A_{a} A_{b}) (\partial_{c} A_{d} + i A_{c} A_{d}) d x^{4}

Where the A fields are matrices which generate the gauge group. Then Dirac's equations are:

S_{D} = \int e ψ^{†} e^{a} σ (\partial_{a} + i A_{a} . τ + i Ω_{a}) ψ + M (ψ ε ψ + ψ^{†} ε ψ^{†}) d x^{4}

with:

Ω_{a} = ε e_{a} e^{b} e^{c} \partial_{b} e_{c}

and

τ

is a generator of a gauge group, such as SU(n). M is a matrix which gives the masses.

ψ

is a collumn of particles. And

σ . x = [\begin{matrix} x + t & y + i z \\ y - i z & x - t \end{matrix}]

ε = [\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix}]

What we don't know: Why gauge group is

S U (3) \times S U (2) \times U (1)

. Where the particle masses, CKM matrix and PMNS matrix gets their values. Finally there is a Higgs field:

S_{H} = \int \sqrt{- g} (g^{a b} (\partial_{a} + i A_{a}) ϕ (\partial_{b} + i A_{b}) ϕ + V (ϕ)) d x^{4}

Supersymmetry

Super Yang mills in 10 dimensions for any gauge group G:

S = \int tr {- \frac{1}{2} F_{a b} F^{a b} - i ψ^{α} γ_{α β}^{a} (\partial_{a} + i g A_{a}) ψ^{β}}

Supergravity in 11 dimensions:

S = \int \sqrt{- g} (R + F^{a b c d} F_{a b c d} - \frac{1}{6} ε^{a b c d e f g h i j k} A_{a b c} F_{d e f g} F_{h i j k}) d x^{11}

+ \int \sqrt{- g} (ψ_{a}^{†} γ^{a b c} (\partial_{b} + i Ω_{b}) ψ_{c} + (F_{a b c d} + F_{a b c d}^{*}) (ψ_{e}^{†} γ^{a b c d e f} ψ_{f} + 12 ψ^{† a} γ^{b c} ψ^{d})) d x^{11}

With Supersymmetry (guessing):

δ A_{a b c} = γ_{a b c d} ψ^{d}

δ ψ^{a} = γ^{a b c d e} \partial_{b} A_{c d e} + γ^{a b c} γ^{n} \partial_{b} e_{c}^{n}

δ e_{a}^{n} = γ^{n} ψ_{a}