Gauge fields and Kaluza-Klein theory

Correspondance potential A_m - connection and field F_m_n - curvature

In a loop (+dx,+dy,-dx,-dy) the phase increases by :
A_x dx + (A_y + ś_x A_y dx) dy - (A_x + ś_y A_x dy) dx - A_y dy
= (ś_x A_y - ś_y A_x) dx dy
= F_x_y dx dy

Kaluza-Klein theory

The vielbeins e^a_m are the matrices of base changing between curved and locally flat coordinates.
The metric tensor g_m_n in curved coordinates is :
g_m_n = h_a_b e^a_m e^b_n
where h_a_b is the metric tensor of the flat space-time (1 -1 -1 -1 on the diagonal, 0 elsewhere).

According to Kaluza-Klein theory, gauge fields are equivalent to gravitation (curvature of space-time according to general relativity) in compactified dimensions.

The vielbein is :

e^a_m = ( e'^a'_m         0   )
     ( e''^a''_n'' A^n''_m    e''^a''_m'' )

where e'^a'_m is the vielbein of the space-time, e''^a''_m'' is the vielbein of the compactified dimensions, and A^n''_m is the potential.

We can number the dimensions of space-time from 0 to 3 (0=t, 1=x, 2=y, 3=z) and the compactified dimensions from 4 to d-1, and consider all the tensor defined for indices varying from 0 to d-1, completed with zeros, for example :

A^m_m = ( 0      0 )
      ( A^m_m   0 )

with indices m = 0..3 and m = 4..d-1; A^m_m is 0 if m is not in 0..3 or m is not in 4..d-1.

With this convention, we have :
e^a_m = e'^a_m + e''^a_n Aⁿ_m + e''^a_m
= e'^a'_m + e''^a_n (Aⁿ_m + dⁿ_m)
= e'^a_m + e''^a_m + e''^a_m A^m_m

Then the metric tensor in curved coordinates is :
g_m_n = h_a_b e^a_m e^b_n
= h_a_b (e'^a_m + e''^a_m + e''^a_m A^m_m) (e'^b_n + e''^b_n + e''^b_n Aⁿ_n)
= h_a_b e'^a_m e'^b_n + h_a_b e'^a_m e''^b_n + h_a_b e'^a_m e''^b_n Aⁿ_n + h_a_b e''^a_m e'^b_n + h_a_b e''^a_m e''^b_n + h_a_b e''^a_m e''^b_n Aⁿ_n + h_a_b e''^a_m A^m_m e'^b_n + h_a_b e''^a_m A^m_m e''^b_n + h_a_b e''^a_m A^m_m e''^b_n Aⁿ_n
Since h_a_b e'^a_m e''^b_n = 0 because h_a_b is not 0 only if a = b, e'^a_m is not zero only if a is in 0..3 and e''^b_n is not zero only if b is in 4..d-1, and same for h_a_b e''^a_m e'^b_n we have :
g_m_n = h_a_b e'^a_m e'^b_n + h_a_b e''^a_m e''^b_n + h_a_b e''^a_m e''^b_n Aⁿ_n + h_a_b e''^a_m A^m_m e''^b_n + h_a_b e''^a_m A^m_m e''^b_n Aⁿ_n
= g'_m_n + g''_m_n + g''_m_n Aⁿ_n + g''_m_n A^m_n + g''_m_n A^m_m Aⁿ_n