are internal indices, F is a curvature on the orthogonal group S O ( 3 , 1 ) and the connection variables (the gauge fields) are denoted by A a i . The symbol ϕ ϕ i j k l is the Lagrangian multiplier and ϵ ϵ i j k l is the antisymmetric symbol valued over S O ( 3 , 1 ) .