1+1 sector of 3+1 gravity

The rank-1 sector of classical Ashtekar gravity is considered, motivated by the degeneracy of the metric along the Wilson Lines in quantum loop states. It is found that the lines behave like (1 + 1)-dimensional spacetimes with a pair of massless Dirac fields propagating along them. If the lines form a congruence of closed loops, the holonomy must be the same on all the loops. The inclusion of matter and extension to supergravity are also considered.