The Chern-Simons theory has been sucessful to describe the quantum fractional
hall effect. Here another kind of topological field theory - mutual
Chern-Simons theory would be talked about. The mutual Chern-Simons theory has
U(1)*U(1) gauge structure with T and P symmetry, of which the mutual semion
statistics is the key point.
The doped Mott insulator is an example for the mutual Chern-Simons theory in
Higgs phase. It is shown that from a bosonic slave technique, holon and spinon
have mutual Chern-Simons statistics. A global phase diagram for the doped Mott
insulator is obtained.
Z2 topological order is an example for the mutual Chern-Simons theory in gapped
phase. It is shown that the mutual Chern-Simons theory has the same topological
properties as those for Z2 topological order, such as the topological
degeneracy, mutual semion statistics and the edge states.