  Intertwined multiple Chern-Simons gauge fields induce matrix statistics among
particles. We analyse this theory on a torus, focusing on the vacuum structure
and the Hilbert space. The theory can be mimicked, although not completely, by
an effective theory with one Chern-Simons gauge field. The correspondence
between the Wilson line integrals, vacuum degeneracy and wave functions for
these two theories are discussed. Further, it is obtained in both of these
cases that the two total momenta and Hamiltonian commute only in the physical
Hilbert space.
