  The Ising model in two dimensions with special toroidal boundary conditions
is analyzed. These boundary condition, which we call duality twisted boundary
conditions, may be interpreted as inserting a specific defect line ("seam") in
the system, along non-contractible circles of the cylinder, before closing it
into a torus. We derive exact expressions for the eigenvalues of the transfer
matrix for the critical ferromagnetic Ising model on the M x N square lattice
wrapped on the torus with a specific defect line. As result we have obtained
analytically the partition function for the Ising model with such boundary
conditions. In the case of infinitely long cylinders of circumference L with
duality twisted boundary conditions we obtain the asymptotic expansion of the
free energy and the inverse correlation lengths. We find that the ratio of
subdominant finite-size correction terms in the asymptotic expansion of the
free energy and the inverse correlation lengths should be universal. We verify
such universal behavior in the framework of perturbating conformal approach by
calculating the universal structure constant Cn1n for descendent states
generated by the operator product expansion (OPE) of the primary fields. For
such states the calculations of the universal structure constants is difficult
task, since its involve the knowledge of the four-point correlation function,
which in general does not fix by conformal invariance except for some
particular cases, including the Ising model.
