By viewing it as a free-fermion six-vertex model, the dimer model is solved exactly on the torus and the strip.
The dimers are rotated by 45 degrees with respect to the usual orientation with dimers parallel to the bonds
of the square lattice. The number of periodic configurations on an $M\times N$ rectangular lattice is obtained
exactly. We also obtain the modular invariant partition function on the torus and finitized characters on the strip.
From finite-size corrections in the continuum scaling limit, we find the central charge $c=-2$ and the
conformal weights $\Delta_s=((2-s)^2-1)/8$ with $s=1,2,3,\ldots$. We explicitly exhibit the appearance of
nontrivial Jordan cells and assert that the dimer model is described by a nonunitary and logarithmic conformal
field theory as first argued by Izmailian, Priezzhev, Ruelle and Hu in 2005.
Alexi Morin Duchesne