  A lattice model of critical dense polymers is solved exactly for arbitrary
system size on the torus. More generally, an infinite family of lattice loop
models is studied on the torus and related to the corresponding
Fortuin-Kasteleyn random cluster models. Starting with a cylinder, the
commuting periodic single-row transfer matrices are built from the periodic
Temperley-Lieb algebra extended by the shift operators Omega and Omega^{-1}. In
this enlarged algebra, the non-contractible loop fugacity is alpha and the
contractible loop fugacity is beta. The torus is formed by gluing the top and
bottom of the cylinder. This gives rise to a variety of non-contractible loops
winding around the torus. Because of their nonlocal nature, the standard matrix
trace does not produce the proper geometric torus. Instead, we introduce a
modified matrix trace for this purpose. This is achieved by using a
representation of the enlarged periodic Temperley-Lieb algebra with a parameter
v that keeps track of the winding of defects on the cylinder. The transfer
matrix representatives and their eigenvalues thus depend on v. The modified
trace is constructed as a linear functional on planar connectivity diagrams in
terms of matrix traces Tr_d (with a fixed number of defects d) and Chebyshev
polynomials of the first kind. For critical dense polymers, where beta=0, the
transfer matrix eigenvalues are obtained by solving a functional equation in
the form of an inversion identity. The solution depends on d and is subject to
selection rules which we prove. Simplifications occur if all non-contractible
loop fugacities are set to alpha=2 in which case the traces are evaluated at
v=1. In the continuum scaling limit, the corresponding conformal torus
partition function obtained from finite-size corrections agrees with the known
modular invariant partition function of symplectic fermions.
