Relations between Potts and RSOS models on a torus

We study the relationship between Q-state Potts models and staggered RSOS models of the A\_{p-1} type on a torus, with sqrt(Q)=2 cos(pi / p). In general the partition functions of these models differ due to clusters of non-trivial topology. However we find exact identities, valid for any temperature and any finite size of the torus, between various modified partition functions in the two pictures. The field theoretic interpretation of these modified partition function is discussed.