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We classify the sign of the critical Casimir force between two finite objectsseparated by a large distance in the two dimensional systems that can bedescribed by conformal field theory (CFT). In particular, we show that as faras the smallest scaling dimension present in the spectrum of the system issmaller than one, the sign of the force is independent of the shape of theobjects and can be determined by the elements of the modular $S$-matrix of theCFT. The provided formula for the sign of the force indicates that the force isalways attractive for equal boundary conditions independent of the shape of theobjects. However, different boundary conditions can lead to attractive orrepulsive forces. Using the derived formula, we prove the known resultsregarding the Ising model and the free bosons. As new examples, we givedetailed results regarding the Q=3-states Potts model and the compactifiedbosons. In particular, for the latter model we show that Dirichlet boundarycondition does not always lead to an attractive force.

Classification of the sign of the critical Casimir force in two dimensional systems at asymptotically large separations