  An exact statistical mechanical derivation is given of the critical Casimir
interactions between two defects in a planar lattice-gas Ising model. Each
defect is a group of nearest-neighbor spins with modified coupling constants.
Such a system can be regarded as a model of a binary liquid mixture with the
molecules confined to a membrane and the defects mimicking protein inclusions
embedded into the membrane. As suggested by recent experiments, certain
cellular membranes appear to be tuned to the proximity of a critical demixing
point belonging to the two-dimensional Ising universality class. Therefore one
can expect the emergence of critical Casimir forces between membrane
inclusions. These forces are governed by universal scaling functions, which we
derive for simple defects. We prove that the scaling law appearing at
criticality is the same for all types of defects considered here.
