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Thermal fluctuations of a critical system induce long-ranged Casimir forcesbetween objects that couple to the underlying field. For two dimensional (2D)conformal field theories (CFT) we derive an exact result for the Casimirinteraction between two objects of arbitrary shape, in terms of (1) the freeenergy of a circular ring whose radii are determined by the mutual capacitanceof two conductors with the objects' shape; and (2) a purely geometric energythat is proportional to conformal charge of the CFT, but otherwisesuper-universal in that it depends only on the shapes and is independent ofboundary conditions and other details.

Conformal Field Theory of Critical Casimir Interactions in 2D