Thermal fluctuations of a critical system induce long-ranged Casimir forces between objects that couple to the underlying field. For two dimensional (2D) conformal field theories (CFT) we derive an exact result for the Casimir interaction between two objects of arbitrary shape, in terms of (1) the free energy of a circular ring whose radii are determined by the mutual capacitance of two conductors with the objects' shape; and (2) a purely geometric energy that is proportional to conformal charge of the CFT, but otherwise super-universal in that it depends only on the shapes and is independent of boundary conditions and other details. 03.70.+k, 42.25.Fx Objects embedded in a medium constrain its natural fluctuations, resulting in fluctuation-induced forces [1]. The most naturally occurring examples result from modification of electromagnetic fluctuations, manifested variously in van der Waals interactions [2] (between atoms and molecules) to Casimir forces (between conducting plates) [3]. While fluctuations of the latter are primarily quantum in origin, thermal fluctuations of correlated fluids lead to similar interactions, most notably at a critical point (where correlation lengths are macroscopic) [4, 5]. Critical fluctuation-induced forces have been observed in helium [6] and in binary liquid mixtures [7-9] Critical fluctuations of a binary mixture were recently employed to manipulate and assemble colloidal particles [10].Biological membranes are mainly composed of mixtures of lipid molecules, and could potentially be poised close to a critical point demixing point [11,12], in the two-dimensional Ising universality class. It has been suggested that membrane concentration fluctuations could thus lead to critical Casimir forces between inclusions on such membranes, motivating computation of such forces between discs embedded in the critical Ising model [13]. Membranes (and interfaces) also undergo thermal shape fluctuations governed by the energy costs of bending (and surface tension) [14]. Modification of these fluctuations have also been proposed as a source of interactions amongst inclusions on membranes [15,16], possibly accounting for patterns of colloidal particles at an interface [17]. There is extensive literature on this topic, and the interested reader can consult recent publications [18,19]. Yet another entropic force is proposed to act between surface/membrane bio-adhesion bonds [20].Conformal field theories (CFTs) have proved highly successful in studies of two dimensional (2D) systems at criticality [21,22]. Various boundary conditions have been examined for Ising (or 3-state Potts) model on a cylinder [23]. Connections to Casimir forces between parallel plates [24,25] and spheres [26,27] have been explored. Non-spherical particles at large separations have been studied with the small particle operator expansion [28,29]. However, a general formulation for interactions between two (or more) objects of arbitrary shape embedded in a CFT appears to be lacking. Some special cases recently studied inc...