CFT from the arc point of view and structural relations to planar algebras

One approach to study moduli spaces is to use arcs. Taking this approach, we defined the arc operad, which among other things models correlation functions on over a certain partial compactification of moduli space. Using a careful analysis, we then were able to restrict the theory to moduli space and to extend it to the open/closed case. An example of these correlation functions are actions on the Hochschild cochain complex of a (Frobenius) algebra. These operations include Deligne’s conjecture, its cyclic version and string topology. These actions are built from two main ingredients, a discretization and the co-simplicial structure. On this discretized level of actions there is a striking analogy of our theory with that of planar algebras, whose deeper connection we will elucidate and explore.