Global symmetry breaking and defect conformal manifolds

In the presence of exactly marginal operators, one can deform a conformal field theory to other such theories, forming a space of theories living on what's known as the conformal manifold. In a theory with conformal defects, one may be able to retain the bulk theory but transform the defect along what I'll call the "defect conformal manifold". This requires the existence of exactly marginal defect operators. A simple setting where this happens is in the presence of a global symmetry broken by the defect. The broken symmetry current gives rise to an exactly marginal defect operator and a defect conformal manifold which is the symmetry breaking coset. This simple observation lets us derive an exact identity for integrated 4-point functions of the marginal defect operator which is highly nontrivial. We implement this in several models where the 4-point functions have been previously calculated.