Factorizing Defects from Generalized Pinning Fields

We introduce generalized pinning fields in conformal field theory that model a large class of critical impurities at large distance, enriching the familiar universality classes. We provide a rigorous definition of such defects as certain unbounded operators on the Hilbert space and prove that when inserted on codimension-one surfaces they factorize the spacetime into two halves. The factorization channels are further constrained by symmetries in the bulk. As a corollary, we solve such critical impurities in the 2D minimal models and establish the factorization phenomena previously observed for localized mass deformations in the 3D <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi mathvariant="normal">O</a:mi> <a:mo stretchy="false">(</a:mo> <a:mi>N</a:mi> <a:mo stretchy="false">)</a:mo> </a:math> model.