Exact results in defect conformal field theories**

Abstract

Defect Conformal Field Theories describe critical points of Quantum Field Theories excited or modified in the neighbourhood of a large p-dimensional submanifold. We elucidate the constraints of conformal invariance on correlation functions and we provide both recurrence relations, and in some cases exact results, for the conformal blocks with scalar external operators. MotivationsConformal defects appear in a variety of situations of theoretical and phenomenological interest. Boundaries and interfaces [14,15] provide a prototypical example of codimension one conformal defects, while Wilson and 't Hooft operators [12], as well as surface operators [5,11] are examples of higher codimension defects, which play a role in gauge theories. Low energy example of defects include vortices [6], magnetic-like impurities in spin systems [3], localized particles acting as sources for the order parameter of some bosonic system [1], higher dimensional descriptions of theories with long range interactions [16] etc. Moreover, conformal defects provide a useful tool to study the quantum Entanglement [2]. Given the importance of conformal defect in physics, our main goal [4] is to provide a toolbox to compute correlation functions and conformal blocks for CFT with conformal defects. The latter are important to generalize the conformal bootstrap program, recently extended to codimension one defects [8,10,13], to higher codimension defects. Correlation functions and CFT dataWe consider correlation functions of local operators constrained by SO(p + 1, 1) × SO(q) symmetry. This is the symmetry preserved by a flat p−dimensional extended operator inserted in R d , which is usually denoted as a conformal defect (of codimension q ≡ d − p).