Effective Field Theory of Conformal Boundaries

We introduce an effective field theory (EFT) for conformal impurity by considering a pair of transversely displaced impurities and integrating out modes with mass inversely proportional to the separation distance. This EFT captures the universal signature of the impurity seen by a heavy local operator. We focus on the case of conformal boundaries and derive universal formulas from this EFT for the boundary structure constants at high energy. We point out that the more familiar thermal EFT for conformal field theory is a special case of this EFT with distinguished conformal boundaries. We also derive, for general conformal impurities, non-positivity and convexity-like constraints on the Casimir energy which determines the leading EFT coefficient.