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On Marginal Operators in Boundary Conformal Field Theory

The presence of a boundary (or defect) in a conformal field theory allows one to generalize the notion of an exactly marginal deformation. Without a boundary, one must find an operator of protected scaling dimension $Δ$ equal to the space-time dimension $d$ of the conformal field theory, while with a boundary, as long as the operator dimension is protected, one can make up for the difference $d-Δ$ by including a factor $z^{Δ-d}$ in the deformation where $z$ is the distance from the boundary. This coordinate dependence does not lead to a reduction in the underlying $SO(d,1)$ global conformal symmetry group of the boundary conformal field theory. We show that such terms can arise from boundary flows in interacting field theories. Ultimately, we would like to be able to characterize what types of boundary conformal field theories live on the orbits of such deformations. As a first step, we consider a free scalar with a conformally invariant mass term $z^{-2} ϕ^2$, and a fermion with a similar mass. We find a connection to double trace deformations in the AdS/CFT literature.