Fermions in Boundary Conformal Field Theory : Crossing Symmetry and $ε$-Expansion

We use the equations of motion in combination with crossing symmetry to constrain the properties of interacting fermionic boundary conformal field theories. This combination is an efficient way of determining operator product expansion coefficients and anomalous dimensions at the first few orders of the $\epsilon$ expansion. Two necessary ingredients for this procedure are knowledge of the boundary and bulk spinor conformal blocks. The bulk spinor conformal blocks are derived here for the first time. We then consider a number of examples. For $\phi$ a scalar field and $\psi$ a fermionic field, we study the effects of a $\phi \bar \psi \psi$ coupling in $4- \epsilon$ dimensions, a $\phi^2 \bar \psi \psi$ coupling in $3 -\epsilon$ dimensions, and a $(\bar \psi \psi)^2$ coupling in $2+\epsilon$ dimensions. We are able to compute some new anomalous dimensions for operators in these theories. Finally, we relate the anomalous dimension of a surface operator to the behavior of the charge density near the surface.