General properties of the boundary renormalization group flow for supersymmetric systems in 1+1 dimensions

We consider the general supersymmetric one-dimensional quantum system with boundary, critical in the bulk but riot at the boundary. The renormalization group (RG) flow on the space of boundary conditions is generated by the boundary beta functions beta(a)(lambda) for the boundary coupling constants lambda(a). We prove a gradient formula theta In z/theta lambda(a) = -g(ab)(S)beta(b) where z(lambda) is the boundary partition function at given temperature T = 1/beta, and g(ab)(S)(lambda) is a certain positive-definite metric on the space of supersymmetric boundary conditions. The proof depends on canonical ultraviolet behavior at the boundary. Any system whose short distance behavior is governed by a fixed point satisfies this requirement. The gradient formula implies that the boundary energy, -theta In z/theta beta = -T beta(a)theta(a) In z, is nonnegative. Equivalently, the quantity ln z (lambda) decreases under the RG flow.