We consider the general supersymmetric one-dimensional quantum system with boundary, critical in the bulk but not at the boundary. The renormalization group (RG) flow on the space of boundary conditions is generated by the boundary beta functions ßa(?) for the boundary coupling constants ?a. We prove a gradient formula ? ln z/??a = -gSabßb where z(?) is the boundary partition function at given temperature T = 1/ß, and gSab(?) 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, -? ln z/?ß = - Tßa?a ln z, is nonnegative. Equivalently, the quantity ln z(?) decreases under the RG flow. © 2009 International Press.
|Number of pages||28|
|Journal||Advances in Theoretical and Mathematical Physics|
|Publication status||Published - Dec 2009|