Duality and Reversal of Renormalization Group Ow

The coupling of a boundary spin operator to the two dimensional Ising model is known to lead to renormalization group ow to a theory with xed or Dirichlet boundary conditions at large distance from the boundary. Using the fact that spin and disorder operators are dual under a Kramers{Wannier or T-duality transformation, we show that coupling a boundary disorder operator results in an RG ow to free or Neumann boundary conditions. It seems that RG ow is reversed under the duality transformation. We reconcile this result with the conventional wisdom about RG ow and with the g-theorem of AAeck and Ludwig by noting that there are diierent possible deenitions of free or xed boundary conditions in the Ising theory, with diierent ground state degeneracies. In the two dual calculations, we show that the RG ow is restricted to diierent (dual) sectors of the theory with diierent free and xed boundary states. Within each sector, the ow is always towards the conformally invariant boundary state with lowest ground state degeneracy.