therefore commute with both ${\cal L}_0$ and $\bar L_0$, and hence also with the closed-string Hamiltonian. Thus, chiral defects can be taken freely to the boundary of the cylinder, where they induce RG flows of boundary states.

Defect lines that are both conformal and chiral correspond to operators that commute with both the left and the right Virasoro algebras, $$\label{topo}
\left[\; {L}_n  \; ,  \; {\cal O}_{\rm chir}(A^*, Y^*)\; \right]\; =
\left[\; {\bar L}_n  \; ,  \; {\cal O}_{\rm chir}(A^*,Y^*)\;
\right]\; = 0\ .$$ We will call such defect lines topological, since their action only depends on the homotopy class of the integration contour. The corresponding operator will be denoted for short by ${\cal O}_{\rm top}$. The conditions (\[topo\]), together with a Cardy constraint of integral multiplicities, have been used in ref. [@Petkova:2000ip] to give an algebraic definition of (topological) defect lines. Notice that the Cardy constraint is automatically obeyed provided ${\cal O}_{\rm top}$ is the transfer matrix for some local defect. Note also that ${\cal O}_{\rm top}$ need not be continuously connected (in D-brane configuration space) to the identity operator. What can be asserted, following ref. [@Graham:2003nc], is that if ${|\!|}{\cal B}{\rangle\!\rangle}\to {|\!|}{\cal B}^\prime{\rangle\!\rangle}$ is an allowed boundary RG 