in the coupling constant. In section 5 we renormalise this operator and study its infrared fixed points. We verify explicitly at low orders that at the stable infrared fixed points, the operator commutes with the current algebra, and that its spectrum is given by generalised quantum dimensions. Finally in section 6 we pull back these operators to the boundary, and discuss the induced boundary RG flow. We end with some open questions.

Transfer matrices and defect lines
==================================

Consider a stack of $n$ identical D-branes in a target space parametrised by $X^M$. Let $n {|\!|}{\cal B}{\rangle\!\rangle}\ $ be the conformal boundary state of the branes, with $  A$ and $  Y$ their matrix-valued gauge and coordinate fields. Switching on non-zero backgrounds for these fields leads to the following (formal) modification of the boundary state : $$\label{defn}
n\;{|\!|}{\cal B}{\rangle\!\rangle}\ \longrightarrow \;
{\rm Tr} \;  {\rm P} \; 
{\rm exp}\left( i \int_0^{2\pi}\hskip -0.1mm d\sigma \left[ A_M(X)\;
\partial_\sigma X^M + {1\over 2\pi
\alpha^\prime}\;   Y_M(X)\;
\partial_\tau X^M\right]  \right) \; {|\!|}{\cal B}{\rangle\!\rangle}\ .$$ Here the integral runs over the boundary of the half-infinite cylinder that is parametrised by $\sigma\in [0,2\pi]$ and $\tau\in [0,\infty)$, the trace is over the Chan-Patton indices of $A$ and $Y$, and 