is possible that it fails at the quantum level.

A closely-related question is the following. The path-ordered exponential in eq. (\[defn\]) is the closed-string dual of the transfer matrix, or evolution operator in the open channel. Let us denote this operator by ${\cal O}(A,Y)$. If we were to quantise $X^M$ with $\sigma$ as time, then ${\cal O}$ would be the evolution operator in the interaction representation, around the theory defined by the boundary condition ${\cal B}$.[^5] From this perspective, only the action of ${\cal O}$ on the boundary state ${|\!|}{\cal B}{\rangle\!\rangle}$ is really specified. Extending the action of the path-ordered exponential to [*all*]{} closed-string states requires a push-forward of $A$ and $Y$ to the whole target manifold. This makes it, in turn, possible to push the integration contour from the boundary to the interior of the worldsheet, where boundary conditions are no more effective (see figure 1). Reversing this operation amounts to taking the limit $$\label{limit}
{\rm lim}_{\varepsilon\to 0}\; {\cal O}(A,Y)\; 
e^{i\varepsilon (L_0+\bar L_0)} {|\!|}{\cal B}{\rangle\!\rangle}\ ,$$ where $L_0+\bar L_0$ is the Hamiltonian in the closed-string channel. If the limit is non-singular, it should only depend on the pullbacks of $A$ and $Y$ to the tangent and normal bundles of 