P denotes path ordering. The factor ${2\pi \alpha^\prime}$, conventional in the string-theory literature, could have been absorbed in the definition of $Y$. The worldsheet fields $X^M(\sigma,\tau)$ are quantised in the closed-string channel, [*i.e.*]{}  with $\tau$ being the time-like coordinate. The boundary at $\tau=0$ is thus spacelike and (barring singularities at coincident points) all worldsheet fields in expression (\[defn\]) commute. The path ordering is, of course, still non-trivial because the background fields are matrix-valued.

The attentive reader will have noticed that we wrote $A$ and $Y$ as one-form fields, defined on the entire target manifold, ${\cal M}$, even though they live, a priori, only in the tangent (or normal) bundle of the D-brane worldvolume. What we implicitly assume is that the freedom in choosing this ‘push forward’ does not matter, because of the boundary conditions imposed by ${|\!|}{\cal B}{\rangle\!\rangle}$. Consider for example a D-instanton at the origin, whose boundary state satisfies the conditions $X^M(\sigma, 0) {|\!|}{\cal B}{\rangle\!\rangle}\, =\, 0$. Ignoring problems of operator ordering, one would conclude that the integrand in (\[defn\]) reduces to $ Y_M(0)  \partial_\tau   X^M/2\pi \alpha^\prime $, so that only the value of $Y$ at the origin really matters. This is of course a semiclassical argument, and it 