Do the fields in string theory only live on the world line of the string? Say for instance we have a $D0$ brane, and this $D0$ brane is charged under the KK field. Does the strength of the KK field only exist on the worldline of the $D0$ brane? Same goes for strings: do the charged SUGRA fields live on the world sheet of the string?
Or does the field strength of the $D0$ brane or string extend outwards in space (like the classic EM field generated by a charged particle)? My apologies if this seems confusing.
 A: The fields in string theory aren't restricted to the worldvolume any more than the electromagnetic field is restricted to the worldline of a classical charged particle: they extend throughout spacetime. It pays to consider the coupling of a 1D particle to the background electromagnetic gauge potential: the action is augmented by the interaction term
$$
\int \mathrm{d}\tau\ A_\mu(X)\dot{X}^\mu
$$
which is precisely the pull-back of the gauge one-form $A_\mu\mathrm{d}x^\mu$ onto the worldline, though one does not immediately spot this. The same thing happens for the background fields in string theory, be it the Kalb-Ramond field or the graviton: while they naturally carry spacetime indices, they enter the string action/vertex operators through the pull-back of the spacetime field onto the worldvolume, and corresponding coupling terms are introduced into the Polyakov action. Here's an example for the Kalb-Ramond field:
$$
S_\text{int}=\frac{1}{4\pi\alpha'}\int\mathrm{d}^2\sigma\sqrt{g} \ i\epsilon^{\alpha\beta}B_{\mu\nu}(X)\partial_aX^\mu\partial_bX^\nu
$$
where the pullback is a lot more evident. Now consider the essentially analogous scenario where the D-branes act as sources for the RR-fields. The RR-fields/potentials are again defined on the spacetime, and the worldvolume of the Dp-branes can couple locally to the $(p+1)$-form background gauge potential, once again via the pullback (don't be misled by the terminology, the worldvolume of a Dp-brane is also $(p+1)$-dimensional). For simplicity, consider the electric coupling (in the absence of the mixing Kalb-Ramond field, to keep things simple):
$$
S_\text{int} = \frac{e_p}{(p+1)!}\int \mathrm{d}^{p+1}\sigma \ A_{\mu_1...\mu_{p+1}}\partial_0 X^{\mu_1}...\partial_p X^{\mu_{p+1}}
$$
[See e.g. Equation (6.54); Becker, Becker, Schwarz, String Theory and M-Theory]
When you see the dimensions of the gauge fields in the RR-sector of the theory under consideration, you can immediately determine which Dp-branes are allowed to carry the RR-charges via the electric and magnetic coupling. Accordingly, the Bianchi identity of the RR gauge fields is violated in the presence of the sources:
$$
\mathrm{d}G_{p}=0\longrightarrow\mathrm{d}G_{p}=\mathcal{J}_{p+1}
$$
Note that this the exterior derivative on the target spacetime. Similar to how a local charge source in electrodynamics modifies the field strength outside of its worldline due to the p-form Maxwell equation, the charged D-branes thus act as sources for the RR-field not just on the brane itself, but also in the bulk.
