NS5-brane. The flux captured in this region is half of that generated by the D6-brane, so that a net one-half quantum of D4 charge reaches infinity along the NS5-brane. This last statement is true in each of the stages (i,ii,iii,iv), though only in stage (iii) are all of the relevant parts of the NS5-brane visible in figure 1. In stage (iv), the neck has narrowed so as to become difficult to resolve and all that remains is a D4-brane string stretching between an NS5-brane and a D6-brane.

The above picture seems to follow from general properties of the supergravity field equations, but it is important to check them by studying exact solutions in detail. This will be particularly clear in a moment when we discuss the 6-brane ‘puzzle,’ which is an apparent exception to the above story. The known exact solutions come in several forms, the first of which [@NOYY] considers a test D2-brane in the background created by a six-brane. This case is particularly tractable using the M-theory description in which we have an M2-brane in a Kaluza-Klein monopole background. In this case, any holomorphic curve represents a BPS configuration of the M2-brane. By moving the M2-brane past the 