Shape of branes pulled by strings

We examine the system where a string stretches between a pair of $D$-branes, and study the bending of the $D$-brane caused by the tension of the string. If the distance between the pair of $D$-branes is sent to infinity, the tension of the string stretching between them is strong enough to pull the spike all the way to infinity. We study the shape of these spikes when the branes are a finite distance apart using two different methods. First, we consider a string stretched between a pair of $D2\mathrm{}$-branes in type-IIA theory by going to the M-theory limit in which all of these branes are M-theory two-branes embedded along a holomorphic curve. Second, we consider a $D$-string stretched between a pair of $D3\mathrm{}$-branes in type-IIB theory and infer the geometry of the $D3\mathrm{}$-brane embeddings from the configuration of the adjoint scalar field in the magnetic monopole solution of Prasad and Sommerfield. The case of a fundamental string stretching between a pair of $D3\mathrm{}$-branes follows from $S$ duality. The energy of these configurations matches the expected value based on the fundamental string and $D$-string tensions.