The stability of the shell of D6-D2 branes in a ${\cal N}=2$ supergravity solution

Abstract

The stability of the shell of wrapped D6-branes on K3 is investigated from the point of view of supergravity. We first construct an effective energy-momentum tensor for the shell under the reasonable conditions and show that supersymmetric solutions satisfy Israel's junction conditions at arbitrary radius of the shell. Next we study the perturbation of the whole system including the self-gravity of the shell. It is found that in spite of the existence of wrapped D6-branes with negative tension, there is no eigenmode whose frequencies of the shell and the fields are imaginary numbers, at any radius of the shell. Furthermore, when the radius of the shell is less than the enhan\c{c}on radius, resonances are produced, and this indicates a kind of ``instability'' of the system. This can even classically explain why the shell is constructed at the enhan\c{c}on radius.