Casimir dynamics: Interactions of Surfaces with codimension >1 due to Quantum Fluctuations

We study the Casimir force between defects (branes) of co-dimension larger than 1 due to quantum fluctuations of a scalar field $\varphi$ living in the bulk. We show that the Casimir force is attractive and that it diverges as the distance between the branes approaches a critical value $L_c$. Below this critical distance $L_c$ the vacuum state $\varphi =0$ of the theory is unstable, due to the birth of a tachyon, and the field condenses.