Collapsing D-Branes in Calabi-Yau Moduli Space: I

We study the quantum volume of D-branes wrapped around various cycles in Calabi-Yau manifolds, as the manifold's moduli are varied. In particular, we focus on the behaviour of these D-branes near phase transitions between distinct low energy physical descriptions of the resulting string theory. Whereas previous studies have solely considered quantum volumes in the context of two-cycles in perturbative string theory or D-branes in the specific example of the quintic hypersurface, we work more generally and find qualitatively new features. On the mathematical side, as we briefly note, our work has some interesting implications for certain issues in arithmetics.