We study the quantum volume of D-branes wrapped around various cycles in Ca
labi-Yau manifolds, as the manifold's moduli are varied. In particular, we
focus on the behaviour of these D-branes near phase transitions between dis
tinct low energy physical descriptions of the resulting string theory. Wher
eas previous studies have solely considered quantum volumes in the context
of two-cycles in perturbative string theory or D-branes in the specific exa
mple of the quintic hypersurface, we work more generally and find qualitati
vely new features. On the mathematical side, as we briefly note, our work h
as some interesting implications for certain issues in arithmetic. (C) 2001
Published by Elsevier Science B.V.