  The Gopakumar-Vafa (GV) formula expresses certain couplings that arise in
Type IIA compactification to four dimensions on a Calabi-Yau manifold in terms
of a counting of BPS states in M-theory. The couplings in question have
applications to topological strings and supersymmetric black holes. In this
paper, we reconsider the GV formula, taking a close look at the Schwinger-like
computation that was suggested in the original GV work. The goal is to
understand the background that must be used in this computation, the role
played by the extended supersymmetry of this background, and how the
computation gives a holomorphic result though superficially depending only on
particle masses. We also examine in a similar way the Ooguri-Vafa (OV) formula,
which is an extension of the GV formula to include D4-branes.
