In order to understand the relationship between the Gromov-Witten
invariants of a Calabi-Yau 3-fold X and the enumerative geometry of X,
one needs to know how multiple covers of curves contribute to the
invariants.
In these lecture notes, we survey some old and new results about multiple
cover formulas. We also define ``BPS invariants'' in terms of the
Gromov-Witten invariants via the formula of Gopakumar and Vafa. These
invariants are conjecturally integer valued and we show that the known
multiple-cover formulas for the Gromov-Witten invariants indeed lead to
integral contributions to the BPS invariants, sometimes in subtle
ways. These integrality predictions lead to conjectural congruence
properties of Hurwitz numbers. We prove a few of these congruences in the
last section.
Ultimately, we hope the understanding of the contribution of curves in X
to the BPS invariants of X will lead to an intrinsic geometric definition
of the BPS invariants and that the Gopakumar-Vafa formula can be proven as
a theorem (rather than a definition).