  We study BPS states which arise in compactifications of M-theory on
Calabi-Yau manifolds. In particular, we are interested in the spectrum of the
particles obtained by wrapping M2-brane on a two-cycle in the CY manifold X. We
compute the Euler characteristics of the moduli space of genus zero curves
which land in a holomorphic four-cycle $S \subset X$. We use M. Kontsevich's
method which reduces the problem to summing over trees and observe the
discrepancy with the predictions of local mirror symmetry. We then turn this
discrepancy into a supporting evidence in favor of existence of extra moduli of
M2-branes which consists of the choice of a flat U(1) connection recently
suggested by C. Vafa and partially confirm this by counting of the arbitrary
genus curves of bi-degree (2,n) in $\IP^1 \times \IP^1$ (this part has been
done together with Barak Kol). We also make a conjecture concerning the
counting of higher genus curves using second quantized Penner model and discuss
possible applications to the string theory of two-dimensional QCD.
