In the woods of M-theory

We study BPS states which arise in compactifications of M-theory onCalabi-Yau manifolds. In particular, we are interested in the spectrum of theparticles obtained by wrapping M2-brane on a two-cycle in the CY manifold X. Wecompute the Euler characteristics of the moduli space of genus zero curveswhich land in a holomorphic four-cycle $S \subset X$. We use M. Kontsevich'smethod which reduces the problem to summing over trees and observe thediscrepancy with the predictions of local mirror symmetry. We then turn thisdiscrepancy into a supporting evidence in favor of existence of extra moduli ofM2-branes which consists of the choice of a flat U(1) connection recentlysuggested by C. Vafa and partially confirm this by counting of the arbitrarygenus curves of bi-degree (2,n) in $\IP^1 \times \IP^1$ (this part has beendone together with Barak Kol). We also make a conjecture concerning thecounting of higher genus curves using second quantized Penner model and discusspossible applications to the string theory of two-dimensional QCD.Comment: harvmac, 14 pp., v2. references added, typos correcte