  Following the work of Lima et al on the exact evaluation of the
nonperturbative contribution to the superpotential from open-membrane instanton
in Heterotic M-Theory, we evaluate systematically the contribution to the
superpotential of a membrane instanton obtained by wrapping of a single M2
brane, once, on an isolated supersymmetric 3-cycle in a G_2-holonomy manifold.
We then try to relate the results obtained to those sketched out in Harvey and
Moore. We also do a heat-kernel asymptotics analysis to see whether one gets
similar UV-divergent terms for (one or both of) the bosonic and fermionic
determinants indicative of (partial) cancelation among them. The answer is in
the affirmative, as expected by the supersymmetry of the starting membrane
action. This work is a first step, both, in extending the work of Harvey and
Moore to the evaluation of nonperturbative superpotentials for non-rigid
supersymmetric 3-cycle wrappings, and in understanding the large N
Chern-Simons/closed type-A topological string theory duality of Gopakumar and
Vafa from M theory point of view.

High Energy Physics - Theory