  We begin with a discussion on two apparently disconnected topics - one
related to nonperturbative superpotential generated from wrapping an M2-brane
around a supersymmetric three cycle embedded in a G_2-manifold evaluated by the
path-integral inside a path-integral approach of [1], and the other centered
around the compact Calabi-Yau CY_3(3,243) expressed as a blow-up of a degree-24
Fermat hypersurface in WCP^4[1,1,2,8,12]. For the former, we compare the
results with the ones of Witten on heterotic world-sheet instantons [2]. The
subtopics covered in the latter include an N=1 triality between Heterotic, M-
and F-theories, evaluation of RP^2-instanton superpotential, Picard-Fuchs
equation for the mirror Landau-Ginsburg model corresponding to CY_3(3,243),
D=11 supergravity corresponding to M-theory compactified on a `barely' G_2
manifold involving CY_3(3,243) and a conjecture related to the action of
antiholomorphic involution on period integrals. We then show an indirect
connection between the two topics by showing a connection between each one of
the two and Witten's MQCD [3]. As an aside, we show that in the limit of
vanishing "\zeta", a complex constant that appears in the Riemann surfaces
relevant to definining the boundary conditions for the domain wall in MQCD, the
infinite series of [4] used to represent a suitable embedding of a
supersymmetric 3-cycle in a G_2-mannifold, can be summed.
