  We study the cosmological information contained in the Minkowski Functionals
(MFs) of weak gravitational lensing convergence maps. We show that the MFs
provide strong constraints on the local type primordial non-Gaussianity
parameter f_NL. We run a set of cosmological N-body simulations and perform
ray-tracing simulations of weak lensing, to generate 100 independent
convergence maps of 25 deg^2 field-of-view for f_NL = -100, 0 and 100. We
perform a Fisher analysis to study the degeneracy among other cosmological
parameters such as the dark energy equation of state parameter w and the
fluctuation amplitude sigma_8. We use fully nonlinear covariance matrices
evaluated from 1000 ray-tracing simulations. For the upcoming wide-field
observations such as Subaru Hyper Suprime-Cam survey with the proposed survey
area of 1500 deg^2, the primordial non-Gaussianity can be constrained with a
level of f_NL ~ 80 and w ~ 0.036 by weak lensing MFs. If simply scaled by the
effective survey area, a 20000 deg^2 lensing survey using Large Synoptic Survey
Telescope will give constraints of f_NL ~ 25 and w ~ 0.013. We show that these
constraints can be further improved by a tomographic method using source
galaxies in multiple redshift bins.
