  Cosmological defects result from cosmological phase transitions in the early
Universe and the dynamics reflects their symmetry-breaking mechanisms. These
cosmological defects may be probed through weak lensing effects because they
interact with ordinary matters only through the gravitational force. In this
paper, we investigate global textures by using weak lensing curl and B modes.
Non-topological textures are modeled by the non-linear sigma model (NLSM), and
induce not only the scalar perturbation but also vector and tensor
perturbations in the primordial plasma due to the nonlinearity in the
anisotropic stress of scalar fields. We show angular power spectra of curl and
B modes from both vector and tensor modes based on the NLSM. Furthermore, we
give the analytic estimations for curl and B mode power spectra. The amplitude
of weak lensing signals depends on a combined parameter $\epsilon^{2}_{v} =
N^{-1}\left( v/m_{\rm pl} \right)^{4}$ where $N$ and $v$ are the number of the
scalar fields and the vacuum expectation value, respectively. We discuss the
detectability of the curl and B modes with several observation specifications.
In the case of the CMB lensing observation without including the instrumental
noise, we can reach $\epsilon_{v} \approx 2.7\times 10^{-6}$. This constraint
is about 10 times stronger than the current one determined from the Planck. For
the cosmic shear observation, we find that the signal-to-noise ratio depends on
the mean redshift and the observing number of galaxies as $\propto z^{0.7}_{\rm
m}$ and $\propto N^{0.2}_{\rm g}$, respectively. In the study of textures using
cosmic shear observations, the mean redshift would be one of the key design
parameters.
