  The spatial texture of internal degree of freedom of electrons has profound
effects on the properties of materials. Such texture in real space can manifest
as an emergent magnetic field (or Berry curvature), which is expected to induce
interesting valley/spin-related transport phenomena. Moir\'e pattern, which
emerges as a spatial variation at the interface of 2D atomic crystals, provides
a natural platform for investigating such real space Berry curvature effects.
Here we study moir\'e structures formed in homobilayer transition metal
dichalcogenides (TMDs) due to twisting, various uniform strain profiles, and
their combinations, where electrons can reside in either layer with the layer
index serving as an internal degree of freedom. The layer pseudo-spin exhibits
vortex/antivortex textures in the moir\'e supercell, leading to a giant
geometric magnetic field and a scalar potential. Within a geometric picture,
the moir\'e magnetic field is found as the cross product of the gradients of
the out-of-plane pseudo-spin and the in-plane pseudo-spin orientation
respectively. We discover dual roles of uniform strain: Besides being a cause
of the moire atomic texture in the homobilayer, it also contributes a
pseudo-gauge potential that modifies the local phase of interlayer coupling.
Consequently, strain can be employed to tune the in-plane pseudo-spin texture,
while interlayer bias tunes the out-of-plane pseudo-spin, and we show how the
moir\'e magnetic field's spatial profile, intensity, and flux per supercell can
be engineered. Through the geometric scalar correction, the landscape of the
scalar potential can also be engineered along with the moir\'e magnetic field,
forming distinct effective lattice structures. These properties render TMD
moir\'e structures promising to build tunable flux lattices for transport and
topological material applications.
