In long-wavelength moir\'e patterns of homobilayer semiconductors, the layer
pseudospin of electrons is subject to a sizable Zeeman field that is spatially
modulated from the interlayer coupling in moir\'e. By interference of this
spatial modulation with a homogeneous but dynamically tunable component from
out-of-plane electric field, we show that the spatial-temporal profile of the
overall Zeeman field therefore features a topological texture that can be
controlled in an ultrafast timescale by a terahertz field or an interlayer
bias. Such dynamical modulation leads to the emergence of an in-plane electric
field for low energy carriers, which is related to their real space Berry
curvature -- the moir\'e magnetic field -- through the Faraday's law of
induction. These emergent electromagnetic fields, having opposite signs at the
time reversal pair of valleys, can be exploited to manipulate valley and spin
in the moir\'e landscape under the control by a bias pulse or a terahertz
irradiation.Comment: To appear in Natural Science