  We extend the recent study of 3d tt* geometry to 5d (4d) N=1 (N=2)
supersymmetric theories. We show how the amplitudes of tt* geometry lead to an
extension of topological strings and Nekrasov partition functions to twistor
space, where the north/south poles of the twistor sphere lead to
topological/anti-topological string amplitudes. However, the equator of the
twistor sphere is the physical region for the amplitudes, where the unitarity
of physical amplitudes is respected. We propose that the amplitudes for line
operators studied by Gaiotto, Moore and Neitzke lead to difference equations
for the twistorial topological string partition function, extending the
familiar one for the monodromy of Branes in the context of the standard
topological strings. This setup unifies tt* geometry, topological strings and
the hyperKahler geometry formulated by GMN into a single framework.
