  We define twistorial topological strings by considering tt* geometry of the
4d N=2 supersymmetric theories on the Nekrasov-Shatashvili half-Omega
background, which leads to quantization of the associated hyperKahler
geometries. We show that in one limit it reduces to the refined topological
string amplitude. In another limit it is a solution to a quantum
Riemann-Hilbert problem involving quantum Kontsevich-Soibelman operators. In a
further limit it encodes the hyperKahler integrable systems studied by GMN. In
the context of AGT conjecture, this perspective leads to a twistorial extension
of Toda. The 2d index of the half-Omega theory leads to the recently introduced
index for N=2 theories in 4d. The twistorial topological string can
alternatively be viewed, using the work of Nekrasov-Witten, as studying the
vacuum geometry of 4d N=2 supersymmetric theories on T^2x I where I is an
interval with specific boundary conditions at the two ends.
