Twistorial Topological Strings and a tt* Geometry for N=2 Theories in 4d

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.