Torsion and Supersymmetry in Omega-background

We study the dimensional reduction of ten-dimensional super Yang-Mills theory in curved backgrounds with torsion. We examine the parallel spinor conditions and the constraints for the torsion parameters which preserve supersymmetry and gauge symmetry in four dimensions. In particular we examine the ten-dimensional Omega-background with the torsion which is identified with the R-symmetry Wilson line gauge fields. After the dimensional reduction, we obtain the Omega-deformed N=4 super Yang-Mills theory. Solving the parallel spinor conditions and the torsion constraints, we classify the deformed supersymmetry associated with the topological twist of N=4 supersymmetry. We also study deformed supersymmetries in the Nekrasov-Shatashvili limit.