Historically the cohomological field theory first has been introduced as a twisted version of global space-time supersymmetric quantum field theory, specifically the N = 2 supersymmetric Yang-Mills theory in four dimensional space-time [1]. The global space-time supersymmetry, by definition, requires the existence of a spinor which is constant everywhere on the space-time manifold M . A spinor does exist on a spin manifold. A spin manifold, however, rarely admits a constant spinor. The canonical way overcoming the above difficulty is localizing the supersymmetry, which procedure almost magically introduces (super-)gravity into the picture. There is a second option called twisting, meaning that one defines a new Lorentz symmetry group by a suitable combination of the original Lorentz symmetry with an internal global symmetry of the theory. As a result, the supercharges transform differently under the new Lorentz symmetry, which typically includes some components which transform as scalars. Such a scalar component Q, which is nilpotent Q = 0, is regarded as a supercharge of the twisted theory. The resulting theory is well-defined on an arbitrary space-time since there are no global obstruction for a scalar and enjoys general covariance without gravity. The path integral of the theory depends only on the global cohomology of Q, provided that one uses Q-invariant observables, which property coined the adjective cohomological [2]. A twisted theory is closely related to the underlying space-time supersymmetric theory. Namely the path integral of the twisted theory computes a certain chiral (or BPS) sector of physical amplitudes [3][4]. This is due to the trivial holonomy of flat space-time where the physical theory is usually defined. Then