We develop the geometric formulation for SUSY theories proposed by Frenkel, Losev and Nekrasov.The formalism is based on localization on instanton moduli spaceand it's deformation and leads to a rich family of non-perturbatively well-defined QFTs. Among these theories are the Morse theory, $\beta\gamma-bc$ system and A-model type theories, but with more effort much widerrange of theories may be covered. The advantage of geometric formalism is it's explicit target space coordinate-independence. We develop further the geometric formalism and study in detail the relation of this formalism with the conventional ones. We study the ways to define polyvector field observables in a coordinate-independent wayand compare it to free-field methods.
We investigate the local limit of polyvector fields: for holomorphic polyvector fields we find a nice symmetric regularization prescription,while for non-holomorphic vector fields there arises an interesting dependence on the angleof point-splitting.
Comparing the geometric calculation with free-field approach to first-order non-linear sigma-models based on holomortices, we show the origin of conditionally convergent integrals in the latter approach and prove the way to deal with them.