Elements of Fedosov geometry in Lagrangian BRST Quantization

A Lagrangian formulation of the BRST quantization of generic gauge theories in general irreducible non-Abelian hypergauges is proposed on the basis of the multilevel Batalin--Tyutin formalism and a special BV--BFV dual description of a reducible gauge model on the symplectic supermanifold $\mathcal{M}_0$ locally parameterized by the antifields to Lagrangian multipliers and the fields of the BV method. The quantization rules are based on a set of nilpotent anticommuting operators $\Delta^\mathcal{M}, \mathcal{V}^\mathcal{M}, \mathcal{U}^\mathcal{M}$ defined through both odd and even symplectic structures on a supersymplectic manifold $\mathcal{M}$, locally representable as an odd (co)tangent bundle over $\mathcal{M}_0$, provided by the choice of a flat Fedosov connection and by means of a compatible with it non-symplectic metric on $\mathcal{M}_0$. The generating functional of Green's functions is constructed in general coordinates on $\mathcal{M}$ with help of operators of contracting homotopies for $\mathcal{V}^\mathcal{M}, \mathcal{U}^\mathcal{M}$. The gauge independence of the S-matrix is proved. The Ward identity is derived